NETWORKED SYSTEM of CRYSTAL OSCILLATORS a Thesis

NETWORKED SYSTEM OF CRYSTAL OSCILLATORS

A Thesis

Presented to the

Faculty of

San Diego State University

In Partial Fulﬁllment

of the Requirements for the Degree

Master of Science in Applied Mathematics

with a Concentration in

Dynamical Systems

Steven Isaac Reeves

Spring 2016

iii

In solving a problem of this sort, the grand thing is to be able to reason backward.

– Sherlock Holmes v

ABSTRACT OF THE THESIS

NETWORKED SYSTEM OF CRYSTAL OSCILLATORS by Steven Isaac Reeves Master of Science in Applied Mathematics with a Concentration in Dynamical Systems San Diego State University, 2016

The dynamics of a system of coupled crystal oscillators (CCOST) is examined with the aim of developing a stable precision timing device. Symmetry is used to establish the existence and stability of collective patterns of oscillations in the CCOST device. We investigate N identical crystal oscillators, where each is described by a two-mode nonlinear oscillator circuit that exhibits SO(2) × SO(2)-symmetry. The coupling is assumed to be identical, and two different topologies, unidirectional and bidirectional, are considered. The unidirectional topology leads to a network with SO(2) × SO(2) × ZN -symmetry. On the other hand, the bidirectional topology yields a network with SO(2) × SO(2) × DN -symmetry. The possible patterns of oscillation are classiﬁed using these symmetries and their respective isotropy subgroups. The effects of noise on the unidirectional CCOST device will be investigated with respect to phase error reduction. Phase error reduction of certain patterns will be tested against an uncoupled-averaged control group. This work will instruct on the design rules for the proposed precision timing device. vi

TABLE OF CONTENTS PAGE ABSTRACT ...... v LIST OF TABLES...... vii LIST OF FIGURES ...... viii ACKNOWLEDGMENTS ...... xi CHAPTER 1 Introduction ...... 1 2 Crystal Oscillators ...... 4 2.1 Two-Mode Oscillator Model ...... 5 2.2 Averaging ...... 6 2.3 Stability ...... 8 3 Coupled Crystal Oscillator System...... 12 3.1 Averaged Equations ...... 12 3.2 Symmetries of Collective Patterns ...... 16 3.2.1 The unidirectional case ...... 17 3.2.2 The bidirectional case ...... 21 3.3 Isotypic Decomposition...... 28 3.4 Spectrum of Eigenvalues and Linear Stability ...... 31 4 Numerical Continuation ...... 38 4.1 Unidirectionally Coupled Ring ...... 38 4.2 Bidirectionally Coupled Ring ...... 44 5 Phase Error ...... 48 5.1 Uncoupled Control Group ...... 50 5.2 Unidirectional Coupling ...... 51 6 Discussion and Future Work...... 54 BIBLIOGRAPHY ...... 57 APPENDICES A Averaging...... 60 vii

LIST OF TABLES PAGE 2.1 Classiﬁcation of solutions of Eq. (2.6) based on isotropy subgroups...... 9 3.1 One-dimensional irreducible representations and spatial subgroups of the bifurcating periodic solutions...... 22

3.2 Isotropy subgroups of periodic solutions bifurcating from DN Hopf bifurcation at a Γ-simple subspace consisting of the sum of isomorphic two-dimensional absolutely irreducible representations of DN ...... 23 3.3 Spatial subgroups of the rotating waves bifurcating simultaneously from the symmetric Hopf bifurcation point...... 24 0 3.4 Possible spatial subgroups K depending on which element of K ﬁxes the kernel (dg)(0,λ)...... 27 viii

LIST OF FIGURES PAGE 2.1 (Left) Schematic of a quartz crystal. (Right) A quartz crystal behaves like circuit composed of an inductor, capacitor and resistor, so it oscillates with a precise resonant frequency when it is subjected to an electric field...... 4 2.2 Two-mode crystal oscillator circuit. A second set of spurious RLC components (R2,L2,C2) are introduced by parasitic elements. [37] [33] ...... 5 2.3 Time series solutions of a single two-mode crystal oscillator model (2.4). (Top left) Two parameter bifurcation diagram indicating the regions of existence of the time-series solutions as a function of parameters R1/a and R2/a. (Top right) Unstable Mixed Mode (i1(t), i2(t)), (bottom left) Stable Mode 1 (i1(t), 0) , (bottom right) Stable Mode 2 (0, i2(t)). Parameter values are: −4 −4 R1 = 30.9Ω, R2 = 181.1Ω, L1 = 5.2 × 10 H, L2 = 2.6 × 10 H, −13 −14 8 C1 = 1.0 × 10 F , C2 = 2.5 × 10 F , a = 939, b = 3 × 10 ...... 11 3.1 CCOST concept with unidirectionally coupled crystal oscillators...... 12 3.2 Rings of N = 16 oscillator units representing a 2π periodic so- 8 lution with spatial subgroups K = D2(κ, (π, γ )) (left) and K = 4 D2(κ, (π, γ )) (right) corresponding respectively to the cases m = 1 and m = 2. The various colors represent different waveforms and the numeric labels the relative phases between cells of same color, with 02 meaning oscillation at twice the frequency of the other units...... 26 3.3 Rings of N = 16 units representing a 2π-periodic solution with spatial 0 8 0 4 subgroups K = Z2(π, γ ) (left) and K = Z2(π, κγ ) (right) corresponding to the cases m = 1 and m = 2 respectively. The various colors represent different waveforms and the numeric labels the relative phases between cells of same color. Phase shifts between different colors are uncorrelated and 02 means oscillation at twice the frequency of the other units...... 29 4.1 Bifurcation diagrams for a ring of N = 3 crystal oscillators coupled unidirectionally in a ring configuration. The ring exhibits Z3-symmetry, i.e., cyclic permutations of three crystal oscillators. Parameter values are: R1 = 30.9Ω, −4 −4 −13 R2 = 181.1Ω, L1 = 5.2 × 10 H, L2 = 2.6 × 10 H, C1 = 1.0 × 10 F , −14 C2 = 2.5 × 10 F , a = 939, b = 3E08...... 39 4.2 Bifurcation diagrams for a ring of N = 4 crystal oscillators coupled unidirectionally in a ring configuration. The ring exhibits Z4-symmetry, i.e., cyclic permutations of four crystal oscillators. Parameter values: same as in Fig. 4.1 ...... 41 ix

4.3 Bifurcation diagrams for a ring of N = 5 crystal oscillators coupled unidirectionally in a ring configuration. The ring exhibits Z5-symmetry, i.e., cyclic permutations of five crystal oscillators. Parameter values: same as in Fig. 4.1 ...... 42 4.4 Bifurcation diagrams for a ring of N = 6 crystal oscillators coupled unidirectionally in a ring configuration. The ring exhibits Z6-symmetry, i.e., cyclic permutations of six crystals. Parameter values: same as in Fig. 4.1...... 44 4.5 Bifurcation diagrams for a ring of N crystal oscillators coupled biidirectionally in a ring configuration. The diagrams depict the existence and stability of various branches of collective oscillations that appear, mainly via Hopf bifurcations, as a function of coupling strength λ. The ring exhibits DN -symmetry, i.e., cyclic permutations of three crystal oscillators. Parameter values are: −4 −4 R1 = 30.9Ω, R2 = 181.1Ω, L1 = 5.2 × 10 H, L2 = 2.6 × 10 H, −13 −14 8 C1 = 1.0 × 10 F , C2 = 2.5 × 10 F , a = 939, b = 3 × 10 ...... 45 4.6 Two-parameter bifurcation diagrams for a ring of N crystal oscillators coupled biidirectionally in a ring configuration. Parameter values are: R1 = 30.9Ω, −4 −4 −13 R2 = 181.1Ω, L1 = 5.2 × 10 H, L2 = 2.6 × 10 H, C1 = 1.0 × 10 F , −14 8 C2 = 2.5 × 10 F , a = 939, b = 3 × 10 ...... 47 1 5.1 This figure displays 22MHz rotating wave solution, RW1 , of the unidirectional CCOST model, with 3 nodes and coupling strength λ = 0.99. Above, the −7 current, Xi, is plotted over a period of 1.0 × 10 seconds. Below, one noise function is displayed over the same duration...... 49 5.2 The uncoupled-averaged (λ = 0) phase error as a function of array size. The top section displays the average,√ maximum, and minimum phase errors of 50 samples for each N.A 1/ N scaling is observed. Below, the range of phase error values are displayed as a function of N. As N increases, we also see a general reduction in the sample range...... 50 5.3 The phase error with respect to the Synchronized Pattern with λ = −0.99 as a function of N. The top section of the figure displays the mean, maximum, and minimum√ phase errors for a 50 sample data set for each N. The average data fits a 1/ N reduction curve. Below, the range of possible values in the sample is shown. For most N, the range is decreasing, with notable exceptions of N = 7 and N = 10...... 52

5.4 The phase errors for the RW1 pattern as a function of the number of oscillators in the CCOST device when λ = 0.99. Above, the average, maximum and minimum phase errors of a sample of 50 simulations for each N is plotted. Additionally, the average data values ﬁt a 1/N scaling. Below, the range of values is displayed as a function of N. The range decreases monotonically, illustrating the strength of this pattern...... 52 x

5.5 The phase errors with respect to the RW2 pattern as a function of N with λ = 0.99. The top section illustrates the average, maximum, and minimum phase errors of a 50 sample collection for each N. The data does not present a reduction for these values of N. Below, the range of values is represented as a function of N. While we do see a reduction in range, when N > 6 the mean values do not follow this trend...... 53 xi

ACKNOWLEDGMENTS I am incredibly thankful for my thesis committee chair and advisor, Dr. Antonio Palacios, for helping me in becoming a better researcher and mathematician. In addition, I am grateful for my thesis committee members, Dr. Peter Blomgren and Dr. Satish Sharma, without their support this thesis would not have been possible. Much of the calculations and support for this work is from the Office of Naval Research, and SPAWAR Systems Center Pacific. I thank the Office of Naval Research for funding for me and this work. I owe much to SPAWAR Systems Center Pacific and I thank Dr. Patrick Longhini and Dr. Visirath In in particular for their guidance and contributions to this work. I recognize the engineers at SPAWAR for their contribution to the experimental side of this project. I further acknowledge Dr. Ricardo Carretero and the Nonlinear Dynamical Systems group at San Diego State for giving me the tools to tackle a problem of this sort. 1

CHAPTER 1

INTRODUCTION

Precise time is crucial to a variety of economic activities around the world. Communication systems, electrical power grids, and financial networks all rely on precision timing for synchronization and operational efficiency. The free availability of GPS [2] time has enabled cost savings for industrial and scientific developments that depend on precise time and has led to significant advances in capability. For example, wireless telephone and data networks use GPS time to keep all of their base stations in synchronization. This allows mobile handsets to share limited radio spectrum more efficiently. Similarly, digital broadcast radio services use GPS time to ensure that the signals from all radio stations arrive at receivers in lockstep, so that listeners can tune between stations with minimum delay. However, even the most sophisticated satellite navigation equipment cannot operate in every environment. And even under perfect weather and environmental conditions, mechanical failure can still occur and hinder accessibility. In fact, many of the 32 satellites in the GPS constellation are operating past their intended lifespan or suffering from equipment failure. There have been a few launch incidents in past years, and the Air Force, which maintains the 30-year-old network, is overburdened with competing space priorities. Thus it is reasonable to wonder what would happen if the U.S. Global Positioning System is not available due to environmental or to complete mechanical failure. Regardless of the causes we could predict that if GPS were to fail completely, the cost would be severe. Civil aviation, trucking, shipping, and telecommunications would be rendered stationary, and countless other industries would be affected. Internet activity would slow to a crawl, because many internet service providers rely on precise GPS time stamps to route data. Agribusiness and commercial fishing could be blinded, causing food prices to skyrocket. For these reasons, GPS modernization has now become an ongoing initiative of the U.S. Government with new capabilities to meet growing military, civil and commercial needs [19]. Nevertheless, GPS service can degrade quickly when the signal is denied, impaired or otherwise unavailable. This thesis studies the possibility of developing a compact, high-precision, timing system using a Coupled Crystal Oscillator System or CCOST for short. The aim is to study the collective response of N crystal oscillators, coupled in some fashion, with the goal of creating robust stable oscillations to achieve high precision timing. Further, this work will provide a complete classification of the various patterns of collective behavior that are 2 created, mainly, through symmetry-breaking bifurcations, as well as the regions of existence and stability of each pattern, and finally examine the phase error reduction of a coupled system compared to the uncoupled standard. The results from this study will aid future simulations, design and fabrication tasks. Historically, the inability to determine longitude accurately made navigation on the open seas difficult and treacherous. In the Renaissance era, when Europe began its exploration, determining longitude required comparing the time at the current location with the time at a known location, say the Greenwich meridian. However, no shipboard clocks could determine time to an accuracy sufficient for navigational purposes. Heads of several seafaring nations offered great prizes for a solution to the problem of longitude. In the early 18th century, the Longitude Prize offered by Britain led to the development of the ship’s chronometer. [28] This device was so amazingly workable that it remained in use unchanged in its essential elements until the electronic era of the early 20th century. Following World War I and the development of the electronic oscillator and radio communications, the U.S. Navy took an ever more active role in the development of emerging Precision Time and Time Interval (PTTI) technologies. The U.S. Naval Observatory (USNO), the Naval Research Laboratory (NRL) and, after World War II, the Office of Naval Research (ONR), the Defense Advanced Research Projects Agency (DARPA) and the National Science Foundation [23] were important players in the development of the technology that makes up the current state of the art in PTTI [5]. The advances that had been made in high-frequency electronics during World War II radar research set the stage for the development of atomic clocks. In 1942 the Joint Chiefs of Staff established a Radio Propagation Laboratory at the National Bureau of Standards (NBS), now the National Institute of Standards and Technology (NIST). The Radio Propagation Laboratory developed the world’s first atomic clock in 1948. This clock was based on the measurement of a spectroscopic absorption line in ammonia. Because its stability was no better than that of high-quality quartz oscillators, the ammonia system was quickly abandoned for the greater potential accuracy of the cesium atomic beam device. At the heart of this device was a microwave cavity design developed in 1948 by Norman Ramsey of Harvard University (Ramsey received the Nobel Prize for this work in 1989.) Immediately following the launch of the first artificial Earth-orbiting satellite, Sputnik, by the Soviet Union in 1957, the Navy set up the Naval Space Surveillance System (NAVSPASUR). In 1964, Roger Easton of the NRL put forward a concept for an improved navigation system that would orbit precision clocks. Signals from such a satellite could provide more precise navigation as well as precise time signals that were available worldwide. To achieve this goal, NRL started programs to develop improved quartz frequency standards 3 suitable for spaceflight. Soon thereafter, the Timation program, which involved atomic clocks in space, was established. These space-qualified atomic clocks were then used in the Global Positioning System (GPS), GPS became a joint service program in 1973, with the Air Force designated executive agent for the system. Since 1976 the length of a second has been defined as the frequency of a specific resonant mode of the cesium atom. The frequency inaccuracy of the cesium clocks is approximately 8.64ns/day. In comparison the typical watch crystal has an error of about 20ppm, which is about 1.73s/day. The increased accuracy and precision comes with a price. Typical cost range of crystal oscillators are in the few dollars while a cesium clock can be in the order of $40,000. Currently, the accuracy of the NIST atomic clock, called NIST-F2 [27], is on the order of 1x10−16, making it on the order of three times as accurate as its predecessor NIST-F1, which had served as the standard since 1999. Both clocks use a ‘fountain’ of cesium atoms to determine a precise measure of a second. The key operational difference is that F1 operates near room temperature whereas the atoms in F2 are shielded within a much colder environment [16]. This thesis is organized as follows. The introduction provides an informative description for the general reader of the state-of-the-art in the development of precise timing devices, including, of course, atomic clocks. In Chapter 2 we introduce the physics behind crystal oscillators followed by a description of the equivalent electric circuit representation. A two-mode mathematical model of the circuit and its dynamic behavior is also presented in great detail. In Chapter 3 a comprehensive analysis of the collective response of a network of crystal oscillators is conducted. This network has two coupling topologies: a unidirectional topology and bidirectional topology forming ring structures. The former case leads to a system with ZN -symmetry, where ZN is the cyclic group of permutations of N objects. The latter case produces a system with DN -symmetry, where DN is the dihedral group of symmetries of an N-sided polygon. To classify the possible patterns of oscillations, we exploit the symmetry of the network to identify those patterns that can arise from local primary bifurcations at the equilibrium solution via equivariant Hopf bifurcations. We show that the periodic solutions are rotating waves, but with the phase shift patterns on the units in the ring having the structure of discrete rotating waves. Then, we study the existence of additional patterns that emerge via secondary bifurcations from rotating waves. In Chapter 4 computational bifurcation analysis of the collective response of a CCOST system with the aid of the continuation software package AUTO [11] is computed. Chapter 5 features the simulation of phase error of the unidirectional coupling topology against the uncoupled baseline. Lastly, this thesis concludes with a discussion on the results and research to come. 4

CHAPTER 2

CRYSTAL OSCILLATORS

A crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions [30, 40]. Almost any object made of an elastic material could be used like a crystal, with the appropriate transducers, since all objects have natural resonant frequencies of vibration. For example, steel is very elastic and has a high speed of sound, so it was often used in mechanical filters before quartz. Resonant frequency depends on the size, shape, elasticity, and speed of sound within the material. High-frequency crystals are typically cut in the shape of a simple, rectangular plate. Low-frequency crystals, such as those used in digital watches, are typically cut in the shape of a tuning fork. For applications not needing very precise timing, a low-cost ceramic resonator is often used in place of a quartz crystal. When a crystal of quartz is properly cut and mounted, it can be made to distort in an electric field by applying a voltage to an electrode near or on the crystal. This property is known as electrostriction or inverse piezoelectricity. When the field is removed, the quartz will generate an electric field as it returns to its previous shape, and this can generate a voltage. The result is that a quartz crystal behaves like a circuit composed of an inductor, capacitor and resistor, with a precise resonant frequency [41], see Fig. 2.1.

Figure 2.1. (Left) Schematic of a quartz crystal. (Right) A quartz crystal behaves like circuit composed of an inductor, capacitor and resistor, so it oscillates with a precise resonant frequency when it is subjected to an electric field. 5 Quartz has the further advantage that its elastic constants and its size change in such a way that the frequency dependence on temperature can be very low. The specific characteristics will depend on the mode of vibration and the angle at which the quartz is cut (relative to its crystallographic axes). Therefore, the resonant frequency of the plate, which depends on its size, will not change much, either. This means that a quartz clock, filter or oscillator will remain accurate. For critical applications the quartz oscillator is mounted in a temperature-controlled container, called a crystal oven, and can also be mounted on shock absorbers to prevent external mechanical vibrations. Next we review the analysis of a single crystal oscillator model conducted in [22]. This presentation is motivated, however, from the group-theoretical and numerical perspectives.

2.1 TWO-MODE OSCILLATOR MODEL The crystal oscillator circuit sustains oscillation by applying a voltage signal from the quartz resonator, amplifying it, and feeding it back to the resonator. The rate of expansion and contraction of the quartz is the resonant frequency, and is determined by the cut and size of the crystal as mentioned in the previous paragraph. When the energy of the generated output frequencies matches the losses in the circuit, an oscillation can be sustained. The frequency of the crystal can be slightly adjusted by modifying the attached capacitances. A varactor, a diode with capacitance depending on applied voltage, is often used in voltage-controlled crystal oscillators, VCOs. The analog port of the VCO chip is modeled by a nonlinear resistor R−, see Fig. 2.2,

Figure 2.2. Two-mode crystal oscillator circuit. A second set of spurious RLC components (R2,L2,C2) are introduced by parasitic elements. [37] [33]

that obeys the voltage-current relationship

v = −ai + bi3, 6 where a and b are constant parameters. A major reason for the wide use of crystal oscillators is their high Q factor. This is a dimensionless parameter that indicates how under-damped an oscillator is. For a crystal oscillator, it can be deﬁned as the ratio of the resonant frequency with respect to the half-power bandwidth, i.e., the bandwidth over which the power of vibration is greater than half the power at the resonant frequency. Higher Q indicates that the oscillations die out more slowly. A typical Q value for a quartz oscillator ranges from 104 to 106, compared to perhaps 102 for an LC oscillator. The inductance of the leads connecting the crystal to the VCO port is represented by

Lc. In addition, parasitic elements can be represented by a series resonator (L2, C2, R2) connected in parallel with the nonlinear resistor. The resulting circuit, depicted in Fig. 2.2, forms a two-mode resonator model. Applying Kirchhoff’s voltage law yields the following governing equations

2 d ij dij 1 2 di1 di2 Lj 2 + Rj + ij = [a − 3b(i1 + i2) ] + , (2.1) dt dt Cj dt dt where j = 1, 2 and Lc has been included in L1. Letting x1 = i1, x2 = di1/dt, x3 = i2 and 2 T x4 = di2/dt, ω0j = 1/LjCj and X = [x1, x2, x3, x4] , the model equations (2.1) can be rewritten as dX = F (X) ≡ AX + N (X), (2.2) dt where     0 1 0 0   0 0  a − R a  −3b  −ω2 1 0     (x + x )2(x + x )  01  N1  1 3 2 4   L1 L1     L1  A =   , N (X) =   =   .  0 0 0 1   0   0      a a − R   −3b  2 2  N  2  0 −ω02 2 (x1 + x3) (x2 + x4) L2 L2 L2

The terms AX and N (X) represent the linear and nonlinear terms, respectively, which, together, govern the behavior of the two-mode crystal oscillator.

2.2 AVERAGING

In what follows we assume nonresonance conditions among ω01 and ω02, so that there ω p are no nonzero integers p and q for which 01 = . Using the Van der Pol transformation ω02 q 7

T U = Φ(t)X, where U = [u1, . . . , u4] and Φ(t) is the co-rotating frame of reference:

 1  cos ω1t − sin ω1t 0 0 ω1  1  − sin ω1t − cos ω1t 0 0  Φ(t) =  ω1  ,  1   0 0 cos ω2t − sin ω2t  ω2  1 0 0 − sin ω2t − cos ω2t ω2

Eq. (2.2) can be rewritten as

dU = εF (U, t), (2.3) dt where the derivatives are explicitely given by

1 u˙ = Ω u cos(ω t) − u sin(ω t) − N sin(ω t) 1 ω 1 1 1 2 1 1 1 1 1 v˙1 = Ω1 u1 cos(ω1t) − u2 sin(ω1t) − N1 cos(ω1t) ω1 1 u˙ = Ω u cos(ω t) − u sin(ω t) − N sin(ω t) 2 ω 2 3 2 4 2 2 2 2 1 v˙2 = Ω2 u3 cos(ω2t) − u4 sin(ω2t) − N2 cos(ω2t), ω2

2 2 2 2 with ω01 − ω1 = εΩ1 and ω02 − ω2 = εΩ2. Averaging over the periods T1 = 2π/ω1 and

T2 = 2π/ω2 we arrive at the simpliﬁed equation

dU = εF¯(U) ≡ εAU¯ + εN¯ (U), (2.4) dt

1 Z T where F¯(U) = lim F (U, t)dt and T →∞ T 0

a − R1 Ω1    − 0 0 u1 2 2 2 2 2L 2ω [u1 + u2 + 2(u3 + u4)]  1 1  L1   Ω a − R   u   1 1 0 0   2 [u2 + u2 + 2(u2 + u2)]   3b  1 2 3 4  ¯  2ω1 2L1  ¯ L1  A =   , N (X) = − u .  a − R2 Ω2  8  3 2 2 2 2   0 0 −   [u3 + u4 + 2(u1 + u2)]  2L 2ω  L2   2 2   u  Ω a − R  4 2 2 2 2   2 2  [u3 + u4 + 2(u1 + u2)] 0 0 L2 2ω2 2L2

Equation (2.4) can be interpreted as a representation of the original model Eq. (2.2) with respect to two rotating frames of reference, one rotating with speed ω1 and one with 8

speed ω2. We invite the reader to see the appendix for the full averaging calculation. Observe

that in this rotating coordinate system, u1 = u2 = 0 and u3 = u4 = 0 are two invariant subspaces so that the two modes of oscillation of the crystal decouple from one another.

Using complex coordinates: z1 = u1 + u2 i and z2 = u3 + u4 i, and re-scaling time, the model Eq. (2.4) can be expressed as

3b 2 2 z˙1 = (µ1 + τ1i)z1 − (|z1| + 2|z2| )z1 8L1 3b 2 2 z˙2 = (µ2 + τ2i)z2 − (|z2| + 2|z1| )z2, 8L2

where µ1 = (a − R1)/(2L1), τ1 = Ω1/(2w1), µ2 = (a − R2)/(2L2) and τ2 = Ω2/(2w2). 1 1 −iτ1t −iτ2t Finally, letting z˜1 = √ z1e and z˜2 = √ z2e , we arrive, after dropping the˜marker, 8 8 at the following set of equations in complex coordinates:

3b z˙ = µ z − (|z |2 + 2|z |2)z 1 1 1 L 1 2 1 1 (2.5) 3b 2 2 z˙2 = µ2z1 − (|z2| + 2|z1| )z2. L2

Observe that in these coordinates, Eq. (2.5) commutes with a 2-Torus 2 4 2 T = SO(2) × SO(2), which acts on R = C diagonally by

iθ1 iθ2 (θ1, θ2) · (z1, z2) = (e z1, e z2),

2 2 where (θ1, θ2) ∈ T and (z1, z2) ∈ C .

2.3 STABILITY

The linearization of Eq. (2.5) about (z1, z2) = (0, 0) produces two pairs of complex eigenvalues a − R1 Ω1 σ1 = ± i 2L1 ω1 a − R2 Ω2 σ2 = ± i. 2L2 ω2

It follows that when both R1/a > 1 and R2/a > 1 the trivial equilibrium is asymptotically stable. In fact, we show below that in this region there it is the only solution so

U = (0, 0, 0, 0) is, actually, globally asymptotically stable. When R1/a < 1, while R2/a > 1 is ﬁxed, the trivial solution loses stability and a limit cycle emerges, restricted to the invariant

subspace u3 = u4 = 0, via a standard Hopf bifurcation. This solution is called Mode 1

in [22]. Since u3 = u4 = 0 is invariant, it corresponds to only one current in the crystal 9 oscillating while the other remains zero, i.e., of the form (i1(t), 0). A similar scenario leads to

Mode 2 ,(0, i2(t)), along the invariant subspace u1 = u2 = 0, when R2/a < 1 while R1/a > 1 is now held ﬁxed. In order to get a more complete picture of the regions of existence and stability of these modes of oscillations, we convert Eq. (2.4) to polar coordinates through the 2 2 3 4 transformation: ρ1 = u1 + u2 and ρ2 = u1 + u2. We get

(a − R ) 3b ρ˙ = 1 ρ − ρ ρ2 + 2ρ2 1 2L 1 8L 1 1 2 1 1 (2.6) (a − R2) 3b 2 2 ρ˙2 = ρ2 − ρ2 ρ2 + 2ρ1 . 2L2 8L2

The phase equations decouple from the amplitude equations due to the nonresonance conditions, thus they are not included in (2.6). It is important to note that now the amplitude equations (2.6) commute only with the standard action of the Z2 × Z2 symmetry group in the plane [14], which is what remains of the 2-torus T2 phase-shift symmetries. This type of reﬂectional symmetry commonly appears in engineering applications of nonlinear oscillators. In particular, it is found in classical systems such as: the pendulum equations [42], Dufﬁng oscillators [18, 32] and Van der Pol oscillators [4, 9, 17, 35, 36]. In recent works, the same type of odd symmetry has appeared in modern systems that include: vibratory gyroscopes [3, 8, 34, 39, 38] and energy harvesting systems [6, 7, 24]. Equilibria of

Isotropy Solution Type

Z2 × Z2 (ρ1, ρ2) = (0, 0) Trivial Steady State r ! 4(a − R ) (1, −1) (ρ , ρ ) = 1 , 0 Mode 1: Periodic solution, period T = 2π/ω Z2 1 2 3b 01 r ! 4(a − R ) (−1, 1) (ρ , ρ ) = 0, 2 Mode 2: Periodic solution, period T = 2π/ω Z2 1 2 3b 02 ∗ ∗ 1 (ρ1, ρ2) = (ρ1, ρ2) Mixed-Mode: Invariant 2-torus Table 2.1. Classification of solutions of Eq. (2.6) based on isotropy subgroups. equations (2.6) correspond to steady states, limit cycles,and invariant two-tori in the original model Eq. (2.3). Specifically, there are four types of solutions and they can be classified based on their isotropy subgroups of Z2 × Z2, shown in Table 2.1. Modes 1 and 2 exist only if

R1/a < 1 and R2/a < 1, respectively. As indicated above, these periodic solutions appear as primary bifurcations off the trivial steady state via standard Hopf bifurcations. The bifurcation of either branch could be supercritical or subcritical depending on how the parameters R1/a 10 and R2/a are varied. For instance, if R2/a > 1 is held ﬁxed while R1/a decreases below 1 then Mode 1 appears via a supercritical Hopf bifurcation as the trivial solution loses stability.

If we then follow the unstable branch of the trivial solution while decreasing R2/a a

subcritical Hopf bifurcation leads to Mode 2 as R2/a crosses 1. A similar scenario leads to a

supercritical Hopf bifurcation for Mode 2 followed by a subcritical one for Mode 1 as R1/a is

varied ﬁrst and then R2/a. ∗ ∗ The mixed-mode solution (ρ1, ρ2) exists only inside the region (R1/a, R2/a) bounded

by R1/a < R2/(2a) + 1/2 < 1 and R2/a < R1/(2a) + 1/2 < 1. This is a two-frequency solution that lies on an invariant 2-torus. This 2-torus appears through a secondary bifurcation from the two pure modes. Generically, this family of 2-tori is either always stable or always unstable. For our model of crystal oscillators it is the latter case since the linearization of ∗ ∗ Eq. (2.6) about (ρ1, ρ2) yields positive real parts of eigenvalues. In this same region, the real parts of the eigenvalues of the linearization of Eq. (2.6) about Mode 1 and Mode 2 are both negative, leading to bistability between Modes 1 and 2. The time-series solutions and corresponding phase spaces for each individual pure mode and mixed-mode oscillation are shown in Fig. 2.3. The regions of existence and stability are depicted in the two-parameter bifurcation diagram also included in Fig. 2.3. In summary, Modes 1 and 2 can be stable. Bistability among these two modes is found in the region bounded by

R1/a < R2/(2a) + 1/2 < 1 and R2/a < R1/(2a) + 1/2 < 1, which is also the same region where the stable mixed-mode solution exists and is always unstable. The common point

where R1/a = R2/a = 1 corresponds to a codimension-two torus bifurcation at which two pairs of eigenvalues of the linearization of Eq. (2.2) about the trivial solution cross the

imaginary axis, i.e., a Hopf-Hopf bifurcation. The boundary curves R1/a = R2/(2a) + 1/2

and R2/a = R1/(2a) + 1/2, which are the locus of the secondary bifurcations that lead to a change of stability for Modes 1 and 2, respectively, meet at the torus bifurcation point. The presence of the two-frequency oscillations in the mixed mode is shown in the time-series plot shown in Fig. 2.3. A 3D phase-space visualization of the associated torus is also shown in Fig. 2.3. However, we emphasize that this mixed-mode solution is always unstable so it does not appear after transient integration. 11

(0,i ) (0,0) 1 2 (0,i ) x 10−3 0.8 2 2 1

/a 0.6

3 0 1 x

R −1 0.4 (i ,0) 1 −2 2 0.2 (i ,i ) (i ,0) 1 2 1 1 1 0.5 5 0 x 10 0 0 −1 −3 0 0.2 0.4 0.6 0.8 1 −0.5 x 10 −2 x −1 R /a 2 x 2 1 x 10−3

2 x x 1 1 x 0.01 x 3 3 1

0 0

−0.01 −1

−2 −0.02 400 500 600 700 800 800 820 840 860 880 t t

Figure 2.3. Time series solutions of a single two-mode crystal oscillator model (2.4). (Top left) Two parameter bifurcation diagram indicating the regions of existence of the time-series solutions as a function of parameters R1/a and R2/a. (Top right) Unstable Mixed Mode (i1(t), i2(t)), (bottom left) Stable Mode 1 (i1(t), 0) , (bottom right) Stable Mode 2 (0, i2(t)). Parameter values −4 −4 −13 are: R1 = 30.9Ω, R2 = 181.1Ω, L1 = 5.2 × 10 H, L2 = 2.6 × 10 H, C1 = 1.0 × 10 F , −14 8 C2 = 2.5 × 10 F , a = 939, b = 3 × 10 . 12

CHAPTER 3

COUPLED CRYSTAL OSCILLATOR SYSTEM

This chapter studies the collective behavior of a Coupled Crystal Oscillator System (CCOST) made up of N ∈ N, assumed to be identical, crystal oscillators. We consider ﬁrst the case of unidirectional coupling in a ring fashion, as is shown schematically in Fig. 3.1.

The spatial symmetry of the ring is described by the group ZN of cyclic permutations of N objects. Since the dynamics of each node is still described by Eq. (2.5) with internal symmetry SO(2) × SO(2) then, effectively, the entire network symmetry is given by the group

Γunidir = SO(2) × SO(2) × ZN .

In the case of bidirectional coupling, the spatial symmetry is captured by the dihedral group DN of permutations of an N-gon, so the network symmetry is the group

Γbidir = SO(2) × SO(2) × DN .

In both cases, unidirectional and bidirectional coupling, we analyze the collective behavior of the network for arbitrary ring size N using classiﬁcation of periodic solutions via their symmetry groups. Moreover, by using the isotypic decomposition of the phase space under the action of the group of symmetries we can study the linearized system of equations and calculate analytical expressions for the critical eigenvalues that lead to a wide range of symmetry-breaking bifurcations to patterns of collective behavior.

f2 f1

f3 fN

f4 f5

Figure 3.1. CCOST concept with unidirectionally coupled crystal oscillators. 13

3.1 AVERAGED EQUATIONS Applying Kirchhoff’s law to the CCOST network with unidirectional coupling yields the following governing equations

d2i di 1 L k,j + R k,j + i = a − 3bi + i − λi + i 2 k,j dt2 k,j dt C k,j k,1 k,2 k+1,1 k+1,2 k,j (3.1) di di di di k,1 + k,2 − λ k+1,1 + k+1,2 , dt dt dt dt

where k = 1, 2,...,N, j = 1, 2. Since we assume identical components in each crystal

oscillator, then the set of parameters reduces to: Lk,1 = L1, Lk,2 = L2, Rk,1 = R1, Rk,2 = R2,

Ck,1 = C1 and Ck,2 = C2. Eq. (3.1) can be rewritten in matrix form as follows

dX k = F (X ) − λB X + N (X ,X ), (3.2) dt k k+1 k k+1

0 0 T T where Xk = [ik1, ik1, ik2, ik2] = [xk1, . . . , xk4] , F (Xk) is the internal dynamics of each individual crystal as is described by Eq. (2.2). B is the matrix for the linear coupling terms while the vector N contains nonlinear coupling terms, they are given by

    0 0 0 0 0      a a   3b os os 2 es es os es  0 0  − [(xk − λxk+1) (xk − λxk+1) − xk xk ]  L1 L1   L1  B =   , N (X) =   , 0 0 0 0   0       a a    0 0  −3b os os 2 es es os es  L L [(xk − λxk+1) (xk − λxk+1) − xk xk ] 2 2 L2

os es where xk = xk1 + xk3 and xk = xk2 + xk4. Using a similar set of Van der Pol

transformations and after averaging over the periods T1 = 2π/ω1 and T2 = 2π/ω2 we get the following set of equations:

dU k = εAU¯ − ελBU¯ + εN¯ (U ,U ), (3.3) dt k k+1 k k+1 14

T ¯ where Uk = [uk1, . . . , uk4] , A is the same matrix associated with the linear components of a single crystal oscillator, see Eq. (2.4), and  a  0 0 0   2L N1  1 a   0 0 0      N2 ¯  2L1  ¯   B =  a  , N (U) =   ,  0 0 0  N3  2L2     a  0 0 0 N4 2L2

where N1,..., N4 contain nonlinear terms in Uk and Uk+1. The actual expressions are too long to be listed here, however, the full averaging is included in the second portion of the

appendix. Complexifying again: zk1 = uk1 + uk2i and zk2 = uk3 + uk4i, we arrive at the network equations for a unidirectionally coupled CCOST system in the following form

3b 2 2 z˙k1 = (µk1 + iτk1)zk1 − ξ1zk+1,1 − |zk1| + 2|zk2| zk1+ 8L1 3bλ 2 2 (|zk1| + 2|zk2| )zk+1,1 + zk1z¯k+1,1 +z ¯k1zk+1,1 zk1 + 2 zk2z¯k+1,2 +z ¯k2zk+1,2 zk1 − 8L1 2 3bλ 2 2 |zk+1,1| + 2|zk+1,2| zk1 + zk1z¯k+1,1 +z ¯k1zk+1,1 zk+1,1 + 2 zk2z¯k+1,2 +z ¯k2zk+1,2 zk+1,1 8L1 3 3bλ 2 2 |zk+1,1| + 2|zk+1,2| zk+1,1 , 8L1

3b 2 2 z˙k2 = µk2 + iτk2 zk2 − ξ2zk+1,2 − 2|zk1| + |zk2| zk2+ 8L1 3bλ 2 2 2|zk1| + |zk2| zk+1,2 + zk2z¯k+1,2 +z ¯k2zk+1,2 zk2 + 2 zk1z¯k+1,1 +z ¯k1zk+1,1 zk2 − 8L1 2 3bλ 2 2 2|zk+1,1| + |zk+1,2| zk2 + zk2z¯k+1,2 +z ¯k2zk+1,2 zk+1,2 + 2 zk1z¯k+1,1 +z ¯k1zk+1,1 zk+1,2 8L1 3 3bλ 2 2 2|zk+1,1| + |zk+1,2| zk+1,2 . 8L1 (3.4) In the bidirectional coupling case, Kirchhoff’s voltage law yields the following governing equations

2 d ik,j dik,j 1 Lk,j 2 + Rk,j + ik,j = dt dt Ck,j 2 a − 3b ik,1 + ik,2 − λ ik+1,1 + ik+1,2 − λ ik−1,1 + ik−1,2 (3.5) di di di di di di k,1 + k,2 − λ k+1,1 + k+1,2 − λ k−1,1 + k−1,2 , dt dt dt dt dt dt 15 where k = 1, 2,...,N, j = 1, 2. Identical components in each crystal oscillator are assumed again and rewrite Eq. (3.1) in matrix form

dX k = AX − λB( X + X ) + N (X ,X ,X ). (3.6) dt k k−1 k+1 k−1 k k+1

Matrices A and B are the same as in the unidirectional case and the governing equations is very similar except that now the nonlinear terms are given by   0    3b os os os 2 es es es  − [(xk − λxk+1 − λxk−1) (xk − λxk+1 − λxk−1)]  L1  N (X) =   ,    0     −3b os os os 2 es es es  [(xk − λxk+1 − λxk−1) (xk − λxk+1 − λxk−1)] L2

os es where xk = xk1 + xk3 and xk = xk2 + xk4. Using a similar set of Van der Pol

transformations and after averaging over the periods T1 = 2π/ω1 and T2 = 2π/ω2 we get the following set of equations:

dU k = εAU¯ − ελB¯( U + U ) + εN¯ (U ,U ,U ), (3.7) dt k k−1 k+1 k−1 k k+1

where A¯ and B¯ are the same matrices as in the unidirectional coupling case. The nonlinear ¯ terms N (Uk−1,Uk,Uk+1) are too long to be listed here, but the full averaging is written in the

last portion of the appendix. Again, we complexify: zk1 = uk1 + uk2i and zk2 = uk3 + uk4i,

and let ξk1 = aλ/(2Lk1) and ξk2 = aλ/(2Lk2) to arrive at the network equations for a bidirectionally coupled CCOST system in the following form 16

3b 2 2 z˙k1 = (µk1 + iτk1)zk1 − ξk1 zk−1,1 + zk+1,1 − |zk1| + 2|zk2| zk1+ 8L1 3bλ 2 2 (|zk,1| + 2|zk,2| ) zk−1,1 + zk+1,1 + zk1z¯k−1,1 +z ¯k1zk−1,1 + zk1z¯k+1,1 +z ¯k1zk+1,1 zk,1+ 8L1 2 zk,2z¯k−1,2 +z ¯k2zk−1,2 + zk,2z¯k+1,2 +z ¯k2zk+1,2 zk1 − 2 3bλ 2 2 2 2 |zk−1,1| + |zk+1,1| + 2(|zk−1,2| + |zk+1,2| ) zk,1 + zk−1,1z¯k+1,1 +z ¯k−1,1zk+1,1 zk1+ 8L1 2 zk−1,2z¯k+1,2 +z ¯k−1,2zk+1,2 zk1 + zk1z¯k−1,1 +z ¯k1zk−1,1 + zk1z¯k+1,1 +z ¯k1zk+1,1 zk−1,1 + zk+1,1 + 2 zk2z¯k−1,2 +z ¯k,2zk−1,2 + zk,2z¯k+1,2 +z ¯k2zk+1,2 zk−1,1 + zk+1,1 + 3 3bλ 2 2 2 2 |zk−1,1| + |zk+1,1| + 2(|zk−1,2| + |zk+1,2|) zk−1,1 + zk+1,1 + 8L1 zk−1,1z¯k+1,1 +z ¯k−1,1zk+1,1 + 2(zk−1,2z¯k+1,2 +z ¯k−1,2zk+1,2) zk−1,1 + zk+1,1 ,

3b 2 2 z˙k2 = (µk2 + iτk2)zk2 − ξk2 zk−1,2 + zk+1,2 − |zk2| + 2|zk1| zk2+ 8L2 3bλ 2 2 (|zk2| + 2|zk1| ) zk−1,2 + zk+1,2 + zk2z¯k−1,2 +z ¯k,2zk−1,2 + zk,2z¯k+1,2 +z ¯k,2zk+1,2 zk,2+ 8L1 2 zk1z¯k−1,1 +z ¯k1zk−1,1 + zk1z¯k+1,1 +z ¯k1zk+1,1 zk2 − 2 3bλ 2 2 2 2 |zk−1,2| + |zk+1,2| + 2(|zk−1,1| + |zk+1,1| ) zk,2 + zk−1,2z¯k+1,2 +z ¯k−1,2zk+1,2 zk,2+ 8L2 2 zk−1,1z¯k+1,1 +z ¯k−1,1zk+1,1 zk2 + zk2z¯k−1,2 +z ¯k2zk−1,2 + zk,2z¯k+1,2 +z ¯k2zk+1,2 zk−1,2 + zk+1,2 + 2 zk1z¯k−1,1 +z ¯k1zk−1,1 + zk1z¯k+1,1 +z ¯k1zk+1,1 zk−1,2 + zk+1,2 + 3 3bλ 2 2 2 2 |zk−1,2| + |zk+1,2| + 2(|zk−1,1| + |zk+1,1|) zk−1,1 + zk+1,1 + 8L2 zk−1,2z¯k+1,2 +z ¯k−1,2zk+1,2 + 2(zk−1,1z¯k+1,1 +z ¯k−1,1zk+1,1) zk−1,2 + zk+1,2 . (3.8)

3.2 SYMMETRIESOF COLLECTIVE PATTERNS We are interested in classifying the possible patterns of oscillations that can arise in a network of crystal oscillators such as (3.4) and (3.8). To do this, we use a family of results from equivariant bifurcation theory as described below.

Consider the action of the Abelian group SO(2) × SO(2) × ZN given by

iθ1 iθ2 iθ1 iθ2 iθ1 iθ2 (γ, θ1, θ2).(z11, z12, . . . , zN1, zN2) = (e z21, e z22, . . . , e zN1, e zN2, e z11, e z12), (3.9) 17 where γ ∈ ZN and (θ1, θ2) ∈ SO(2) × SO(2). In the case of bidirectional coupling,we need the additional action of the order two element κ ∈ DN :

κ.(z11, z12, . . . , zN1, zN2) = (z11, z12, zN1, zN2, zN−1,1, zN−1,2, ··· , z21, z22), (3.10)

The form of equations (3.4) is such that the system can be projected to two identical subsystems. From the action of the SO(2) group in (3.9) and (3.10) on each oscillator, we can consider the ﬁxed-point subspace in each case; that is, the subspaces ﬁxed by (1, θ1, 0) and

(1, 0, θ2). Letting Zk = (zk1, zk2), those are

V1k = {Zk | zk2 = 0} and V2k = {Zk | zk1 = 0}

˜ and they are ﬂow-invariant. Thus, for i = 1, 2, Vi = Vi1 ⊕ · · · ⊕ ViN is ﬂow-invariant with

respect to the SO(2)(θi) × ZN action and with respect to SO(2)(θi) × Dn in the bidirectional ˜ case. Therefore, we obtain identical differential equation systems restricted to Vi (i = 1, 2) and so stability and bifurcation results are equally valid for both subsystems. We focus our ˜ discussion on V1 to keep the notation as simple as possible. We denote the differential ˜ equation subsystem in V1 by ˙ Q1 = F (Q1, λ) (3.11)

N where Q1 = (z11, z21, . . . , zN1) ∈ C are the coordinates of the phase space and λ ∈ R is the coupling parameter. We begin by identifying the oscillatory patterns that can arise from local bifurcation at the equilibrium solution via equivariant Hopf bifurcation [14]. We show that the periodic solutions obtained have the form of rotating waves [21]. Then, we look at the possible local secondary bifurcations from rotating waves.

3.2.1 The unidirectional case

Suppose that the linearization L0 = (dF )0 has a pair of eigenvalues crossing the imaginary axis with nonzero speed at ±iω. Then, the eigenspace Eiω is generically a

Γ-simple representation of SO(2) × ZN ; that is, Eiω is the sum of two (isomorphic) absolutely irreducible representations, or it is a non-absolutely irreducible representation, see [14]. The irreducible representations of Abelian groups are one-dimensional complex spaces. Without loss of generality, we can assume that the frequency is ω = 1.

The group SO(2) acts diagonally on each component of Q1, therefore the irreducible

representations of SO(2) × ZN are obtained directly by starting with the irreducible representations of ZN . If γ is a generator of ZN , then it acts irreducibly by complex rotation: let z ∈ C and ζ = exp(2πi/N), then γ.z = ζmz for some m = 0,...,N − 1 and we label 18 those irreducible representations by Wm for m = 0,...,N − 1. Note that Wj, Wk are

nonisomorphic representations of SO(2) × ZN for j 6= k. We notice that for m = 0 the action is trivial and if N is even, then the action for m = N/2 is γ.z = −z, corresponding to the

alternating representation. For ZN only, the cases m = 0 and m = N/2 can be considered as real irreducible representations. However, the SO(2) action (θ.z = eiθz) forces all irreducible representations to be of complex type; that is, they are non-absolutely irreducible.

Thus, the action of SO(2) × ZN on its irreducible representations is given by

(θ, γ).z = e2πim/N+iθz

for γ a generator of ZN and θ ∈ SO(2). The eigenspace Ei generically corresponds to a Wm irreducible representation on which we must also allow for the action of phase shift symmetries given by S1. Let ψ ∈ S1, then ψ.z = eiψz. This implies that the group 1 SO(2) × ZN × S acts on Ei via

(θ, γ, ψ).z = e2πim/N+iθ+iψz

Therefore, the irreducible representation Wm is ﬁxed by the group

k 1 Σm = {(θ, γ , −θ − 2πkm/N) | k = 0,...,N − 1, θ ∈ SO(2)} ⊂ SO(2) × ZN × S

with

¯ k ZN/d := Σm ∩ (SO(2) × ZN ) = {(−2πkm/N, γ , 0) | k = 0,...,N − 1}

isomorphic to ZN/d where d = gcd(m, N). Applying the Equivariant Hopf Theorem to this situation yields the following result. Proposition 3.1. Suppose that the linearization of (3.11) at the origin has a pair of purely imaginary eigenvalues crossing the imaginary axis with nonzero speed at ±i. Then, generically, there exists a unique branch of periodic solutions with period near 2π bifurcating from the origin and the symmetry group of the periodic solution is given by the pair ¯ 1 (H,K) = (Σm, ZN/d) where H/K ' S . Therefore, the periodic solution is a rotating wave ¯ with ZN/d spatial symmetry. We now describe the type of waveforms given by Proposition 3.1 in system (3.3). Let

X(t) = (i1(t), . . . , iN (t)) 19 ¯ be a rotating wave with symmetry (Σm, ZN/d). The effect of the continuous part, given by elements of the form (θ, 1, −θ), on X(t) guarantees the rotating wave nature of X(t); that is, −iθ e ij(t) = ij(t − θ) for j = 1,...,N. We set SOg(2) = {(θ, 1, −θ) | θ ∈ [0, 2π)}. The spatial symmetry group action on X(t) is given by

−2πim/N −2πim/N −2πim/N (−2πm/N, γ, 0).X(t) = (e i2(t), e i3(t), . . . , e i1(t))

= (i2(t − 2πm/N), i3(t − 2πm/N), . . . , i1(t − 2πm/N)).

and so (−2πm/N, γ, 0).X(t) = X(t) implies

X(t) = (X(N/d)1(t),...,X(N/d)d(t)) (3.12)

where for all j = 1, . . . , d,

X(N/d)j(t) = (i1(t), i1(t + 2πm/N), . . . , i1(t + 2π((N/d) − 1)m/N))

with the phase shifts taken mod 2π and the subscript N/d identiﬁes the size of the vector.

Example 3.1. Consider the case N = 8 with m = 6 (i.e. d = 2) so that Xλ(t) is a ¯ one-parameter family of rotating waves with isotropy subgroup (Σ6, Z4). Using the formula (3.12), one can verify that

X(t) = (i1(t), i1(t + 3π/2), i1(t + π), i1(t + π/2), i1(t), i1(t + 3π/2), i1(t + π), i1(t + π/2)).

¯ Remark 3.2. It is important to note that the spatial symmetry group of the rotating wave ZN/d generates the phase shifts in the coupled network, but it does not correspond to the spatial symmetry group observed in the waveform itself. In the case of Example 3.1, the spatial ¯ symmetry of the rotating wave is Z4, but the spatial symmetry that one can see in the 4 waveform is isomorphic to Z2, generated by the order two element γ . In order to avoid confusion in the interpretation of spatial symmetry, we consider only the spatial symmetry ¯ group of the rotating wave ZN/d in our discussions.

Bifurcation from rotating waves We now investigate symmetry-breaking bifurcations from the rotating waves obtained in Proposition 3.1. Bifurcations from a rotating wave X(t) can be

analyzed within the framework developed by Krupa [21]. Let X0 = X(0), then X(t) can be

seen as the SOg(2)-group orbit of X(0); X(t) = SOg(2)X(0). Let NX(0) be the space of vectors normal to X(t). Because X(t) is a one-dimensional manifold, we have

dim NX(0) = 2N − 1. 20

Let Xλ(t) be a one-parameter family of rotating waves obtained from Proposition 3.1 ¯ with symmetry group (Σm, ZN/d). The bifurcations of Xλ(t) are in one-to-one ¯ 2N−1 2N−1 correspondence with bifurcations of a ZN/d-equivariant vector ﬁeld g : R × R → R where Xλ(t) corresponds to a trivial equilibrium (0, λ) for g. Suppose that an eigenvalue of

(dg)(0,λ) crosses at 0 with nonzero speed at λ = λ0. Then, generically, the kernel of (dg)(0,λ0) ¯ is an absolutely irreducible representation of ZN/d [14]. The irreducible representations of ¯ ZN/d are described above and lead to one-dimensional complex irreducible representations, except for the absolutely irreducible representations in the case m = 0 or if m = N/2d for d even. This leads to the following result. ¯ Proposition 3.2. Suppose that Xλ(t) with symmetry group (Σm, ZN/d) has a bifurcation at

λ = λ0 such that (dg)(0,λ0) has a zero eigenvalue. Then, generically, either Xλ(t) has a symmetry-preserving saddle-node bifurcation, or if N/d is even then a branch of rotating ¯ wave solutions Yλ(t) with symmetry group (Σm˜ , ZN/2d) bifurcates from Xλ(t) where m˜ = 2m mod N. ¯ Proof. Generically, the kernel of (dg)(0,λ) is one-dimensional. Suppose that ZN/d acts trivially ¯ on the kernel of (dg)(0,λ). Then, the centre manifold is fixed by ZN/d and the bifurcating ¯ solution preserves the symmetry group. If ZN/d acts nontrivially on (dg)(0,λ), then ¯ (−2πm/N, γ) acts by −1 and so ZN/2d fixes the kernel. Therefore, the centre manifold is also ¯ ¯ fixed by ZN/2d and the bifurcating solution has spatial subgroup ZN/2d. The waveform of periodic solutions Y (t) bifurcating from X(t) is given by

Y (t) = (X(N/2d)1(t),...,X(N/2d)(2d)(t)).

Thus, bifurcation to periodic solutions from X(t) can only occur generically for the irreducible representations of real type. Bifurcation from X(t) via the complex irreducible representations generically leads to tori supporting quasi-periodic ﬂows. The drift given by ¯ ¯ the group orbit is determined by the dimension of the maximal torus in N(ZN/d)/ZN/d. ¯ ¯ ¯ Because the group is Abelian, N(ZN/d) = Σm and so dim N(ZN/d)/ZN/d = 1. Example 3.3. We continue Example 3.1 with one-parameter family of rotating waves

Xλ(t) = (i1(t), i1(t + 3π/2), i1(t + π), i1(t + π/2), i1(t), i1(t + 3π/2), i1(t + π), i1(t + π/2)).

Suppose that (dg)(0,λ) has a one-dimensional kernel on which (3π/2, γ) acts nontrivially. 2 ¯ Then, h(π, γ )i ⊂ Z4(3π/2, γ) ﬁxes the kernel and the bifurcating periodic solution Y (t) has ¯ 2 isotropy subgroup (Σ4, Z2(π, γ )). The bifurcating rotating wave has waveform

Y (t) = (i1(t), i1(t + π), i1(t), i1(t + π), i1(t), i1(t + π), i1(t), i1(t + π)). 21 3.2.2 The bidirectional case As above, we suppose that the linearization has a pair of purely imaginary eigenvalues

(assumed to be ±i) and consider the eigenspace Ei which is generically Γ-simple.

The DN irreducible representations are all absolutely irreducible. We now proceed to the analysis using the direct product theory developed by Dionne et al. [10]. The irreducible

representations of the SO(2) × DN action are tensor products of irreducible representations U N of the SO(2) action on C (that is, U = C) and irreducible representations V of Dn on R .

The form of the Γ-simple eigenspace Ei depends on the absolutely irreducible representation

of DN . The bifurcating branches from the Equivariant Hopf Theorem are obtained by ﬁnding

the C-axial subgroups of the action of SO(2) × DN on Ei. We seek for C-axial subgroups of 1 SO(2) × DN × S and for this, we need the homomorphism

1 1 1 Θ: SO(2) × S × DN × S → SO(2) × DN × S

deﬁned by Θ(θ, φ, g, ψ) = (θ, g, φ + ψ). It is shown in [10] that if Aφ ⊂ SO(2) × S1 and ψ 1 B ⊂ DN × S are C-axial twisted subgroups, then the subgroup φ ψ 1 A ×˙ B ⊂ SO(2) × DN × S is also C-axial where

Aφ×˙ Bψ = Θ(Aφ × Bψ).

Now, the only (nontrivial) twisted subgroup of the SO(2) action is given by SO^(2) = {(θ, −θ) | θ ∈ [0, 2π)} and so we only need to determine the twisted subgroups of

DN . 1 Proposition 3.3. If a periodic solution has isotropy subgroup SO^(2)×˙ Σ ⊂ SO(2) × DN × S with the elements of Σ of the form (σ, ψ). Then, the spatial subgroup is isomorphic to Σ, and we denote it by Σ.

Proof. Elements of SO^(2)×˙ Σ have the form (θ, γ, −θ + ψ). Therefore, elements with zero phase shifts are obtained for θ = ψ. Thus, the spatial subgroup is made up of the elements

(ψ, γ) ∈ SO(2) × DN and so the spatial subgroup is isomorphic to Σ.

For the one-dimensional trivial and alternating representations of DN , we have

Ei = U ⊗ V ' C. In the case of the trivial representation of DN , this means only the SO(2) action on C is present and a unique branch of rotating waves bifurcates. If N is odd and V is the alternating representation with κ acting nontrivially, then SO(2) × Z2(κ) acts on Ei and

(π, κ) is the only element ﬁxing Ei. This leads to a unique branch of bifurcating periodic solutions with isotropy subgroup

1 SO^(2)×˙ Z2(κ, π) := {(θ, κ, −θ + π) | θ ∈ SO(2), (κ, π) ∈ DN × S } 22 ¯ which contains the spatial subgroup Z2 = h(π, κ)i ⊂ SO(2) × DN . In the case N even, two other situations arise from γ acting by −1: the subgroups SO(2)×˙ Z2(γ) and

SO(2)×˙ D2(γ, κ) acting on Ei. The analysis is similar to the previous case and the isotropy subgroup of the branches of bifurcating solutions are

1 SO^(2)×˙ Z2(γ, π) := {(θ, γ, −θ + π) | θ ∈ SO(2), (γ, π) ∈ DN × S }

with spatial subgroup Z2 = h(π, γ)i, and

1 SO^(2)×˙ D2((γ, π), (κ, π)) := {(θ, σ, −θ+π) | θ ∈ SO(2), (σ, π) ∈ DN ×S with σ ∈ {γ, κ}}

with spatial subgroup D2 = h(π, γ), (π, κ)i. Proposition 3.4. Suppose that the linearization of (3.11) at the origin has a pair of purely

imaginary eigenvalues crossing the imaginary axis with nonzero speed at ±i and Ei = U ⊗ V

where V is a one-dimensional absolutely irreducible representation of DN . Then, generically, there exists a unique branch of periodic solutions with period near 2π bifurcating from the origin and the symmetry group of the periodic solution with symmetry group (H, K) where H = SO^(2)×˙ K and K is found in Table 3.1. Therefore, the periodic solution is a rotating wave with spatial symmetry group K.

V K Trivial 1 κ acts nontrivially Z2(π, κ) γ acts nontrivially D2(κ, (π, γ)) κ and γ act nontrivially D2((π, κ), (π, γ)) Table 3.1. One-dimensional irreducible representations and spatial subgroups of the bifurcating periodic solutions.

The waveform for the rotating wave X(t) with symmetry SO^(2) has no restriction.

The waveforms for the rotating waves with K = Z2(π, κ) is

X(t) = (i1(t), i2(t), . . . , i(N+1)/2(t), i(N+1)/2(t + π), . . . , i2(t + π)) (3.13)

where i1(t) oscillates at twice the frequency, forced by the κ symmetry. For the case

K = D2(κ, (π, γ)) the rotating wave X(t) = (i1(t), . . . , iN (t)) is such that ij+1(t) = ij(t + π)

for all j = 2,...,N taken mod N. For K = D2((π, κ), (π, γ)), X(t) is as in the previous case, but with oscillations at twice the frequency. 23

For the two-dimensional representations of DN , following [10] we describe the ˆ C-axial isotropy subgroups. There are three: a rotating wave B3, and two discrete standing waves Bˆ1 and Bˆ2 whose expression depends on the parity of N. We deﬁne k 2πkm 1 eN/d = 1, γ , − ∈ SO(2) × DN × S | 0 ≤ k ≤ N − 1 Z N

where again d = gcd(m, N). For the standard representation of DN , we have m = 1 which is the case done in [10]. Recall that the cases m > 1 correspond to the other nonisomorphic two

dimensional irreducible representations of DN . Table 3.2 shows the subgroups as subsets of 1 SO(2) × DN × S .

N odd N even (with 2m|N) ˆ N/(2m) B1 Z2(1, κ, 0) Z2(1, κ, 0) ⊕ Z2(1, γ , π) ˆ N/(2m) B2 Z2(1, κ, π) Z2(1, κγ, 0) ⊕ Z2(1, γ , π) ˆ B3 Zed Zed

Table 3.2. Isotropy subgroups of periodic solutions bifurcating from DN Hopf bifurcation at a Γ-simple subspace consisting of the sum of isomorphic two-dimensional absolutely irreducible representations of DN .

ˆ We automatically have that SO^(2)×˙ Bj are C-axial isotropy subgroups and those are the only C-axial isotropy subgroups as we now show. We use Proposition 6.5 from [10] in our 1 −1 context and let P ⊂ SO(2) × DN × S be a C-axial isotropy subgroup, Q = Θ (P ) and ˆ 1 ˆ 1 ˆ B = Q ∩ (DN × S ), A = Q ∩ (SO(2) × S ). The result states that if dimC FixU (A) = 1, then A,ˆ Bˆ are C-axial and P = Aˆ×˙ Bˆ. Because U' C, then Aˆ must either ﬁx C or only {0}. ˆ ˆ ˆ But FixU (A) ⊗C FixV (B) ⊃ FixU⊗CV (Q) 6= {0} which means dimC FixU (A) = 1. Automatically all the C-axial subgroups are the ones already described as the ×˙ products above. The results are summarized here. Proposition 3.5. Suppose that the linearization of (3.11) at the origin has a pair of purely imaginary eigenvalues crossing the imaginary axis with nonzero speed at ±i. Then, generically, three branches of rotating waves with period near 2π bifurcate from the origin ˆ with isotropy subgroup (H,K) = (SO^(2)×˙ Bi, Ki) for i = 1, 2, 3 where Ki is given in Table 3.3 where m 6= N. ˆ In particular, note that SO^(2)×˙ B3 = Σm from Proposition 3.1 and for N odd, ˆ SO^(2)×˙ B2 = SO^(2)×˙ Z2(κ, π) obtained in Proposition 3.4. Therefore, the waveforms are given by (3.12) and (3.13), respectively. 24

C-axial subgroup Ki for N odd Ki for N even (with 2m|N) ˆ N/(2m) SO^(2)×˙ B1 Z2(κ) D2(κ, (π, γ )) ˆ N/(2m) SO^(2)×˙ B2 Z2(π, κ) D2(κγ, (π, γ )) ˆ ¯ ¯ SO^(2)×˙ B3 Zd(−2πm/N, γ) Zd(−2πm/N, γ) Table 3.3. Spatial subgroups of the rotating waves bifurcating simultaneously from the symmetric Hopf bifurcation point.

ˆ For N odd, the waveform of rotating waves with isotropy subgroups SO^(2)×˙ B1 is

X(t) = (i1(t), i2(t), . . . , i(N+1)/2(t), i(N+1)/2(t), . . . , i2(t)).

If N is even, the waveform depends on whether N ≡ 0 mod 4 or N ≡ 2 mod 4 and on the value of m. First, the components of X(t) are partitioned in γN/2m orbits

n o ik(t), i N (t), . . . , i N (t) (3.14) k+( 2m ) k+(2m−1)( 2m )

with successive elements of the orbit being π out of phase. Then, the κ symmetry either leaves the orbit invariant, or merges two orbits. Instead of working out completely the combinatorics of the phase shifts on these N/2m orbits, we look at an example of the D2(κ, (π, γ )) case to illustrate the various waveforms N/2m possible. The case D2(κγ, (π, γ )) can be done using a similar analysis as below. Example 3.4. Let N = 16, therefore we consider the cases m = 1, 2, 4 with spatial subgroups 8 4 2 D2(κ, (π, γ )), D2(κ, (π, γ )) and D2(κ, (π, γ )) respectively. We perform explicit computations using (3.14) in the case m = 1. The γ8 orbits contain two elements

{i1+k(t), i9+k(t)} = {i1+k(t), i1+k(t + π)}

with k = 0,..., 7. The κ symmetry is the reﬂection across the axis going through the vertices 1 and 9, therefore, perpendicular to the axis going through the vertices 5 and 13. The κ

symmetry leaves invariant the orbits i1+k for k = 0 and k = 4. The other orbits are paired

using the κ symmetry. For k = 1, 2, 3, the orbit of i1+k is interchanged with the orbit of i9−k:

κ.i1+k(t) = i17−k(t) and κ.i9+k(t) = i9−k(t). Thus obtaining orbits of four elements

{i1+k(t), i9−k(t), i9+k(t), i17−k(t)} = {i1+k(t), i1+k(t + π), i1+k(t + π), i1+k(t)}. 25

For m = 2, the γ4 orbits have four elements. Two of them are

{i1(t), i5(t), i9(t), i13(t)} = {i1(t), i1(t + π), i1(t), i1(t + π)}

{i3(t), i7(t), i11(t), i15(t)} = {i3(t), i3(t + π), i3(t), i3(t + π)}

and both of them are κ invariant. However, for the second one, this means i3(t) oscillates at

twice the frequency of the other oscillators because κ.i3(t) = i15(t) and κ.i11(t) = i7(t) leads

to i3(t) = i3(t + π). The two other orbits of four elements are merged using the κ symmetry last and we have

7 {i2+2j(t)}j=0 = {i2(t), i2(t + π), i2(t + π), i2(t), i2(t), i2(t + π), i2(t + π), i2(t)}.

7 Finally, for m = 4, we have two orbits {ik+2j(t)}j=0 for k = 1, 2. In each case,

ik+2j(t) = ik+2(j−1)(t + π).

However, for k = 2, the κ symmetry forces the oscillators to have twice the frequency of the oscillators in the other orbit. Figure 3.2 shows schematic diagrams of the phase shifts in 2π-periodic solutions with 8 N = 16 oscillator units. The spatial subgroups are D2(κ, (π, γ )) on the left and 4 D2(κ, (π, γ )) on the right. The colors represent various waveforms and the numeric labels the relative phase shifts between the cells. Note that the phase shifts between different colors

are uncorrelated; e.g. on the left, the 0 at units 1 and 2 correspond to i1(t) and i2(t), but

i9(t) = i1(t + π).

Bifurcation from rotating waves As in the previous case, we use the theory from [21] to

obtain bifurcations from the rotating waves obtained above. Let Xλ(t) be a one-parameter family of rotating waves with spatial symmetry group K obtained from Proposition 3.4 or 2N−1 2N−1 Proposition 3.5. The mapping g : R × R → R is therefore K-equivariant. Note that ¯ the rotating wave case with isotropy subgroup Zd is identical to the one studied in the ZN unidirectional setting and bifurcations are given by Proposition 3.2. We classify the possible symmetry-breaking bifurcations beginning with the cases in 2N−1 Proposition 3.4 where K is either isomorphic to Z2 or D2. Hence, the action of K on R yields only trivial and alternating one-dimensional (absolutely) irreducible representations. We obtain the following result.

Proposition 3.6. Suppose that Xλ(t) has isotropy subgroup SO^(2)×˙ K where K = 1,

K = Z2(π, κ), K = D2(κ, (π, γ)) or K = D2((π, κ), (π, γ)). Then, generically, either Xλ(t) 26

5 4 3 5 4 3 0 π 6 0 0 6 π 02 π 0 2 π 0 2 7 π 0 1 7 02 0 1

8 π 0 16 8 0 0 16

9 π 0 15 9 0 02 15 10 π 0 10 0 π 2 π π π 14 0 π π 14 11 12 13 11 12 13

Figure 3.2. Rings of N = 16 oscillator units representing a 2π periodic solution with 8 4 spatial subgroups K = D2(κ, (π, γ )) (left) and K = D2(κ, (π, γ )) (right) corresponding respectively to the cases m = 1 and m = 2. The various colors represent different waveforms and the numeric labels the relative phases between cells of same color, with 02 meaning oscillation at twice the frequency of the other units. has a symmetry-preserving saddle-node bifurcation or a pitchfork bifurcation to a branch of 0 0 rotating waves Yλ(t) with isotropy group SO^(2) × K where K = 1 if K = Z2(π, κ) or 0 K ' Z2 if K = D2(κ, (π, γ)) or K = D2((π, κ), (π, γ)).

Proof. The symmetry-preserving bifurcation occurs if (dg)(0,λ) corresponds to a trivial representation of K. If K = Z2(π, κ) with (dg)(0,λ) corresponding to an alternating representation, then the center manifold is ﬁxed only by 1 and Yλ(t) is a rotating wave with 0 K = 1. If K = D2(κ, (π, γ)) or K = D2((π, κ), (π, γ)) and (dg)(0,λ) corresponds to an alternating representation, then one of the following isotropy subgroups acts trivially on the center manifold: Z2(κ), Z2(π, κ), Z2(π, γ) or Z2(κγ). Therefore, Yλ(t) has isotropy subgroup 0 0 SO^(2) × K where K is one of the Z2 subgroups just described. 0 The waveform of the symmetry-breaking bifurcating solution with K = Z2(π, κ) is 0 given by (3.13). If K = Z2(κ), then

X(t) = (i1(t), i2(t), . . . , i(N/2)(t), i(N/2)+1(t), i(N/2)(t), . . . , i1(t)) (3.15)

with i1(t) and i(N/2)+1(t) oscillating at twice the frequency of the other oscillators. If 0 K = Z2((π, γ)), then

X(t) = (i1(t), i2(t), . . . , iN/2(t), i1(t + π), i2(t + π), . . . , iN/2(t + π)). 27

0 If K = Z2(κγ) then the waveform X(t) can be obtained from (3.15) by cyclically permuting with j 7→ (j + N/4) mod N for j = 1,...,N. We now consider symmetry-breaking bifurcations from the rotating waves SO^(2)×˙ Bj.

If N is odd, the cases for B1 and B2 are relatively straightforward. In this case,

(dg)(0,λ) is Z2 symmetric and therefore bifurcations occur from the trivial representation or from the alternating representation; that is, a pitchfork bifurcation. This is summarized in the next result.

Proposition 3.7. Suppose N is odd and that Xλ(t) has isotropy subgroup SO^(2)×˙ Bj for j = 1 or j = 2. Then, generically, either Xλ(t) has a symmetry-preserving saddle-node bifurcation or has a pitchfork bifurcation to a branch of rotating waves Yλ(t) with isotropy subgroup SO^(2) ˆ The remaining cases are for N even. For j = 1, 2, K = Bj are Abelian groups and the kernel (dg)(0,λ) is generically a one-dimensional absolutely irreducible representation: the N/2m trivial or an alternating representation. If Xλ(t) has spatial subgroup D2(κ, (π, γ )), the alternating irreducible representations correspond to only one group element ﬁxing (dg)(0,λ), N/2m N/2m either κ , (π, γ ) or (π, κγ ). Similarly, if Xλ(t) has spatial subgroup N/2m D2(κγ, (π, γ )), then the alternating representation is ﬁxed by only one group element: (π, κ), (π, γN/2m) or κγN/2m. This leads to the following result. ˆ Proposition 3.8. Suppose that Xλ(t) has isotropy subgroup SO^(2)×˙ Bj where j = 1 or j = 2. Then, generically, either Xλ(t) has a symmetry-preserving saddle-node bifurcation or 0 a pitchfork bifurcation to a branch of rotating waves Yλ(t) with isotropy group SO^(2)×˙ K 0 where K is given in Table 3.4.

0 0 0 K K K K

N/2m N/2m N/2m D2(κ, (π, γ )) Z2(κ) Z2(π, γ ) Z2(π, κγ ) N/2m N/2m N/2m D2(κγ, (π, γ )) Z2(κγ ) Z2(π, γ ) Z2(π, κ)

0 Table 3.4. Possible spatial subgroups K depending on which element of K ﬁxes the kernel (dg)(0,λ).

0 The waveform of rotating waves with Z2(π, κ) and K = Z2(κ) have been found N/2m above, see (3.13) and (3.15). We illustrate the remaining cases of K = D2(κ, (π, γ )) using Example 3.4. 8 Example 3.5. For m = 1, breaking to Z2(π, γ ) leads to eight orbits {ik(t), ik+8(t)} for 8 k = 1,..., 8 where ik+8(t) = ik(t + π). If instead the spatial subgroup is Z2(π, κγ ), this

leads to the following nine orbits: i5(t) and i13(t) are alone in their orbit and oscillate at twice 28 the frequency as the other oscillators, while the seven remaining orbits are {i5−k(t), i5+k(t)} for k = 1, 2, 3, 4 and {i13−k(t), i13+k(t)}, for k = 1, 2, 3 where i5+k(t) = i5−k(t + π) for

k = 1, 2, 3, 4 and i13+k(t) = i13−k(t + π) for k = 1, 2, 3 (we can also write {i5−k(t), i5+k(t)} for k = 1,..., 7 taking indices mod 16). 4 For m = 2, the spatial isotropy subgroup Z2(π, γ ) leads to a wave form with 3 3 {i1+4j(t)}j=0 unchanged and {i3+4j(t)}j=0 = {i3(t), i3(t + π), i3(t), i3(t + π)} while the last orbit breaks into the following two orbits:

{i2(t), i6(t), i10(t), i14(t)} = {i2(t), i2(t + π), i2(t), i2(t + π)}

and

{i4(t), i8(t), i12(t), i16(t)} = {i4(t), i4(t + π), i4(t), i4(t + π)}.

4 Now, for the spatial isotropy subgroup Z2(π, κγ ) we have nine orbits as above with the orbits

i7(t) and i15(t) ﬁxed and oscillating at twice the frequency, with remaining orbits made up of

pairs {i7−k(t), i7+k(t)} for k = 1,..., 7 (with indices taken mod 16) and π phase shifts. 2 In the case m = 4, for spatial subgroup Z2(π, γ ) the two orbits are preserved, but the k = 2 orbit does not oscillate at twice the frequency anymore. Finally, with spatial subgroup 2 Z2(π, κγ ) we have again a family of nine orbits with π phase between each oscillator. This

time, centered around the orbits i8(t) and i16(t). Those are {i8−k(t), i8+k(t)} with k = 1,..., 7 and indices taken mod 16. Figure 3.3 shows a schematic diagram of phase shifts of 2π-periodic solutions with 8 4 N = 16 oscillator units. The spatial subgroups are Z2(π, γ ) on the left and Z2(π, κγ ) on the right. The colors represent various waveforms and the numeric labels the relative phase shifts between the cells. Note that the phase shifts between different colors are uncorrelated.

3.3 ISOTYPIC DECOMPOSITION In this section, we compute the linearization of the CCOST networks (3.4) and (3.8) at the equilibrium solution located at the origin; thus its isotropy subgroup is either

SO(2) × SO(2) × ZN or SO(2) × SO(2) × DN . We denote by J the linearization of the CCOST network near the origin. We consider both cases, unidirectional coupling via 29

5 4 3 5 4 3 6 0 0 0 6 0 0 0 0 0 2 0 0 2 7 0 0 1 7 02 0 1

8 0 π 16 8 π 0 16

9 π π 15 9 π 02 15 10 π π 10 π π π π π 14 π π π 14 11 12 13 11 12 13

Figure 3.3. Rings of N = 16 units representing a 2π-periodic solution with spatial sub- 0 8 0 4 groups K = Z2(π, γ ) (left) and K = Z2(π, κγ ) (right) corresponding to the cases m = 1 and m = 2 respectively. The various colors represent different waveforms and the numeric labels the relative phases between cells of same color. Phase shifts between different colors are uncorrelated and 02 means oscillation at twice the frequency of the other units.

Eq. (3.3) and bidirectional coupling through Eq. (3.7). The linearization yields   AB 04 ······ 04 B       AB ······ 04  BAB 04 ······ 04         ......   04 AB ··· 04   04 . . . .       ......   ......  Junidir =  . . . . .  ,Jbidir =  . . . . .  ,        ......   04 04 04 AB   . . . . 04        B 04 ··· 04 A  04 04 ··· 04 BAB    B 04 04 ··· 04 BA (3.16) 4×4 ¯ ¯ where 04 ∈ R denotes the zero matrix, A = εA and B = εB. We exploit the ZN and DN subgroups of Γunidir and Γbidir to decompose Junidir and Jbidir, respectively, in block diagonal 4N form via the isotopic decomposition of the phase space R . In this new basis, the eigenvalues can be obtained explicitly and their distribution can also be studied for all N ∈ N. 4N The representation of Γunidir and Γbidir on R breaks up into real irreducible representations as shown in [14]. Let ζ = exp (2πi/N) and j 2j (N−1)j T vj = (v, ζ v, ζ v, . . . , ζ v) , j = 0,...,N − 1, be N vectors in C, for some v ∈ R. The 30

N vectors vj form a basis for C , that is

N C = V0 ⊕ V1 ... ⊕ VN−1.

Since each crystal oscillator has an internal phase space of dimension four, consider

the standard basis vectors {e1, e2, e3, e4} and deﬁne

j 2j (N−1)j T vjk = (ek, ζ ek, ζ ek, . . . , ζ ek) ,

for k = 1, 2, 3, 4. This set of 4N vectors form a basis for the complexiﬁed phase space

4N 4 4 4 C = V0 ⊕ V1 ... ⊕ VN−1.

Let =jk and

Furthermore, let ˜· denote a normalized vector. For N odd, let v0 = (v01, v02, v03, v04) and deﬁne a transformation matrix P =

h i v0, =e11,..., =e14,

Similarly, when N is even, let vN/2 = (v(N/2)1, v(N/2)2, v(N/2)3, v(N/2)4) and P =

h i v0, =e11,..., =e14,

M(θ) = I2 ⊗ A + R(θ) ⊗ B,

where ⊗ denotes the matrix tensor product, In is the n × n identity matrix and R(θ) is the rotation matrix   cos θ − sin θ R(θ) =   . sin θ cos θ

Assume N is odd. Then under the transformation V = UP , where P is given by −1 Eq. (3.17), the linearization (3.16) can be written as Junidir = P JunidirP or explicitly as

Junidir = diag (A + B,M (1 (2π/N)) ,M (2 (2π/N)) ,...,M (k (2π/N))) , (3.19) 31 where k = bN/2c. Similarly, when N is even, the transformation matrix (3.18), yields the diagonalization

Junidir = diag (A + B,M (1 (2π/N)) ,M (2 (2π/N)) ,...,M (k (2π/N)) ,A − B) , (3.20) where k = N/2 − 1. For the bidirectional case, let k ∈ {0, 1,..., bN/2c}, θ = (2πk) /N and −1 Mk = A + 2 cos θB. The linearized system (3.16) can then be written as Jbidir = P JbidirP , where for N odd it becomes

Jbidir = diag M0, M1, M1,..., MbN/2c, MbN/2c , (3.21) and when N is even it is

Jbidir = diag M0, M1, M1,..., MN/2−1, MN/2−1, MN/2 . (3.22)

3.4 SPECTRUMOF EIGENVALUES AND LINEAR STABILITY We now calculate the eigenvalues of the linearized matrix J . From Eq. (3.19) - Eq. (3.22), we see immediately that the eigenvalues of the linear system consist of those from the blocks A + B, M(θ), A − B and each Mj block. Direct calculations produce the eigenvalues of these blocks as follows. For the unidirectional case they are:

` 1 ε (aλω` − aω` + R`ω` ± L`Ω`i) ρA+B = − , 2 L`ω` 1 ε (aλω + aω − R ω ± L Ω i) ρ` = ` ` ` ` ` ` , A−B 2 L ω ` ` (3.23) ` ε ε ρM(θ) = − (aλω` cos θ − aω` + R`ω`) ∓ (aλω` sin θ − L`Ω`) i, 2L`ω` 2L`ω`

` ε ε ηM(θ) = − (aλω` cos θ − aω` + R`ω`) ∓ (aλω` sin θ + L`Ω`) i 2L`ω` 2L`ω`

where ` ∈ {1, 2}. For the bidirectional case, the eigenvalues of the block diagonal matrices

Mj are j ε (−2aλ cos θ + a − Rj ± LjΩji) ρ± = , (3.24) 2ωjLj where j = 1, 2 and each eigenvalue has multiplicity two. We now seek a combination of parameter values for the linear system to be stable. For the unidirectional case, this can be done by examining the real components of the eigenvalues of the blocks A + B, M(θ) and A − B. As we can see from the expressions in (3.23), the 32 critical points for the real-parts of the eigenvalues are determined by the parameters a, λ, R` and θ. Because a and R` are physical constants, they are assumed to be positive. Moreover, the physical system requires that R1 6= R2, so without loss of generality, we assume R2 > R1. As for θ, this parameter ranges from 2π/N to kπ/N. But since kπ/N → π as N → ∞, we are only concerned with θ in (0, π). To simplify the notation in the following analysis, we denote θ1 ∈ (0, π/2) and θ2 ∈ (π/2, π). Because of the range of parameter values, we may divide the analysis into three different cases.

Unidirectional Case 1: a < R1 < R2 The critical values of λ for which the real part of the eigenvalues for the blocks A + B, M(θ) and A − B become zero are given by

R a − R R λA+B = 1 − ` , λM(θ) = ` , λA−B = ` − 1. (3.25) c a c a cos θ c a

From |cos θ| < 1, we see that

a − R` a − R` < . (3.26) a a cos θ

Because cos θ1 > 0 and cos θ2 < 0, we may rewrite the inequality in (3.26) more precisely as

a − R a − R R − a a − R ` < ` < 0 < ` < ` . (3.27) a cos θ1 a a a cos θ2

By assumption R2 > R1, the critical values of λ from the A + B block satisfy

a − R a − R 2 < 1 . a a

We observe that for λ > (a − R1)/a, the real-parts of the eigenvalues from the A + B block

are negative and for λ < (a − R2)/a, the real-parts of the eigenvalues from the A + B block

are positive. Similarly, the critical values of λ from the M(θ1) blocks satisfy

a − R a − R 2 < 1 . a cos θ1 a cos θ1

We observe that for λ > (a − R1)/(a cos θ1), the real-parts of the eigenvalues from the

M(θ1) block are negative and for λ < (a − R2)/a, the real-parts of the eigenvalues from these blocks are positive. Based on the inequality in (3.27), we have established that for

λ > (a − R1)/a, the eigenvalues from A + B and the M(θ1) blocks are negative. Applying similar arguments as in the A + B block, we can show that the real parts of

the eigenvalues from the A − B block are negative if λ < (R1 − a)/a. Moreover, we can 33 repeat the arguments used for the M(θ1) blocks to the M(θ2) block and show that the real parts of the eigenvalues are negative if λ < (a − R1)/(a cos θ2). Based on the inequality

in (3.27), we have established that for λ < (R1 − a)/a, the real parts of the eigenvalues from

A − B and the M(θ2) blocks are negative. By direct substitution, we see that if θ = π/2, the real parts of the eigenvalues of the M(π/2) block are negative.

Unidirectional Case 2: a > R2 > R1 Since R2 > R1, we see that

a − R a − R a − R a − R 1 < 2 < 0 < 2 < 1 . a cos θ2 a cos θ2 a a

Direct calculations show that for λ < (a − R1)/(a cos θ2), the eigenvalues from the

M(θ2) blocks have negative real parts and for λ > (a − R2)/(a cos θ2), the eigenvalues have positve real parts. For the A + B block, the real parts of the eigenvalues are negative if

λ > (a − R1)/a and positive if λ < (a − R2)/a. Because of these results, there is no value of

λ for which all real parts of the eigenvalues from the A + B and M(θ2) blocks are

simultaneously negative. Therefore, the origin is always linearly unstable if a > R1 and

a > R2.

Unidirectional Case 3: R1 < a < R2 Because R1 < a, the arguments from the previous case

apply and the origin is also always linearly unstable if R1 < a < R2. We have summarized the linear stability of the system into the following theorem:

Theorem 3.6. We assume a < R1 < R2. Suppose N to be odd, then the origin is linearly stable if a − R a − R 1 < λ < 1 . a 2π N a cos N b 2 c Suppose N is even, the origin is linearly stable if

a − R R − a 1 < λ < 1 . a a

In all other cases, the system is linearly unstable.

Proof. Based on the arguments made above, we see that (a − R1)/a is the lower limit for λ such that all the real parts of the eigenvalues are negative. Suppose N is even, (3.27) shows

that (R1 − a)/a must be the upper limit such that all the eigenvalues have negative real parts. If N is odd, the A − B block is not part of the linear system and the lower limit of linear

stability must be of the form (a − R1)/(a cos θ2) because R1 < R2. Also,

(a − R1)/(a cos(k(2π)/N)) has the smallest magnitude in terms of all the critical values of λ

for the θ2 blocks, therefore it must be upper limit for linear stability. 34 Based on the eigenvalues of the system, we have summarized the possible bifurcations in the following theorem.

Theorem 3.7. We assume R2 > R1 > a. The unidirectionally coupled crystal oscillator system undergoes a symmetry-preserving Hopf bifurcation at

a − R λ∗ = 1 . a

If N is even, the system undergoes a symmetry-breaking Hopf bifurcation at

R − a λ∗ = 1 , a

If N is odd, the system undergoes a symmetry-breaking Hopf bifurcation at

∗ a − R1 λ = 2π N . a cos N b 2 c

Proof. In the symmetry-preserving case, we see from Theorem 3.6 that the system is linearly 2π N stable if (a − R1)/a < λ < (a − R1)/(a cos N b 2 c ). A pair of complex conjugate eigenvalues crosses the imaginary axis with non-zero speed as λ decreases through

(a − R1)/a. Since the crossing eigenvalues corresponds to the A + B block and this block is associated with the symmetry-preserving subspace, the system undergoes a symmetry-preserving Hopf bifurcation. Similarly, in the symmetry-breaking case, the system is linearly stable if 2π N (a − R1)/a < λ < (a − R1)/(a cos N b 2 c ). If N is even, a pair of complex conjugate eigenvalues crosses the imaginary axis with non-zero speed as λ increases through

(R1 − a)/a. Because the crossing eigenvalue corresponds to the A − B block and this block is associated with the symmetry-breaking subspace, the system undergoes a symmetry-breaking Hopf bifurcation. If N is odd, the A − B block is not part of the linear system and as shown in Theorem 3.6, the lower limit for linear stability is 2π N (a − R1)/(a cos( N b 2 c )). Since a pair of complex conjugate eigenvalues corresponding to N the M(b 2 c(2π)/N) block crosses the imaginary axis with non-zero speed at 2π N λ = (a − R1)/(a cos( N )b 2 c) and this block is associated with the symmetry-breaking subspace, therefore a symmetry-breaking Hopf bifurcation occurs.

For the bidirectional case, we assume again that R2 > R1. Because of the range of j,

we see that θ ∈ [0, π]. To simplify the notation, we again let θ1 ∈ [0, π/2) and θ2 ∈ (π/2, π]. Because our interest in understanding the effects of the coupling parameter λ, we may divide the possible combinations of the parameters into three different cases. 35

Bidirectional Case 1: a < R1 < R2 Based on the expression in (3.24), the critical points for linear stability, in terms of λ, are a − R λ = i 2a cos θ for i = 1, 2. Since |cos θ| ≤ 1, we see that

a − R a − R i < 0 < i . (3.28) 2a cos θ1 2a cos θ2

Based on the value of j, the value of θ corresponding to each Mj block must either be in

[0, π/2) or (π/2, π]. To begin, we analyze the Mj blocks with θ ∈ [0, π/2). By assumption

R2 > R1, the critical values of λ from the Mj block satisfy

a − R a − R 2 < 1 < 0. 2a cos θ 2a cos θ

We observe that for λ > (a − R1)/(2a cos θ), the real-parts of all the eigenvalues are

necessarily negative while for λ < (a − R2)/2(a cos θ), they are all positive. Since the order of the critical points is based on the values of j, the greatest critical point must occur for

j = 0. Based on the ordering of the critical values, we see that for λ > (a − R1)/(2a), all

eigenvalues of Mj are linearly stable.

For θ ∈ (π/2, π], the critical values of λ of the Mj blocks satisfy

a − R a − R 0 < 1 < 2 . 2a cos θ 2a cos θ

We observe that for λ > (a − R2)/(2a cos θ), the real-parts of the eigenvalues of the

Mj blocks are positive and for λ < (a − R1)/(2a cos θ), the real-parts of the eigenvalues from these blocks are negative. The ordering of the critical values of λ is based on the value of j, so the smallest critical value occurs if j = 2π/NbN/2c. Based on the ordering of all the possible critical values, for the blocks with j values such that θ ∈ (π/2, π], the eigenvalues have negative real parts if a − R1 λ < 2π N . 2a cos N 2 Combining the results for θ ∈ [0, π/2) and θ ∈ (π/2, π], we see that the system is linearly stable if a − R a − R 1 < λ < 1 . 2a 2π N 2a cos N 2 36

Bidirectional Case 2: a > R2 > R1 This implies that

a − R a − R i < 0 < i . 2a cos θ2 2a cos θ1

By direct calculations, we can show that for λ < (a − R1)/(a cos θ1), the real parts of

the eigenvalues from the Mj blocks with θ ∈ [0, π/2) are positive. Similarly, for

θ ∈ (π/2, π], the real parts of the eigenvalues of the corresponding Mj blocks are also positive. Based on these inequalities, there does not exist a value of λ such that real parts of

the eigenvalues of Mj with θ ∈ [0, π/2) and real parts of eigenvalues of Mj with θ ∈ (π/2, π] are negative simultaneously.

Bidirectional Case 3: R1 < a < R2 Since R1 < a, by the arguments from the previous case, the system is also always linear unstable. We have summarized the linear stability of the system into the following theorems.

Theorem 3.8. Suppose a < R1,R2. The system is linearly stable if

a − R a − R 1 < λ < 1 . 2a 2π N 2a cos N 2

In all other cases, the system is linearly unstable.

Theorem 3.9. Assume R2 > R1 > a. For the bidirectionally coupled crystal oscillator system, the system undergoes a symmetry-preserving Hopf bifurcation at

a − R λ∗ = 1 . 2a

Furthermore, the system undergoes a symmetry-breaking Hopf bifurcation at

∗ a − R1 λ = 2π N . 2a cos N 2

Proof. In the symmetry-preserving case, we see from Theorem 3.6 that the system is linearly 2π N stable when (a − R1)/2a < λ < (a − R1)/ 2a cos N 2 . A pair of complex conjugate eigenvalues crosses the imaginary axis with non-zero speed as λ decreases and reaches

(a − R1)/2a. Since the crossing eigenvalues corresponds to the M0 block and this block is associated with the symmetry-preserving subspace, the system undergoes a symmetry-preserving Hopf bifurcation. Similarly, in the symmetry-breaking case, the system is linearly stable when

(a − R1)/a < λ < (a − R1)/(a cos θ2). A pair of complex conjugate eigenvalues crosses the 2π N imaginary axis with non-zero speed as λ increases and reaches (R1 − a)/(2a cos N 2 ). 37

Since the crossing eigenvalues corresponds to the MbN/2c block and this block is associated with the symmetry-breaking subspace, the system undergoes a symmetry-breaking Hopf bifurcation. 38

CHAPTER 4

NUMERICAL CONTINUATION

In this chapter, a computational bifurcation analysis of the collective response of a CCOST network is conducted, including both unidirectional and bidirectional coupling topologies. All bifurcation diagrams are generated with the aid of the continuation software package AUTO [11].

4.1 UNIDIRECTIONALLY COUPLED RING A bifurcation diagram of the emergent behavior in Eq. (3.1) for an array of N = 3 crystal oscillators, as a function of the coupling strength λ, is shown in Fig. 4.1(top). All other parameters are held ﬁxed. As a convention, solid/dashed lines and ﬁlled-in/empty circles correspond to stable/unstable equilibrium points and stable/unstable periodic solutions, respectively in the bifurcations diagrams. Observe the presence of two pairs of back-to-back Hopf bifurcation points along the zero equilibrium point (0, 0, 0, 0, 0, 0), one pair for λ > 0 and one pair for λ < 0. Each bifurcation point leads to a collective pattern of oscillation by the network system. Starting from left-to-right, the two Hopf points, HB1 and HB2, correspond to symmetry-breaking bifurcations, which are associated with the M(θ) block in equations (3.19) and (3.20), and 1 2 they give rise to rotating wave patterns along the branches labeled RW1 and RW1 , M(θ) respectively. The exact location of the Hopf points is obtained from λc in (3.25), which yields λHB1 = −1.934 and λHB2 = −1.614. These patterns are predicted by Proposition 3.1 with N = 3 and m = 1, so that d = gcd(3, 1) = 1. Thus, the emerging waves are discrete ¯ rotating waves with isotropy subgroup (H,K) = (Σ1, Z3), where k ¯ k Σ1 = {(θ, γ , −θ − kπ/3) |k = 0, 1, 2} and so Z3 = {(−kπ/3, γ , 0)|k = 0, 1, 2}. The collective patterns are of the form:

1 RW1 : X(t) = (i1(t), 0, i1(t + T/3), 0, i1(t + 2T/3), 0),

2 RW1 : X(t) = (0, i2(t), 0, i2(t + T/3), 0, i2(t + 2T/3)),

where the subscript corresponds to the value of m and the superscript indicates which one of 1,2 the two modes is active. Observe that both rotating waves RW1 are locally stable for small positive values of coupling strength, around 0 < λ < 1. 39 Z 3

stable 400 unstable

300 IP1 2 200 IP2 ||x|| RW2 100 1 RW1 1

0 HB HB HB HB 1 2 3 4

−3 −2 −1 0 1 2 3 λ Z 3 1000 HB HB HB HB 1 2 3 4

800 1,2 IP1,2 IP1,2 RW1,2 IP 1

600 RW1,2 RW1,2 1,2 1 1 IP a RW1,2 400 1 IP1 200 IP1 1 1 1 IP RW IP RW1 1 1 0 −3 −2 −1 0 1 2 3 λ

Figure 4.1. Bifurcation diagrams for a ring of N = 3 crystal oscillators coupled unidirectionally in a ring conﬁguration. The ring exhibits Z3-symmetry, i.e., cyclic permutations of three −4 crystal oscillators. Parameter values are: R1 = 30.9Ω, R2 = 181.1Ω, L1 = 5.2 × 10 H, −4 −13 −14 L2 = 2.6 × 10 H, C1 = 1.0 × 10 F , C2 = 2.5 × 10 F , a = 939, b = 3E08.

The next pair of Hopf points, HB3 and HB4, in particular, lead to synchronized oscillations along the branches IP 2 and IP 1, respectively. For instance, along the branch IP 1 all crystals oscillate with Mode 1 current in-phase while Mode 2 is quiescent:

1 IP : X(t) = (i1(t), 0, i1(t), 0, i1(t), 0).

Both branches, IP 1 and IP 2 emerge through symmetry-preserving bifurcations that are associated with the A + B block in equations (3.19) and (3.20). Thus the exact location of the Hopf bifurcation points can be obtained directly by substituting the parameter values for A+B R1, R2 and a into the equation for λc in (3.25), which yields λHB3 = 0.807 and

λHB3 = 0.967. Observe that both patterns of synchronization are locally stable for negative values of coupling strength, approximately −2 < λ < 0. But they are, however, unstable for positive values of λ. A critical observation is that the stability regions of the synchronized 1,2 1,2 solutions IP and rotating waves RW1 do not overlap. This observation is critical because 40 the design of a CCOST device will depend greatly on the selection mechanism for a preferred pattern that can minimize timing errors. Furthermore, in an actual device realization it is not possible to control the set of initial conditions even though it is a very simple task with computer simulations. Thus when only one pattern is stable then turning on the device over the interval −2 < λ < 0, for instance, will guarantee the selection of the IP 1,2 solutions. 1,2 Similarly, operating the device over the region 0 < λ < 1 will guarantee that the RW1 patterns are selected. Which pattern is actually selected will depend on the initial conditions and the size of the basins of attraction. This issue will be investigated in greater detail in future works. Figure 4.1(bottom) contains the two-parameter bifurcation diagram that tracks the boundary curves that deﬁne the regions of existence of the collective patterns, IP s and RW s, over the two parameter space (a, λ). Unstable solutions are indicated with a dashed box around the corresponding labels. Observe that the regions of existence of stable synchronized 1 2 solutions IP and stable rotating waves RW1 do not overlap. Figure 4.2 now shows the bifurcation scenario for a slightly larger ring array of N = 4 crystal oscillators coupled again unidirectionally. Qualitatively, the collective patterns of 1,2 oscillations that emerge are the same as those of the N = 3 case. Two rotating waves, RW2 , that emerge at the Hopf points HB1,2 and of the form

1 RW2 : X(t) = (i1(t), 0, i1(t + T/2), 0, i1(t), 0, i1(t + T/2)),

2 RW2 : X(t) = (0, i2(t), 0, i2(t + T/2), 0, i2(t), 0, i2(t + T/2)). 41

Z 4 500 stable unstable 400

300 1 2 RW 1 2 IP 200 RW2 IP2 ||x|| 2 100

0 HB HB HB HB 1 2 3 4 −1.5 −1 −0.5 0 0.5 1 1.5 λ Z 4 1000 HB1 HB2 HB3 HB4 1 1 1 1 RW IP IP RW 2 2

2 RW1,2 RW 1 2 2 1 IP 1,2 RW 500 IP1 IP 2 a

2 RW1 RW 2 2 1 IP2 IP

1 1 1 RW IP RW IP1 IP1 2 2 RW1 0 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 λ

Figure 4.2. Bifurcation diagrams for a ring of N = 4 crystal oscillators coupled unidirectionally in a ring conﬁguration. The ring exhibits Z4-symmetry, i.e., cyclic permutations of four crystal oscillators. Parameter values: same as in Fig. 4.1

Observe that now the subscript indicates m = 2, which implies d = gcd(4, 2) = 2, so ¯ N/d = 2 and the waves have Z2 spatial symmetry. It follows that the waves have isotropy ¯ k subgroup (H,K) = (Σ2, Z2), where Σ2 = {(θ, γ , −θ − kπ) |k = 0,..., 3} and ¯ k Z2 = {(−kπ, γ , 0)|k = 0,..., 3}. The exact location of the Hopf bifurcation points HB1,2 can be obtained again directly by substituting the parameter values for R1, R2 and a into the equations in (3.25). And two synchronized solutions, IP 1,2, that appear at the Hopf points 1 HB3,4. Now only IP is stable over the interval −1 < λ < 0. Perhaps the most visual difference between the two cases, N = 3 and N = 4, is the reﬂectional symmetry that appears in the one- and two-parameter bifurcation diagrams with N = 4. Observe also that there are 1 1 now two branches of solutions that whirl around from the branches IP and RW1 as they come down from the cusp point around ||x|| = 500. The two-parameter bifurcation diagram 42

1 1 of Fig. 4.2(bottom) shows the patterns IP and RW1 to be stable over a large region of parameter space (a, λ). In Fig. 4.3 we illustrate now the one-parameter bifurcations for a ring with N = 5 crystal oscillators.

Z 5

stable unstable 400 2 IP2 IP1 RW2 ||x|| 200 3

RW1 3 RW1 1

HBHB HBHB HB 0 1 2 3 4 5 -3 -2 -1 0 1 2 3 λ Z 5 1000 HB HB HB HB HB 1 2 3 4 5

RW1 RW2 3 3 IP1,2 RW1 1

a 500

1,2 RW2 IP 3 RW2 3 IP1 IP1 0 -3 -2 -1 0 1 2 3 λ Figure 4.3. Bifurcation diagrams for a ring of N = 5 crystal oscillators coupled unidirectionally in a ring conﬁguration. The ring exhibits Z5-symmetry, i.e., cyclic permutations of ﬁve crystal oscillators. Parameter values: same as in Fig. 4.1

1,2 This time there are two distinct rotating waves, RW3 , that appear from the Hopf bifurcation points HB1,2, respectively. These periodic solutions correspond to the case where ¯ m = 3 in Proposition 3.1, which leads to the isotropy subgroup (H,K) = (Σ3, Z5), where k ¯ k Σ3 = {(θ, γ , −θ − k6π/5) |k = 0,..., 4} and Z5 = {(−k6π/5, γ , 0)|k = 0,..., 4}. The collective pattern is:

1 RW3 : X(t) = (i1(t), i1(t + 3T/5), i1(t + T/5), i1(t + 4T/5), i1(t + 2T/5)) ,

2 RW3 : X(t) = (i2(t), i2(t + 3T/5), i2(t + T/5), i2(t + 4T/5), i2(t + 2T/5)) . 43

1 2 The wave RW3 is stable over the interval −0.7 < λ < 0 while RW3 is stable on the interval 0 < λ < 1. Again, there are also two synchronized collective modes of oscillations 2 1 IP and IP that emerge from symmetry-preserving Hopf bifurcation points HB3 and HB4, respectively, and both are stable over the interval −1 < λ < 0. At the last bifurcation point, 1 HB5, a rotating wave RW1 appears via Hopf symmetry-breaking bifurcation and it corresponds to the “standard” rotating wave with m = 1 and isotropy subgroup ¯ k (H,K) = (Σ1, Z5), where Σ1 = {(θ, γ , −θ − k2π/5) |k = 0,..., 4} and ¯ k Z5 = {(−k2π/5, γ , 0)|k = 0,..., 4}. That is:

1 RW1 : X(t) = (i1(t), i1(t + T/5), i1(t + 2T/5), i1(t + 3T/5), i1(t + 4T/5)) ,

where i2(t) = 0 has been omitted for simplicity. This rotating wave solution is stable over the interval −0.6 < λ < 0. The two-parameter region of multi-stability among various patterns of oscillations can be observed in Fig. 4.3. Observe that this region of multi-stability occurs only 3 when λ < 0 while for 0 < λ < 1 only the RW2 pattern is stable. The one-parameter bifurcations for a ring with N = 6 crystal oscillators coupled unidirectionally are shown in Fig. 4.4. Starting from left-to-right, the bifurcations points 1,2 HB1,2 lead to rotating waves RW2 , respectively. These are periodic solutions with isotropy ¯ k subgroup (H,K) = (Σ2, Z3), where Σ2 = {(θ, γ , −θ − k2π/3) |k = 0,..., 5} and ¯ k Z3 = {(−k2π/3, γ , 0)|k = 0,..., 5}. Thus the waveforms are

1 RW2 : X(t) = (i1(t), i1(t + T/3), i1(t + 2T/3), i1(t), i1(t + T/3), i1(t + 2T/3)) .

2 RW2 : X(t) = (i2(t), i2(t + T/3), i2(t + 2T/3), i2(t), i2(t + T/3), i2(t + 2T/3)) .

Then, at the bifurcation points HB3,4 two additional branches of rotating waves, 1,2 ¯ RW3 , appear. In this case, the waves have isotropy subgroup (H,K) = (Σ3, Z2), where k ¯ k Σ3 = {(θ, γ , −θ − kπ) |k = 0,..., 5} and Z2 = {(−kπ, γ , 0)|k = 0,..., 5}. Thus the waveforms are

1 RW3 : X(t) = (i1(t), i1(t + T/2), i1(t), i1(t + T/2), i1(t), i1(t + T/2)) .

2 RW3 : X(t) = (i2(t), i2(t + T/2), i2(t), i2(t + T/2), i2(t), i2(t + T/2)) .

For positive values of λ, we ﬁnd two more Hopf bifurcation points HB5,6 that are associated with the fully synchronized in-phase patterns IP 2 and IP 1, respectively. Finally, at 1,2 the points HB7,8 two branches of standard rotating waves, RW1 , respectively, appear. ¯ According to Proposition 3.1 these waves have isotropy subgroup (H,K) = (Σ1, Z6), where 44

Z 6 600 stable unstable

400 RW1 IP1

2 3 RW2 IP2 3 ||x|| 200 2 1 RW 2 1 RW 2 RW RW 2 1 1 0 HB HB HB HB HB HB HB HB 1 2 3 4 5 6 7 8 −3 −2 −1 0 1 2 3 λ Z 6 1000 HB HB HB HB HB HB HB HB 1 2 3 4 5 6 7 8 IP1 IP1 RW1 2 1,2 2 RW RW 1 1 1 RW 3

RW1 3 500 IP1 RW1,2 a RW1,2 2 1 1 IP 1 RW1 IP 2 RW1 2 1 RW1 IP 2 1 RW1 IP 2 0 −3 −2 −1 0 1 2 3 λ

Figure 4.4. Bifurcation diagrams for a ring of N = 6 crystal oscillators coupled unidirectionally in a ring conﬁguration. The ring exhibits Z6-symmetry, i.e., cyclic permutations of six crystals. Parameter values: same as in Fig. 4.1.

k ¯ k Σ1 = {(θ, γ , −θ − kπ/3) |k = 0,..., 5} and Z6 = {(−kπ/3, γ , 0)|k = 0,..., 5}. Thus the waveforms are

1 RW1 : X(t) = (i1(t), i1(t + T/6), i1(t + 2T/6), i1(t + 3T/6), i1(t + 4T/6), i1(t + 5T/6)) .

2 RW1 : X(t) = (i2(t), i2(t + T/6), i2(t + 2T/6), i2(t + 3T/6), i2(t + 4T/6), i2(t + 5T/6)) .

4.2 BIDIRECTIONALLY COUPLED RING Once again we conduct a computational bifurcation analysis, with the aid of the continuation software package AUTO [11], of the collective patterns of oscillation that emerge in Eq. (3.7). This time we show in one composite Fig 4.5 the one-parameter bifurcation diagrams, for a bidirectionally coupled array with N = 3 up to N = 6 crystal oscillators, as a function of coupling strength λ. The additional reﬂectional symmetry induced by the bidirectional coupling restricts signiﬁcantly (more than in the unidirectional case) the 45 types of branches of oscillations that can occur. Note that in this case, if the Hopf bifurcation occurs for an eigenspace on which the ZN action of DN is not trivial or the alternating representation, then there are multiple branches of periodic solutions emerging from the Hopf bifurcation point. In what follows, we only describe the ones picked up by AUTO. In the

D D 3 4

stable stable 60 unstable 60 unstable

IP1 40 1 1 2 2 40 RW IP RW1 2 1 2 RW2 IP

||x|| 2 ||x|| IP2 20 20 RW2 1 0 HB HB HB HB 0 HB HB HB HB 1 2 3 4 1 2 3 4 −1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 1 −1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 1 λ λ

D 6

stable 60 unstable RW1 1 3 IP

2 40 RW2 2 3 IP ||x|| 20

0 HB HB HB HB 1 2 3 4 −1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 1 λ

Figure 4.5. Bifurcation diagrams for a ring of N crystal oscillators coupled biidirectionally in a ring conﬁguration. The diagrams depict the existence and stability of various branches of collective oscillations that appear, mainly via Hopf bifurcations, as a function of coupling strength λ. The ring exhibits DN -symmetry, i.e., cyclic permutations of three crystal oscilla- −4 −4 tors. Parameter values are: R1 = 30.9Ω, R2 = 181.1Ω, L1 = 5.2 × 10 H, L2 = 2.6 × 10 H, −13 −14 8 C1 = 1.0 × 10 F , C2 = 2.5 × 10 F , a = 939, b = 3 × 10 .

1,2 N = 3 case, for instance, we observe again two branches of discrete rotating waves RW1 :

1 RW1 : X(t) = (i1(t), 0, i1(t + T/3), 0, i1(t + 2T/3), 0),

2 RW1 : X(t) = (0, i2(t), 0, i2(t + T/3), 0, i2(t + 2T/3)).

These solutions are similar to those of the unidirectional case, except that now their ¯ isotropy subgroup is (H,K) = (SO^(2)×˙ Ze3, Z3(−2π/3, γ)), which is associated with the

standard representation of DN , so that m = 1 and γ = exp (2πi/3). We also observe two in-phase periodic solutions IP 1,2 but this time only IP 1, where all the currents in Mode 1 oscillate in phase with one another, i.e., same wave form and same amplitude, is locally asymptotically stable over the interval −1 < λ < 0. 46

1,2 For N = 4 we observe once again two rotating waves, RW2 , that emerge at the Hopf points HB1,2 and of the form

1 RW2 : X(t) = (i1(t), 0, i1(t + T/2), 0, i1(t), 0, i1(t + T/2)),

2 RW2 : X(t) = (0, i2(t), 0, i2(t + T/2), 0, i2(t), 0, i2(t + T/2)).

¯ Their isotropy subgroup is (H,K) = (SO^(2)×˙ Ze2, Z2(−π, γ)), with m = 2 and 1 γ = exp (πi). The discrete rotating wave RW2 is locally asymptotically stable over the interval 0 < λ < 0.5. There are also two branches of in-phase oscillations IP 1,2 of which only IP 1 is locally stable over the interval −0.4 < λ < 0.0. The bifurcation diagram for N = 5 shows two types of discrete rotating waves:

1 RW1 : X(t) = (i1(t), i1(t + T/5), i1(t + 2T/5), i1(t + 3T/5), i1(t + 4T/5)) ,

1 RW3 : X(t) = (i1(t), i1(t + 3T/5), i1(t + T/5), i1(t + 4T/5), i1(t + 2T/5)) .

1 ˙ ¯ RW1 has isotropy subgroup (H,K) = (SO^(2)×Ze5, Z5(−2π/5, γ)), with m = 1 and 1 γ = exp (2πi/5). The second wave RW3 has isotropy subgroup ¯ (H,K) = (SO^(2)×˙ Ze5, Z5(−6π/5, γ)), with m = 3 and γ = exp (6πi/5). And, again, there are two fully synchronized waves IP 1,2 but only the in-phase oscillations of Mode 1 are locally asymptotically stable over the interval −0.6 < λ < 0. For N = 6 we observe only one type of discrete rotating wave of the form:

1 RW3 : X(t) = (i1(t), i1(t + T/2), i1(t), i1(t + T/2), i1(t), i1(t + T/2)) .

2 RW3 : X(t) = (i2(t), i2(t + T/2), i2(t), i2(t + T/2), i2(t), i2(t + T/2)) .

¯ These waves have isotropy subgroup (H,K) = (SO^(2)×˙ Ze2, Z2(π, γ)), with m = 3 and γ = exp (πi). The two branches of synchronized oscillations, IP 1,2, still exist for N = 6 but once again only the one with Mode 1 is locally asymptotically stable when −0.5 < λ < 0. In Fig. 4.6 we now show the two-parameter bifurcation diagrams that are associated with each of the cases, i.e., N = 3,..., 6, discussed above. The diagrams outline the boundary curves that deﬁne the regions of existence (and stability) of the branches of solutions over the two parameter space (a, λ). Observe that the branch of synchronized solutions IP 1 is not always stable for negative values of λ, as it was the case of the unidirectionally coupled ring

array. Another signiﬁcant difference with respect to the ZN -symmetric case is that now there 47 might not be any stable branches of solutions for positive values of λ. For instance, observe that when N = 3 or N = 5 there are no stable solutions for λ > 0. In contrast, in the ZN case we always ﬁnd at least one stable branch of rotating waves for small positive coupling.

D D 4 3 1000 1000 HB HB HB HB 1 2 3 4 HB HB HB HB 1,2 1 2 3 4 RW1,2 IP RW1,2 2 RW2 800 1 1,2 2 2 2 1 RW IP IP IP 1 RW1 1 1 1 1,2 IP RW IP 1,2 2 1,2 RW 600 IP 1 500 1 1 a RW IP a 2

400 1 1 IP 1 1 RW 1,2 RW IP 2 1 RW 1 1 RW 1 RW 1 1 IP1 2 IP 200 IP1 1 RW1 IP 0 1 0 −1 0 1 −1 −0.5 0 0.5 1 λ λ

D 6 1000 HB HB HB HB 1 2 3 4 IP1 RW1 2

2 RW1,2 RW 3 3 1,2 500 IP2 IP

a 2 RW RW1,2 3 3 1,2 IP IP1 2 1 1,2 RW RW RW1,2 IP 3 3 3 2 IP1 IP 1 IP1 RW 0 3 −1 −0.5 0 0.5 1 λ

Figure 4.6. Two-parameter bifurcation diagrams for a ring of N crystal oscillators coupled biidirectionally in a ring conﬁguration. Parameter values are: R1 = 30.9Ω, R2 = 181.1Ω, −4 −4 −13 −14 L1 = 5.2 × 10 H, L2 = 2.6 × 10 H, C1 = 1.0 × 10 F , C2 = 2.5 × 10 F , a = 939, b = 3 × 108. 48

CHAPTER 5

PHASE ERROR

In this Chapter we study, in detail, the response of the unidirectional CCOST array under the effects of noise. In particular, we examine changes in phase, i.e. the phase error, when noise is applied. Phase error reduction will optimize the capabilities of the proposed precision timing device. In order to accurately simulate the experimental signal, the governing equations will be cast as a system of stochastic differential equations with additive colored noise [29] [20]. The updated model will induce phase error, a phenomena observed in constructed oscillators, deﬁned as the drift of the period of oscillation away from the expected period length. Reducing this phase error allows the device to produce precise time measures for a longer duration. The analysis in this chapter will reveal the patterns of oscillation with the best scaling, allowing us to achieve our ultimate goal of constructing an ultra-stable precision timing device composed of relatively inexpensive components. The noise that is injected into the system, is colored in nature. That is, the noise is assumed to be Gaussian, band-limited, having a zero mean, a variance σ2, and has a speciﬁc correlation time, τc. The noise is assumed to not drive the dynamics of the system, this corresponds to τf τc, where τf is the time-constant of each oscillator [12] [31]. This type of noise is consistent with the expectation in electrical signals. In these simulations, uncorrelated noise is assumed for each oscillator in the device, but for simplicity we assume they have the same intensity, D. The above discussion allows us to state the new model equations in Langevin form [12]: d X = AX − λB(f(X ,X )) + N (X ) + η t k k √ k−1 k+1 k k (5.1) ηk 2D dtηk = − + ξk τc τc

where τc, D are correlation time and intensity respectively, A, B, N are as in Chapter 3, f

represents the type of coupling, each ηk describes the noise function applied to the kth oscillator, and ξk is a Gaussian distributed random variable with mean µ = 0, and standard deviation σ, for the kth oscillator. Each colored noise solution is characterized by:hηi(t)i = 0 2 2 and hηi(t)ηj(s)i = (D/τc) × exp [−|t − s|/τc], where D = σ τc /2 [15]. As τc → 0 the noise becomes white, however in practice all noise is band limited [1]. For the purpose of our −3 −7 simulations, τc = 1 × 10 and D = 1 × 10 . Additionally, we raise the resistance parameter R2 from 181.1Ω to 1000Ω. This change forces the model outside of the parameter 49 space in which the 66MHz solution (the parasitic oscillation) exists. The high frequency solution is removed because, this solution is not dominant in the physical experiments. All other parameters remain the same as they are originally described in Chapter 4. The model equations ( 5.1) are numerically integrated using the Euler-Maruyama scheme [12]. Since the stochastic process in Eq. ( 5.1) is additive and independent of the solution X, more advanced stochastic numerical schemes, like the Milstein method, reduce to Euler-Maruyama [25]. Figure 5.1 illustrates a typical solution of this stochastic model with strong positive coupling, where N = 3. Note that only one noise function is plotted.

x 10−3 N=3 1.5

0.5

i 0 X

−0.5

−1

−1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 10−7

x 10−4 5

2 η

−1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time x 10−7

1 Figure 5.1. This ﬁgure displays 22MHz rotating wave solution, RW1 , of the unidirectional CCOST model, with 3 nodes and coupling strength λ = 0.99. Above, the current, Xi, is plotted over a period of 1.0 × 10−7 seconds. Below, one noise function is displayed over the same duration.

Phase error is calculated by ﬁrst locating the zeros of the solution. The zeros are approximated using a standard three point quadratic interpolation method. From the location n of the zeros, the periods of oscillation are calculated. Let {pi}i=1 be the sequence of periods.

Then the phase error is calculated as σp, that is, the phase error is the standard deviation of these periods. When noise is removed from the equation the phase error is 0. The size of the sequence of periods is dictated by the saved integration time. The integration time used for the data sets considered in this thesis was 3.5 × 10−5 seconds. The corresponding period for the 22MHz solution is approximately 4.5 × 10−8 sec. Therefore, the phase error is the sample 50 standard deviation of approximately 778 cycles. The data presented in the following sections displays the mean phase error for 50 simulations for each value of N. The simulation samples were calculated and collected utilizing the High Performance Computing Center (HPC) at SPAWAR Systems Center Paciﬁc, Naval Base Point Loma in San Diego, California.

5.1 UNCOUPLED CONTROL GROUP Naturally, the uncoupled oscillator system will serve as a standard to measure the performance of the patterns generated by our coupling topology. The uncoupled model is a reduction of Eq. ( 5.1) when λ = 0. It has been observed that an averaged uncoupled signal has a phase error reduction of approximately √1 , where N is the number of oscillators in the N particular system. The top portion of ﬁgure 5.2 illustrates the uncoupled-averaged phase error reduction.

×10-13 Uncoupled Control Group; Threads = 50 1 Average 0.8 Minimum Maximum Ave~1/sqrt(N) 0.6

0.4 Phase Error

0.2

0 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9

×10-14 10 spread = Max-Min 9

7 Spread 6

4 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 N

Figure 5.2. The uncoupled-averaged (λ = 0) phase error as a function of array size. The top section√ displays the average, maximum, and minimum phase errors of 50 samples for each N.A 1/ N scaling is observed. Below, the range of phase error values are displayed as a function of N. As N increases, we also see a general reduction in the sample range.

Each point is the average of 50 simulations. The phase error reduction has a ﬁtted √ 1/ N curve, consistent to traditional observations. The bottom of ﬁgure 5.2 displays the max-min spread to illustrate the extreme values of the data. These quantaties show that the range of phase error values is generally decreasing, except for when N = 6. This simulation serves as a baseline to compare the numerical performance of our patterns induced by coupling. 51

5.2 UNIDIRECTIONAL COUPLING We consider the model equation under the unidirectional topology, that is, when f(Xk−1,Xk+1) = Xk+1. Speciﬁcally, the governing equations become:

d X = AX − λB(X ) + N(X ) + η t k k √ k+1 k k (5.2) ηk 2D dtηk = − + ξk τc τc

for all k = 1, ··· ,N, again where N is the total number of oscillators in the system. All other parameters are the same as in Eq. ( 5.1). As shown in chapters 3 and 4, this network experiences different patterns of oscillation depending on the strength of the coupling parameter, λ. As such, a sorting algorithm was developed to separate phase error values depending on which pattern was observed. This algorithm ﬁnds the ﬁrst peak of each oscillator in the system, after transient integration, and calculates the time difference between them. For example, let N = 3, be a

speciﬁed tolerance, and T be the expected period. If td = |tpeak(i) − tpeak(i+1)| and td − T/3 < , for each i ≤ N − 1, then the algorithm sorts the pattern as RW1. However, if

td < for all i < N, then the algorithm sorts the pattern as synchronized. Figure 5.3 illustrates the phase error as a function of the number of oscillators within the device when the coupling is negative. Recall from Chapter 4, that negative coupling will always lead to a stable synchronized solution. Hence, Fig. 5.3(top) shows the phase error reduction with respect to the synchronized solution. Additionally, the average, maximum, and minimum phase error is plotted along with a fitted scaling. The lower plot in Fig. 5.3 features the max-min spread to show the evolution of range of values as N increases. The data shows that inducing the synchronized solution, through negative coupling, yields similar phase error scaling to that of an uncoupled-averaged solution. That is, we see at √ best a 1/ N phase error reduction. Intuitively, one would expect the synchronized device would behave like a single crystal, and the data supports this intuition. Next, rotating waves are considered by utilizing positive coupling, specifically λ = 0.99. There are two types of stable rotating waves that are examined. The first one, where

each oscillator is T/N out of phase which exists generically when N is odd, is denoted RW1. The second pattern, where each oscillator is T/2 out of phase and exists only when N is even, is denoted RW2. As in the synchronized and control cases, 50 samples were collected for each value of N for both rotating wave patterns. The rotating wave RW1 is presented ﬁrst. 52

×10-14 Synchronized; λ =-0.99; Threads = 50 4 Ave ~1/sqrt(N) Average 3 Maximum Minimum 2

Phase Error 1

0 3 4 5 6 7 8 9 10

×10-14 2.5 Spread = Max-Min 2

1.5 spread

0.5 3 4 5 6 7 8 9 10 N

Figure 5.3. The phase error with respect to the Synchronized Pattern with λ = −0.99 as a function of N. The top section of the ﬁgure displays the mean, maximum,√ and minimum phase errors for a 50 sample data set for each N. The average data ﬁts a 1/ N reduction curve. Below, the range of possible values in the sample is shown. For most N, the range is decreasing, with notable exceptions of N = 7 and N = 10.

RW ; λ = 0.99; Threads = 50 ×10-11 1 1.5 Mean Max Min 1 Mean ~1/N

Phase Error 0.5

0 3 4 5 6 7 8 9

×10-11 1.5 Range = Max- Min

1 Range 0.5

0 3 4 5 6 7 8 9 N

Figure 5.4. The phase errors for the RW1 pattern as a function of the number of oscillators in the CCOST device when λ = 0.99. Above, the average, maximum and minimum phase errors of a sample of 50 simulations for each N is plotted. Additionally, the average data values ﬁt a 1/N scaling. Below, the range of values is displayed as a function of N. The range decreases monotonically, illustrating the strength of this pattern. 53

Figure 5.4 displays the phase error for the T/N rotating wave. Notice that RW1 is not generally present when N is even, thus in terms of the simulation the phase error would be artiﬁcially 0. To avoid confusion, the even values of N are omitted. A 1/N phase error reduction is observed for this pattern. The data shows that RW1 performs better than the uncoupled control group. The range of this sample is monotonically decreasing as N increases. This suggests that as N increases the variability of the phase error also improves.

Lastly, the wave form RW2 is examined. Note that RW2 does not exist for odd N. Therefore, the odd values of N are omitted to remove artiﬁcial zeros in phase error.

RW ; λ = 0.99; Threads = 50 ×10-12 2 6 Average 5 Minimum Maximum 4 Average~ 1/N

Phase Error 2

0 4 5 6 7 8 9 10

×10-12 6 Spread = Maximum -Minimum 5

3 Spread 2

0 4 5 6 7 8 9 10 N

Figure 5.5. The phase errors with respect to the RW2 pattern as a function of N with λ = 0.99. The top section illustrates the average, maximum, and minimum phase errors of a 50 sample collection for each N. The data does not present a reduction for these values of N. Below, the range of values is represented as a function of N. While we do see a reduction in range, when N > 6 the mean values do not follow this trend.

The data in Fig. 5.5 does not show a clear curve with respect to phase error reduction. The data concerning spread shows a reduction in the range of values for N > 6; however, the mean values do not follow a monotonic reduction. For example, the spread decreases when N changes from 6 to 8, but the mean phase error has increased. A ﬁtted 1/N curve is plotted to illustrate the discrepancies between the mean phase errors and the curve. 54

CHAPTER 6

DISCUSSIONAND FUTURE WORK

This thesis investigates the dynamic behavior of a network of N identical nonlinear crystal oscillators coupled together, either unidirectionally or bidirectionally, in a ring configuration. The symmetry of the network is captured by the groups SO(2) × SO(2) × ZN and SO(2) × SO(2) × DN , for the unidirectional and bidirectional coupling cases respectively. These symmetry groups are used to classify, according to their isotropy subgroups, the possible patterns of oscillation that can arise in each coupling topology from local primary bifurcations via an equivariant Hopf bifurcation. It is shown that emergent patterns are rotating wave periodic solutions, but with the units in the ring showing discrete rotating wave structure. The isotopic decomposition of the phase space under the action of the group of symmetries is used to study the linear stability properties of emergent behaviors. Additional patterns of oscillations are studied via secondary bifurcation from the primary branches of rotating waves. Numerical simulations are carried out for some representative examples of networks with N = 3 up to N = 6 crystal oscillators. The robustness of the unidirectional network under additive noise has also been investigated with a focus on phase error. The primary interest was the scaling of the phase error reduction. In Chapter 5, we discovered that the unidirectionally coupled network preformed better than the uncoupled control in terms of phase error reduction. However, not all of the patterns contained in the unidirectional coupling do. We saw that the pattern RW1, the standard discrete rotating wave, performed better than the uncoupled standard by an order of 2. This confirms that coupling the oscillators is practical. Furthermore, if one wanted to decrease the phase error by an order of ten, one would simply couple 10 more oscillators and induce the RW1 pattern. However, if one did not couple, 100 oscillators would have to be added to receive a similar reduction. As N grows to be large, coupling becomes less expensive than averaging. The RW1 pattern is the best pattern for phase error reduction; we saw that the synchronized solution performs only as well as the uncoupled case and the RW2 pattern appears to offer no clear scaling. This thesis only examines the phase error reduction of patterns exhibited by the unidirectional coupling topology. The next logical step would be to simulate the phase errors for the patterns shown in the bidirectional case. The DN topology exhibits fewer patterns, however, the synchronized and RW2 patterns still appear. Intuitively, phase error scaling in the DN topology should concur with the Synchronized and RW2 phase error reductions 55 presented in this thesis. This speculation will be verified in the future. If our intuitions prove correct, then the unidirectional coupling scheme would be a better engineering investment among the two. As is seen in chapter 3 and chapter 4, the standard rotating wave either does not exist or is not generically stable in a bidirectional apparatus. The emphasis of this work is on homogeneous networks of crystal oscillators, coupled instantaneously. This emphasis is not exhaustive by any means but it serves to identify directions and tasks for future work. One immediate task involves an analysis of the effects of delay. Indeed, while the mathematical model equations of the network of crystal oscillators assume instantaneous coupling, in practice we must account for the fact that even high-speed, high-precision, circuit components can introduce a delay in the coupling signal. Similarly, we expect uncertainty in networks of crystal oscillators to arise from two sources: fluctuations in parameter values due to material imperfections (inductances, resistors and capacitances) and signal contamination due to noise in the electronics. The former case may lead to non-homogeneities in parameters, which, in turn, translates into differences in the internal dynamics of each crystal oscillator. Thus, it is also important that we consider the effects of non-homogeneous electronic components. The coupling type studied in this thesis is simple additive coupling. Other forms of coupling could be studied. For example, the coupling parameter could be a function of Xk and its neighbours,that is λ = λ(X). There are a multitude of ways to couple electronic circuits, additive coupling (as studied in this thesis) models hard wiring and resistive coupling of electrical circuits. This coincides with how the CCOST device is being developed. However, different coupling schemes may prove to have better phase error reduction properties. Alternative coupling schemes may be studied in the future. An experimental CCOST device with unidirectional coupling is currently being fabricated and tested at SPAWAR Systems Center Pacific. After completion, we will be able to test experimental phase error data with the simulations. Initial reports from the experimental device are consistent with the results presented in this thesis. When the device is fully developed, SPAWAR can test it against the Precise Intermediate-term Computer-controlled Oscillator (PICO) advanced clock, an averaged crystal oscillator device that is corrected by central processing units and low pass filters [26] [13]. Furthermore, the phase error reductions for λ = −0.99, 0, 0.99 were presented in this thesis. In the future, we will construct a phase error lattice varying both λ and N to find the best reductions in terms of λ. Additionally, since other types of patterns exist in Z6 ,and arguably in other non-prime N configurations, a probability distribution function will be generated to help visualize the basins of attraction for each pattern. For example, a unidirectional CCOST device with N = 9 follows the symmetry properties of Z9. Now, Z9 56 contains a non-trivial normal subgroup, Z3. This results in the existence of multiple irreducible representations and the emergence of not only a RW1 but also a RW3 pattern, where each oscillator is T/3 out of phase. It would be useful to discover, for development purposes, the probabilities of accessing one pattern over another. Finally, other symmetric coupling schemes should also be taken into consideration. For instance, networks with all-to-all coupling can also be realized with crystal oscillators. Under new symmetries, different patterns of collective behavior may emerge with different conditions for their existence and stability. All of these tasks are part of ongoing work with the ultimate goal of guiding the design rules and operation of a precision timing device. 57

BIBLIOGRAPHY

[1] G. Adomian. Stochastic systems. 169, 1983. [2] D.W. Allan. The science of timekeeping. Technical Report 1289, Hewlett Packard, 1997. [3] V. Apostolyuk and F. Tay. Dynamics of micromechanical coriolis vibratory gyroscopes. Sensor Lett., 2:252–259, 2004. [4] E.V. Appleton and B. van der Pol. On a type of oscillation-hysteresis in a simple triode generator. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Ser. 6, 43:177–193, 1922. [5] D.H. Auston. An assessment of precision time and time interval science and technology. Technical report, National Research Council of the National Academies, The National Academic Press, Washington D.C., 2002. [6] S.P. Beeby, M.J. Tudor, and N.M. White. Energy harvesting vibration sources for microsystems applications. Measurement Science and Technology, 17:R175–R195, 2006. [7] B.P.Mann and N.D.Sims. Energy harvesting from the nonlinear oscillations of magnetic levitation. J. Sound and Vib., 319:515–530, 2009. [8] N. Davies. Ring of vibratory gyroscopes with coupling along the drive and sense axes. Master’s thesis, San Diego State University, 2011. [9] B. Van der Pol. On “relaxation-oscillations”. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Ser.7, 7(2):978–992, 1926. [10] B. Dionne, M. Golubitsky, and I. Stewart. Coupled cells with internal symmetry: Ii. direct products. Nonlinearity, 9(2):575, 1996. [11] E. Doedel and X. Wang. Auto94: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. Applied Mathematics Report, California Institute of Technology, 1994. [12] C.W. Gardiner. Handbook of Stachastic Methods 3rd Ed. Springer: Complexity. [13] C. Gasquet and P. Witomski. Fourier Analysis and Applications: Filtering, Numerical, Computation, Wavelets. Springer, 1998. [14] M. Golubitsky, I. N. Stewart, and D. G. Schaeffer. Singularities and Groups in Bifurcation Theory Vol. II, volume 69. Springer-Verlag, New York, 1988. [15] B. Hajek. An Exploration of Random Processes for Engineers. 2006. [16] T.P. Heavner, E.A. Donley, F. Levi, G. Costanzo, T.E. Parker, J.H. Shirley, N. Ashby, S. Barlow, and S.R. Jefferts. First accuracy evaluation of nist-f2. Metrologia, 51(3):174, 2014. 58 [17] P. Holmes and D.R. Rand. Bifurcation of the forced van der pol oscillator. Quart. Appl. Math., 35:495–509, 1978. [18] V. In, A. Palacios, A. Bulsara, P. Longhini, A. Kho, Joseph Neff, Salvatore Baglio, and Bruno Ando. Complex behavior in driven unidirectionally coupled overdamped duffing elements. Physical Review E, 73(6):066121, 2006. [19] P. Kaminski. Technology and innovation enablers for superiority in 2030. Technical report, Department of Defense, Office of the Under Secretary of Defense, October 2013. [20] S. Karlin and H.M. Taylor. An Introduction to Stochastic Modeling. Academic Press, 1998. [21] M. Krupa. Bifurcations of relative equilibria. SIAM Journal on Mathematical Analysis, 21(6):1453–1486, 1990. [22] J.J. Laurin, S.G. Zaky, and K.G. Balmain. Emi-induced failures in crystal oscillators. IEEE Transactions on Electromagnetic Compatibility, 33(4):334–342, 1991. [23] D. Lindley. National science foundation: Coping with unusual atomic collisions makes an atomic clock more accurate. [24] A. Matus-Vargas, H.G. Gonzalez-Hernandez, B. Chan, A. Palacios, P-L. Buono, V. In, S. Naik, A. Phipps, and P. Longhini. Dynamics, bifurcations and normal forms in arrays of magnetostrictive energy harvesters with all-to-all coupling. Int. J. Bifurcation and Chaos, 25(2):1550026, 2015. [25] G. N. Milshtejn. Approximate integration of stochastic differential equations. Theory of Probability and Its Applications, 19(3):557000, 1975. [26] B. Montgomery. Keeping precision time when gps signals stop. Cotts Journal Online, 2005. [27] NIST. Nist launches a new u.s. time standard: Nist f2 atomic clock. [28] J.W. Norie. To find the longitude of chronometers or time-keepers. New and Complete Epitome of Practical Navigation, 1816. [29] B. Øksendal. Stochastic Differential Equations. Springer, 2000. [30] A.K. Poddar and U.L. Rohde. Crystal Oscillators, pages 1–38. Wiley Encyclopedia and Electronics Engineering, 2012. [31] M. Lopez S. Wio, R. Deza. An Introduction to Stochastic Processes and Nonequilibrium Statistical Physics. World Scientific Publishing, 2012. [32] G. Sebald, H. Kuwano, D. Guyomar, and B. Ducharne. Simulation of a duffing oscillator for broadband piezoelectric energy harvesting. Smart Materials and Structures, 20:075022, 2011. [33] J.L. Semmlow. Circuits, signals, and systems for bioengineers. Academic Press, 2005. [34] A. Shkel. Type i and type ii micromachined vibratory gyroscopes. In Proceedings of IEEE/ION PLANS, pages 586–593, San Diego, CA, 2006. 59 [35] B. van der Pol. Forced oscillations in a circuit with non-linear resistance (reception with reactive triode). The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Ser. 7, 3:65–80, 1927. [36] B. van der Pol and J. van der Mark. Frequency demultiplication. Nature, 120:363–364, 1927. [37] S.H. Voldman. ESD: Failure Mechanisms and Models. John Wiley and Sons, 2009. [38] H. Vu. Ring of Vibratory Gyroscopes with Coupling along the Drive Axis. PhD thesis, San Diego State University, 2011. [39] H. Vu, A. Palacios, V. In, P. Longhini, and J. Neff. Two-time scale analysis of a ring of coupled vibratory gyroscopes. Physical Rev. E., 81:031108, 2010. [40] J. Wang, R. Wu, J. Du, T. Ma, D. Huang, and W. Yan. The nonlinear thickness-shear ovibrations of quartz crystal plates under a strong electric field. In IEEE International Ultrasonics Symposium Proceedings, volume 10.1109, pages 320–323. IEEE, 2011. [41] M. Warren. The evolution of the quartz crystal clock. Bell System Technical Journal, 27:510–558, 1948. [42] S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems. Springer-Verlag, New York, 1990. 60

APPENDIX A

AVERAGING 61

AVERAGING

AVERAGING AND BIFURCATION FOR N NETWORKS

This appendix is organized to display the averaging for the single crystal ﬁrst, and then the two coupling schemes second. The averaging for the coupling schemes is ordered in accordance to appearance in the thesis, i.e. uncoupled ﬁrst and coupled second. Due to the length of the equations the font size has been decreased within accordance to the SDSU thesis format guidelines. In the single crystal oscillator we present a Van der Pol transformation and a standard split between radial and phase dynamics. Following these presentations we show that the transformations are equivalent. In the averaging for the coupled systems, we only consider Van der Pol transformations.

FIRST METHOD: Van der Pol transformation

Consider the oscillators

2 d i1 di1 i1 2 di1 di2 L1 1 + R1 + = a − 3b(i1 + i2) + dt dt C1 dt dt (A.1) 2 d i2 di2 i2 2 di1 di2 L1 2 + R2 + = a − 3b(i1 + i2) + . dt dt C2 dt dt

Let 0 0 x1 = i1 x2 = i1 x3 = i2 x4 = i2.

Then system (A.1) takes the form

0 x1 = x2 0 1 R1 a 3b 2 x = − x1 − x2 + (x2 + x4) − (x1 + x3) x2 + x4 2 L1C1 L1 L1 L1 (A.2) 0 x3 = x4 0 1 R2 a 3b 2 x = − x3 − x4 + (x2 + x4) − (x1 + x3) x2 + x4 . 4 L2C2 L2 L2 L2

Consider the invertible Van der Pol transformation

ω1 k1 ω1 0 u1 = cos( t)i1 − sin( t)i k1 ω1 k1 1

ω1 k1 ω1 0 v1 = −sin( t)i1 − cos( t)i k1 ω1 k1 1 (A.3)

ω2 k2 ω2 0 u2 = cos( t)i2 − sin( t)i k2 ω2 k2 2

ω2 k2 ω2 0 v2 = −sin( t)i2 − cos( t)i . k2 ω2 k2 2 62

Then,

2 k1 ω1 00 ω1 u˙ 1 = − ω 2 i1 + i1 sin( k t) 1 k1 1

2 k1 ω1 00 ω1 v˙1 = − ω 2 i1 + i1 cos( k t) 1 k1 1 (A.4) 2 k2 ω2 00 ω2 u˙ 2 = − ω 2 i2 + i2 sin( k t) 2 k2 2

2 k2 ω2 00 ω2 v˙2 = − ω 2 i2 + i2 cos( k t). 2 k2 2

Also,

ω1 ω1 i1 = cos( t)u1 − sin( t)v1 k1 k1

0 ω1 ω1 ω1 ω1 i1 = − sin( t)u1 − cos( t)v1 k1 k1 k1 k1 (A.5)

ω2 ω2 i2 = cos( t)u2 − sin( t)v2 k2 k2

0 ω2 ω2 ω2 ω2 i = − sin( t)u2 − cos( t)v2. 2 k2 k2 k2 k2

We consider the following perturbation of system (A.1)

00 2 R1 0 a 0 0 3b 2 0 0 i + ω i1 = − i + (i + i ) − (i1 + i2) (i + i ) 1 01 L1 1 L1 1 2 L1 1 2 (A.6)

00 2 R2 0 a 0 0 3b 2 0 0 i + ω i2 = − i + (i + i ) − (i1 + i2) (i + i ) 2 02 L2 2 L2 1 2 L2 1 2

where ω2 = 1 and ω2 = 1 . 01 L1C1 02 L2C2 Then,

2 2 2 k1 ω1 −k1ω01 0 0 ω1 u˙ 1 = − ω 2 i1 + f1(i1, i2, i1, i2, t) sin( k t) 1 k1 1

2 2 2 k1 ω1 −k1ω01 0 0 ω1 v˙1 = − ω 2 i1 + f1(i1, i2, i1, i2, t) cos( k t) 1 k1 1 (A.7) 2 2 2 k2 ω2 −k2ω02 0 0 ω2 u˙ 2 = − ω 2 i2 + f2(i1, i2, i1, i2, t) sin( k t) 2 k2 2

2 2 2 k2 ω2 −k2ω02 0 0 ω2 v˙2 = − ω 2 i2 + f2(i1, i2, i1, i2, t) cos( k t). 2 k2 2

2 2 2 2 Setting k1 = k2 = 1, ω01 − ω1 = Ω1 and ω02 − ω2 = Ω2 we obtain 63

0 0 u˙ 1 = Ω1 u1cos(ω1t) − v1sin(ω1t) − f1(i1, i2, i , i , t) sin(ω1t) ω1 1 2

0 0 v˙1 = Ω1 u1cos(ω1t) − v1sin(ω1t) − f1(i1, i2, i , i , t) cos(ω1t) ω1 1 2 (A.8) 0 0 u˙ 2 = Ω2 u2cos(ω2t) − v2sin(ω2t) − f2(i1, i2, i , i , t) sin(ω2t) ω2 1 2

0 0 v˙2 = Ω2 u2cos(ω2t) − v2sin(ω2t) − f2(i1, i2, i , i , t) cos(ω2t) ω2 1 2

where

0 0 R1 a f1(i1, i2, i , i , t) = ω1u1sin(ω1t) + ω1v1cos(ω1t) − ω1u1sin(ω1t) + ω1v1cos(ω1t) + ω2u2sin(ω2t) + ω2v2cos(ω2t) 1 2 L1 L1

3b 2 + u1cos(ω1t) − v1sin(ω1t) + u2cos(ω2t) − v2sin(ω2t) ω1u1sin(ω1t) + ω1v1cos(ω1t) + ω2u2sin(ω2t) L1

+ω2v2cos(ω2t) (A.9)

0 0 R2 a f2(i1, i2, i , i , t) = ω2u2sin(ω2t) + ω2v2cos(ω2t) − ω1u1sin(ω1t) + ω1v1cos(ω1t) + ω2u2sin(ω2t) + ω2v2cos(ω2t) 1 2 L2 L2

3b 2 + u1cos(ω1t) − v1sin(ω1t) + u2cos(ω2t) − v2sin(ω2t) ω1u1sin(ω1t) + ω1v1cos(ω1t) + ω2u2sin(ω2t) L2

+ω2v2cos(ω2t) .

Averaging the system (A.8) we get

1 R T 1 R T 2 1 R T 0 0 u˙ 1 = limT →∞ Ω1u1 sin(ω1t)cos(ω1t)dt − limT →∞ Ω1v1 sin (ω1t) − limT →∞ f1(i1, i2, i , i , t)sin(ω1t)dt ω1 T 0 T 0 T 0 1 2

1 R T 2 1 R T 1 R T 0 0 v˙1 = limT →∞ Ω1u1 cos (ω1t)dt − limT →∞ Ω1v1 sin(ω1t)cos(ω1t) − limT →∞ f1(i1, i2, i , i , t)cos(ω1t)dt ω1 T 0 T 0 T 0 1 2 (A.10) 1 R T 1 R T 2 1 R T 0 0 u˙ 2 = limT →∞ Ω2u2 sin(ω2t)cos(ω2t)dt − limT →∞ Ω2v2 sin (ω2t) − limT →∞ f2(i1, i2, i , i , t)sin(ω2t)dt ω2 T 0 T 0 T 0 1 2

1 R T 2 1 R T 1 R T 0 0 v˙2 = limT →∞ Ω2u2 cos (ω2t)dt − limT →∞ Ω2v2 sin(ω2t)cos(ω2t) − limT →∞ f2(i1, i2, i , i , t)cos(ω2t)dt ω2 T 0 T 0 T 0 1 2 or equivalently 64

1 1 R T 0 0 u˙ 1 = − Ω1v1 − limT →∞ f1(i1, i2, i , i , t)sin(ω1t)dt ω1 2 T 0 1 2

1 1 R T 0 0 v˙1 = Ω1u1 − limT →∞ f1(i1, i2, i , i , t)cos(ω1t)dt ω1 2 T 0 1 2 (A.11) 1 1 R T 0 0 u˙ 2 = − Ω2v2 − limT →∞ f2(i1, i2, i , i , t)sin(ω2t)dt ω2 2 T 0 1 2

1 1 R T 0 0 v˙2 = Ω2u2 − limT →∞ f2(i1, i2, i , i , t)cos(ω2t)dt . ω2 2 T 0 1 2

Setting the integrals in the system (A.11) we obtain

1 R1 a 1 R T 3b 0 0 u˙ 1 = − Ω1v1 − ω1u1 + ω1u1 − limT →∞ f11(i1, i2, i , i , t)sin(ω1t)dt ω1 2 2L1 2L1 T 0 L1 1 2

1 R1 a 1 R T 3b 0 0 v˙1 = Ω1u1 − ω1v1 + ω1v1 − limT →∞ f11(i1, i2, i , i , t)cos(ω1t)dt ω1 2 2L1 2L1 T 0 L1 1 2 (A.12) 1 R2 a 1 R T 3b 0 0 u˙ 2 = − Ω2v2 − ω2u2 + ω2u2 − limT →∞ f11(i1, i2, i , i , t)sin(ω2t)dt ω1 2 2L2 2L2 T 0 L2 1 2

1 R2 a 1 R T 3b 0 0 v˙2 = Ω2u2 − ω2v2 + ω2v2 − limT →∞ f11(i1, i2, i , i , t)sin(ω2t)dt . ω1 2 2L2 2L2 T 0 L2 1 2 where

2 f11 = u1cos(ω1t) − v1sin(ω1t) + u2cos(ω2t) − v2sin(ω2t) ω1u1sin(ω1t) + ω1v1cos(ω1t) + ω2u2sin(ω2t) + ω2v2cos(ω2t))sin(ω1t .

Observing that

1 R T 0 0 3b 2 2 2 2 limT →∞ f11(i1, i2, i , i , t)sin(ω1t)dt = ω1u1 u + 2(u + v ) + v T 0 1 2 8L1 1 2 2 1

1 R T 0 0 3b 2 2 2 2 limT →∞ f11(i1, i2, i , i , t)cos(ω1t)dt = ω1v1 u + 2(u + v ) + v T 0 1 2 8L1 1 2 2 1

1 R T 0 0 3b 2 2 2 2 limT →∞ f11(i1, i2, i , i , t)sin(ω2t)dt = ω2u2 2(u + v ) + u + v T 0 1 2 8L2 1 1 2 2

1 R T 0 0 3b 2 2 2 2 limT →∞ f11(i1, i2, i , i , t)sin(ω2t)dt = ω2v2 2(u + v ) + u + v , T 0 1 2 8L2 1 1 2 2

we conclude that the averaged autonomous systems is given by

1 a−R1 3b 2 2 2 2 u˙ 1 = − Ω1v1 + ω1u1 − ω1u1 u + 2(u + v ) + v ω1 2 2L1 8L1 1 2 2 1

1 a−R1 3b 2 2 2 2 v˙1 = Ω1u1 + ω1v1 − ω1v1 u + 2(u + v ) + v ω1 2 2L1 8L1 1 2 2 1 (A.13) 1 a−R2 3b 2 2 2 2 u˙ 2 = − Ω2v2 + ω2u2 − ω2u2 2(u + v ) + u + v ω1 2 2L2 8L2 1 1 2 2

1 a−R2 3b 2 2 2 2 v˙2 = Ω2u2 + ω2v2 − ω2v2 2(u + v ) + u + v . ω1 2 2L2 8L2 1 1 2 2 65

In the following, we have the integrals in the system (A.12) without calculating limT →∞ . R T 2π 1. INTEGRAL: f11sin(ω1t)dt , T = 0 ω1

6 4 3 2 5 7 R T 1 −9πω1 ω2+91πω1 ω2 −91πω1 ω2 +9πω2 3 0 f11sin(ω1t)dt = − 4 6 4 2 2 4 6 u1 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

5 2 3 4 6 1 56cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 −512cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 +72cos(πω2/ω1)sin(πω2/ω1)ω1ω2 2 − 4 6 4 2 2 4 6 u1u2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

2 5 2 2 3 4 2 6 5 2 3 4 6 1 56cos(πω2/ω1) ω1 ω2 −512cos(πω2/ω1) ω1 ω2 +72cos(πω2/ω1) ω1ω2 −56ω1 ω2 +512ω1 ω2 −72ω1ω2 2 − 4 6 4 2 2 4 6 u1v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

3 7 3 5 2 3 3 4 1 −36cos(πω2/ω1) sin(πω2/ω1)ω1 +400cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 −692cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2

3 6 7 5 2 72cos(πω2/ω1) sin(πω2/ω1)ω1ω2 +18cos(πω2/ω1)sin(πω2/ω1)ω1 −200cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

3 4 6 6 4 3 2 5 7 346cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 −36cos(πω2/ω1)sin(πω2/ω1)ω1ω2 −18πω1 ω2+182πω1 ω2 −182πω1 ω2 +18πω2 2 + 6 4 2 2 4 6 u1u2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

2 6 2 4 3 2 2 5 6 4 3 2 5 1 −48cos(πω2/ω1) ω1 ω2+448cos(πω2/ω1) ω1 ω2 −144cos(πω2/ω1) ω1 ω2 +48ω1 ω2−448ω1 ω2 +144ω1 ω2 − 4 6 4 2 2 4 6 u1u2v1 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

4 7 4 5 2 4 3 4 4 6 2 7 1 −72cos(πω2/ω1) ω1 +800cos(πω2/ω1) ω1 ω2 −1384cos(πω2/ω1) ω1 ω2 +144cos(πω2/ω1) ω1ω2 +72cos(πω2/ω1) ω1 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

2 5 2 2 3 4 2 6 −800cos(πω2/ω1) ω1 ω2 +1384cos(πω2/ω1) ω1 ω2 −144cos(πω2/ω1) ω1ω2 + 6 4 2 2 4 6 u1u2v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

6 4 3 2 5 7 6 1 −9πω1 ω2+91πω1 ω2 −91πω1 ω2 +9πω2 2 1 48cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 − 4 6 4 2 2 4 6 u1v1 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 ) ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

4 3 2 5 −448cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 +144cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 6 4 2 2 4 6 u1v1v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

3 7 3 5 2 3 3 4 1 36cos(πω2/ω1) sin(πω2/ω1)ω1 −400cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 +692cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

3 6 7 5 2 −72cos(πω2/ω1) sin(πω2/ω1)ω1ω2 −18cos(πω2/ω1)sin(πω2/ω1)ω1 +200cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

3 4 6 6 4 3 2 5 7 −346cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 +36cos(πω2/ω1)sin(πω2/ω1)ω1ω2 −18πω1 ω2+182πω1 ω2 −182πω1 ω2 +18πω2 2 + 6 4 2 2 4 6 u1v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

5 5 2 5 3 4 5 6 1 288cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 −320cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 +32cos(πω2/ω1) sin(πω2/ω1)ω1ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 ) 66

3 5 2 3 3 4 3 6 −288cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 +320cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 −32cos(πω2/ω1) sin(πω2/ω1)ω1ω2 + 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

5 2 3 4 6 72cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 −224cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 +24cos(πω2/ω1)sin(πω2/ω1)ω1ω2 3 + 6 4 2 2 4 6 u2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

4 6 4 4 3 4 2 5 2 6 1 −36cos(πω2/ω1) ω1 ω2+328cos(πω2/ω1) ω1 ω2 −36cos(πω2/ω1) ω1 ω2 +36cos(πω2/ω1) ω1 ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

2 4 3 2 2 5 6 5 2 6 3 4 −328cos(πω2/ω1) ω1 ω2 +36cos(πω2/ω1) ω1 ω2 2 1 864cos(πω2/ω1) ω1 ω2 −960cos(πω2/ω1) ω1 ω2 + 6 4 2 2 4 6 u2v1 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 ) ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

6 6 4 5 2 4 3 4 4 6 96cos(πω2/ω1) ω1ω2 −1296cos(πω2/ω1) ω1 ω2 +1440cos(πω2/ω1) ω1 ω2 −144cos(πω2/ω1) ω1ω2 + 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

2 5 2 2 3 4 2 6 5 2 3 4 6 504cos(πω2/ω1) ω1 ω2 −704cos(πω2/ω1) ω1 ω2 +72cos(πω2/ω1) ω1ω2 −72ω1 ω2 +224ω1 ω2 −24ω1ω2 2 + 6 4 2 2 4 6 u2v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

5 2 3 4 1 16cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 −144cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 2 − 4 6 4 2 2 4 6 u2v1 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

3 6 3 4 3 3 2 5 1 72cos(πω2/ω1) sin(πω2/ω1)ω1 ω2−656cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 +72cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

6 4 3 2 5 −36cos(πω2/ω1)sin(πω2/ω1)ω1 ω2+328cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 −36cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 6 4 2 2 4 6 u2v1v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

5 5 2 5 3 4 5 6 1 −864cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 +960cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 −96cos(πω2/ω1) sin(πω2/ω1)ω1ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

3 5 2 3 3 4 3 6 864cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 −960cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 +96cos(πω2/ω1) sin(πω2/ω1)ω1ω2 + 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

5 2 3 4 −144cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 +16cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 2 + 6 4 2 2 4 6 u2v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

2 5 2 2 3 4 5 2 3 4 1 16cos(πω2/ω1) ω1 ω2 −144cos(πω2/ω1) ω1 ω2 −16ω1 ω2 +144ω1 ω2 2 − 4 6 4 2 2 4 6 v1 v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

4 6 4 4 3 4 2 5 1 36cos(πω2/ω1) ω1 ω2−328cos(πω2/ω1) ω1 ω2 +36cos(πω2/ω1) ω1 ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

2 6 2 4 3 2 2 5 −36cos(πω2/ω1) ω1 ω2+328cos(πω2/ω1) ω1 ω2 −36cos(πω2/ω1) ω1 ω2 2 + 6 4 2 2 4 6 v1v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

6 5 2 6 3 4 6 6 4 5 2 1 −288cos(πω2/ω1) ω1 ω2 +320cos(πω2/ω1) ω1 ω2 −32cos(πω2/ω1) ω1ω2 +432cos(πω2/ω1) ω1 ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

4 3 4 4 6 2 5 2 2 3 4 3 4 6 −480cos(πω2/ω1) ω1 ω2 +48cos(πω2/ω1) ω1ω2 −144cos(πω2/ω1) ω1 ω2 +16cos(πω2/ω1) ω1 ω2 +144ω1 ω2 −16ω1ω2 3 + 6 4 2 2 4 6 v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 ) 67

R T 2π 2. INTEGRAL: f11cos(ω1t), T = 0 ω2

2 6 2 4 3 2 2 5 2 7 6 R T 1 48cos(πω2/ω1) ω1 ω2−504cos(πω2/ω1) ω1 ω2 +656cos(πω2/ω1) ω1 ω2 −72cos(πω2/ω1) ω2 −48ω1 ω2 0 f11cos(ω1t)dt = − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

4 3 2 5 7 6 4 3 2 5 7 504ω1 ω2 −656ω1 ω2 +72ω2 ) 2 1 −9πω1 ω2+91πω1 ω2 −91πω1 ω2 +9πω2 2 + 6 4 2 2 4 6 u1u2 − 4 6 4 2 2 4 6 u1v1 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 ) ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2

6 4 3 2 5 1 −48cos(πω2/ω1)sin(πω2/ω1)ω1 ω2+504cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 −656cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

7 72sin(πω2/ω1)cos(πω2/ω1)ω2 2 + 6 4 2 2 4 6 u1v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

4 6 4 4 3 4 2 5 4 7 1 108cos(πω2/ω1) ω1 ω2−1128cos(πω2/ω1) ω1 ω2 +1420cos(πω2/ω1) ω1 ω2 −144cos(πω2/ω1) ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

2 6 2 4 3 2 2 5 2 7 −108cos(πω2/ω1) ω1 ω2+1128cos(πω2/ω1) ω1 ω2 −1420cos(πω2/ω1) ω1 ω2 +144cos(πω2/ω1) ω2 2 + 6 4 2 2 4 6 u1u2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

5 2 3 4 1 32cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 −288cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 − 4 6 4 2 2 4 6 u1u2v1 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

3 6 3 4 3 3 2 5 1 −216cos(πω2/ω1) sin(πω2/ω1)ω1 ω2+2256cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 −2840cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

3 7 6 4 3 288sin(πω2/ω1)cos(πω2/ω1) ω2 +108cos(πω2/ω1)sin(πω2/ω1)ω1 ω2−1128cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

2 5 7 1420cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 −144sin(πω2/ω1)cos(πω2/ω1)ω2 + 6 4 2 2 4 6 u1u2v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

2 5 2 2 3 4 5 2 3 4 1 32cos(πω2/ω1) ω1 ω2 −288cos(πω2/ω1) ω1 ω2 −32ω1 ω2 +288ω1 ω2 − 4 6 4 2 2 4 6 u1v1v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

4 6 4 4 3 4 2 5 4 7 1 −108cos(πω2/ω1) ω1 ω2+1128cos(πω2/ω1) ω1 ω2 −1420cos(πω2/ω1) ω1 ω2 +144cos(πω2/ω1) ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

2 6 2 4 3 2 2 5 2 7 108cos(πω2/ω1) ω1 ω2−1128cos(πω2/ω1) ω1 ω2 +1420cos(πω2/ω1) ω1 ω2 −144cos(πω2/ω1) ω2 2 + 6 4 2 2 4 6 u1v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

6 4 3 6 2 5 6 7 4 4 3 1 −864cos(πω2/ω1) ω1 ω2 +960cos(πω2/ω1) ω1 ω2 −96cos(πω2/ω1) ω2 +1296cos(πω2/ω1) ω1 ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

4 2 5 4 7 2 4 3 2 2 5 −1440cos(πω2/ω1) ω1 ω2 +144cos(πω2/ω1) ω2 −504cos(πω2/ω1) ω1 ω2 +704cos(πω2/ω1) ω1 ω2 + 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

2 7 4 3 2 5 7 −72cos(πω2/ω1) ω2 +72ω1 ω2 −224ω1 ω2 +24ω2 3 + 6 4 2 2 4 6 u2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 ) 68

3 7 3 5 2 3 3 4 1 −36cos(πω2/ω1) sin(πω2/ω1)ω1 +328cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 −36cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

7 5 2 3 4 6 18cos(πω2/ω1)sin(πω2/ω1)ω1 −164cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 +18cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 −18πω1 ω2 + 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

4 3 2 5 7 +182πω1 ω2 −182πω1 ω2 +18πω2 2 + 6 4 2 2 4 6 u2v1 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

5 4 3 5 2 5 5 7 1 2592sin(πω2/ω1)cos(πω2/ω1) ω1 ω2 −2880sin(πω2/ω1)cos(πω2/ω1) ω1 ω2 +288sin(πω2/ω1)cos(πω2/ω1) ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

3 4 3 3 2 5 3 7 −2592cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 +2880cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 −288sin(πω2/ω1)cos(πω2/ω1) ω2 + 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

4 3 2 5 7 504cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 −704cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 +72sin(πω2/ω1)cos(πω2/ω1)ω2 2 + 6 4 2 2 4 6 u2v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

2 6 2 4 3 6 4 3 1 −48cos(πω2/ω1) ω1 ω2+432cos(πω2/ω1) ω1 ω2 +48ω1 ω2−432ω1 ω2 2 − 4 6 4 2 2 4 6 u2v1 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

4 7 4 5 2 4 3 4 1 −72cos(πω2/ω1) ω1 +656cos(πω2/ω1) ω1 ω2 −72cos(πω2/ω1) ω1 ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

2 7 2 5 2 2 3 4 72cos(πω2/ω1) ω1 −656cos(πω2/ω1) ω1 ω2 +72cos(πω2/ω1) ω1 ω2 + 6 4 2 2 4 6 u2v1v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

6 4 3 6 2 5 6 7 4 4 3 1 2592cos(πω2/ω1) ω1 ω2 −2880cos(πω2/ω1) ω1 ω2 +288cos(πω2/ω1) ω2 −3888cos(πω2/ω1) ω1 ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

4 2 5 4 7 2 4 3 2 2 5 4320cos(πω2/ω1) ω1 ω2 −432cos(πω2/ω1) ω2 +1440cos(πω2/ω1) ω1 ω2 −1456cos(πω2/ω1) ω1 ω2 + 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

2 7 4 3 2 5 6 4 3 2 5 7 144cos(πω2/ω1) ω2 −144ω1 ω2 +16ω1 ω2 2 1 −9πω1 ω2+91πω1 ω2 −91πω1 ω2 +9πω2 3 + 6 4 2 2 4 6 u2v2 − 4 6 4 2 2 4 6 v1 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 ) ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

6 4 3 1 48cos(πω2/ω1)sin(πω2/ω1)ω1 ω2−432cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 2 − 4 6 4 2 2 4 6 v1 v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

3 7 3 5 2 3 3 4 1 36cos(πω2/ω1) sin(πω2/ω1)ω1 −328cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 +36cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

7 5 2 3 4 6 −18cos(πω2/ω1)sin(πω2/ω1)ω1 +164cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 −18cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 −18πω1 ω2 + 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

4 3 2 5 7 182πω1 ω2 −182πω1 ω2 +18πω2 2 + 6 4 2 2 4 6 v1v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )

5 4 3 1 −864sin(πω2/ω1)cos(πω2/ω1) ω1 ω2 − 4 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 ))

5 2 5 5 7 3 4 3 960sin(πω2/ω1)cos(πω2/ω1) ω1 ω2 −96sin(πω2/ω1)cos(πω2/ω1) ω2 +864cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 + 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 ))

3 2 5 3 7 4 3 −960cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 +96sin(πω2/ω1)cos(πω2/ω1) ω2 −144cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 6 4 2 2 4 6 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 ))

2 5 16cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 3 + 6 4 2 2 4 6 v2 ω2(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω2 )) 69

R T 2π 3. INTEGRAL: f11sin(ω2t), T = 0 ω2

R T 0 f11sin(ω2t)dt =

1 7 − 24cos(πω2/ω1)sin(πω2/ω1)ω 4ω1(ω1−ω2)(3ω1−ω2)(ω1−3ω2)(ω1+3ω2)(3ω1+ω2)(ω1+ω2) 1

5 2 3 4 3 − 224cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 72cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 u1

3 7 3 5 2 + 108cos(πω2/ω1) sin(πω2/ω1)ω1 − 1056cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

3 3 4 3 6 7 + 764cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 72cos(πω2/ω1) sin(πω2/ω1)ω1ω2 − 54cos(πω2/ω1)sin(πω2/ω1)ω1

5 2 3 4 6 + 528cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 − 382cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 36cos(πω2/ω1)sin(πω2/ω1)ω1ω2

6 4 3 2 5 7 2 − 18πω1 ω2 + 182πω1 ω2 − 182πω1 ω2 + 18πω2 u1u2

2 6 2 4 3 2 2 5 6 4 3 2 5 2 + − 24cos(πω2/ω1) ω1 ω2 + 224cos(πω2/ω1) ω1 ω2 − 72cos(πω2/ω1) ω1 ω2 + 24ω1 ω2 − 224ω1 ω2 + 72ω1 ω2 )u1v1

4 7 4 5 2 4 3 4 4 6 2 7 + 108cos(πω2/ω1 ω1 − 1056cos(πω2/ω1) ω1 ω2 + 764cos(πω2/ω1) ω1 ω2 − 72cos(πω2/ω1) ω1ω2 − 108cos(πω2/ω1) ω1

2 5 2 2 3 4 2 6 2 5 7 + 1056cos(πω2/ω1) ω1 ω2 − 764cos(πω2/ω1) ω1 ω2 + 72cos(πω2/ω1) ω1ω2 u1v2 + (288sin(πω2/ω1)cos(πω2/ω1) ω1

5 5 2 5 3 4 5 6 − 2048cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 + 1952cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 192cos(πω2/ω1) sin(πω2/ω1)ω1ω2

3 7 3 5 2 3 3 4 − 288cos(πω2/ω1) sin(πω2/ω1)ω1 + 2048cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 1952cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

3 6 7 5 2 + 192cos(πω2/ω1) sin(πω2/ω1)ω1ω2 + 72cos(πω2/ω1)sin(πω2/ω1)ω1 − 512cos(πω2/ω1)sin(πω2/ω1)ω1 ω2

3 4 2 4 6 4 4 3 4 2 5 + 56cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 u1u2 + − 72cos(πω2/ω1) ω1 ω2 + 656cos(πω2/ω1) ω1 ω2 − 72cos(πω2/ω1) ω1 ω2

2 6 2 4 3 2 2 5 6 7 + 72cos(πω2/ω1) ω1 ω2 − 656cos(πω2/ω1) ω1 ω2 + 72cos(πω2/ω1) ω1 ω2 u1u2v1 + 576cos(πω2/ω1) ω1

6 5 2 6 3 4 6 6 4 7 − 4096cos(πω2/ω1) ω1 ω2 + 3904cos(πω2/ω1) ω1 ω2 − 384cos(πω2/ω1) ω1ω2 − 864cos(πω2/ω1) ω1

4 5 2 4 3 4 4 6 2 7 + 6144cos(πω2/ω1) ω1 ω2 − 5856cos(πω2/ω1) ω1 ω2 + 576cos(πω2/ω1) ω1ω2 + 288cos(πω2/ω1) ω1

2 5 2 2 3 4 2 6 5 2 3 4 6 − 1904cos(πω2/ω1) ω1 ω2 + 1504cos(πω2/ω1) ω1 ω2 − 144cos(πω2/ω1) ω1ω2 − 144ω1 ω2 + 448ω1 ω2 − 48ω1ω2 u1u2v2

5 2 3 4 2 + 16cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 − 144cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 u1v1 70

3 6 3 4 3 3 2 5 + 72cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 656cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 + 72cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

6 4 3 2 5 − 36cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 328cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 − 36cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 u1v1v2

5 7 5 5 2 5 3 4 + − 288sin(πω2/ω1)cos(πω2/ω1) ω1 + 2048cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 1952cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

5 6 3 7 3 5 2 + 192cos(πω2/ω1) sin(πω2/ω1)ω1ω2 + 288cos(πω2/ω1) sin(πω2/ω1)ω1 − 2048cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

3 3 4 3 6 5 2 + 1952cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 192cos(πω2/ω1) sin(πω2/ω1)ω1ω2 − 144cos(πω2/ω1)sin(πω2/ω1)ω1 ω2

3 4 2 7 7 7 5 2 + 16cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 u1v2 + 144sin(πω2/ω1)cos(πω2/ω1) ω1 − 1456sin(πω2/ω1)cos(πω2/ω1) ω1 ω2

7 3 4 7 6 5 7 + 1456sin(πω2/ω1)cos(πω2/ω1) ω1 ω2 − 144sin(πω2/ω1)cos(πω2/ω1) ω1ω2 − 216sin(πω2/ω1)cos(πω2/ω1) ω1

5 5 2 5 3 4 5 6 + 2184cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 2184cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 + 216cos(πω2/ω1) sin(πω2/ω1)ω1ω2

3 7 3 5 2 3 3 4 + 90cos(πω2/ω1) sin(πω2/ω1)ω1 − 910cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 + 910cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

3 6 7 5 2 − 90cos(πω2/ω1) sin(πω2/ω1)ω1ω2 − 9cos(πω2/ω1)sin(πω2/ω1)ω1 + 91cos(πω2/ω1)sin(πω2/ω1)ω1 ω2

3 4 6 6 4 3 2 5 7 3 − 91cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 9cos(πω2/ω1)sin(πω2/ω1)ω1ω2 − 9πω1 ω2 + 91πω1 ω2 − 91πω1 ω2 + 9πω2 u2

6 6 6 4 3 6 2 5 4 6 + − 288cos(πω2/ω1) ω1 ω2 + 320cos(πω2/ω1) ω1 ω2 − 32cos(πω2/ω1) ω1 ω2 + 432cos(πω2/ω1) ω1 ω2

4 4 3 4 2 5 2 6 2 4 3 − 480cos(πω2/ω1) ω1 ω2 + 48cos(πω2/ω1) ω1 ω2 − 216cos(πω2/ω1) ω1 ω2 + 672cos(πω2/ω1) ω1 ω2

2 2 5 6 4 3 2 5 2 − 72cos(πω2/ω1) ω1 ω2 + 72ω1 ω2 − 512ω1 ω2 + 56ω1 ω2 u2v1

8 7 8 5 2 8 3 4 8 6 6 7 + 432cos(πω2/ω1) ω1 − 4368cos(πω2/ω1) ω1 ω2 + 4368cos(πω2/ω1) ω1 ω2 − 432cos(πω2/ω1) ω1ω2 − 864cos(πω2/ω1) ω1

6 5 2 6 3 4 6 6 4 7 + 8736cos(πω2/ω1) ω1 ω2 − 8736cos(πω2/ω1) ω1 ω2 + 864cos(πω2/ω1) ω1ω2 + 504cos(πω2/ω1) ω1

4 5 2 4 3 4 4 6 2 7 2 5 2 − 5096cos(πω2/ω1) ω1 ω2 + 5096cos(πω2/ω1) ω1 ω2 − 504cos(πω2/ω1) ω1ω2 − 72cos(πω2/ω1) ω1 + 728cos(πω2/ω1) ω1 ω2

2 3 4 2 6 2 3 7 + 728cos(πω2/ω1) ω1 ω2 + 72cos(πω2/ω1) ω1ω2 u2v2 + − 36cos(πω2/ω1) sin(πω2/ω1)ω1

3 5 2 3 3 4 7 + 328cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 36cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 + 18cos(πω2/ω1)sin(πω2/ω1)ω1

5 2 3 4 6 4 3 2 5 7 2 − 164cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 18cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 − 18πω1 ω2 + 182πω1 ω2 − 182πω1 ω2 + 18πω2 u2v1 71

5 6 5 4 3 5 2 5 + 576sin(πω2/ω1)cos(πω2/ω1) ω1 ω2 − 640sin(πω2/ω1)cos(πω2/ω1) ω1 ω2 + 64sin(πω2/ω1)cos(πω2/ω1) ω1 ω2

3 6 3 4 3 3 2 5 − 576cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 + 640cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 64cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

6 4 3 2 5 + 144cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 − 448cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 48cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 u2v1v2

7 7 7 5 2 7 3 4 + − 432sin(πω2/ω1)cos(πω2/ω1) ω1 + 4368sin(πω2/ω1)cos(πω2/ω1) ω1 ω2 − 4368sin(πω2/ω1)cos(πω2/ω1) ω1 ω2

7 6 5 7 5 5 2 + 432sin(πω2/ω1)cos(πω2/ω1) ω1ω2 + 648sin(πω2/ω1)cos(πω2/ω1) ω1 − 6552cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

5 3 4 5 6 3 7 + 6552cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 648cos(πω2/ω1) sin(πω2/ω1)ω1ω2 − 198cos(πω2/ω1) sin(πω2/ω1)ω1

3 5 2 3 3 4 3 6 + 2002cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 2002cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 + 198cos(πω2/ω1) sin(πω2/ω1)ω1ω2

7 5 2 3 4 − 9cos(πω2/ω1)sin(πω2/ω1)ω1 + 91cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 − 91cos(πω2/ω1)sin(πω2/ω1)ω1 ω2

6 6 4 3 2 5 7 2 2 6 + 9cos(πω2/ω1)sin(πω2/ω1)ω1ω2 − 9πω1 ω2 + 91πω1 ω2 − 91πω1 ω2 + 9πω2 u2v2 + − 16cos(πω2/ω1) ω1 ω2

2 4 3 6 4 3 3 4 7 4 5 2 4 3 4 + 144cos(πω2/ω1) ω1 ω2 + 16ω1 ω2 − 144ω1 ω2 v1 + − 36cos(πω2/ω1) ω1 + 328cos(πω2/ω1) ω1 ω2 − 36cos(πω2/ω1) ω1 ω2

2 7 2 5 2 2 3 4 2 + 36cos(πω2/ω1) ω1 − 328cos(πω2/ω1) ω1 ω2 + 36cos(πω2/ω1) ω1 ω2 v1 v2

6 6 6 4 3 6 2 5 4 6 + 288cos(πω2/ω1) ω1 ω2 − 320cos(πω2/ω1) ω1 ω2 + 32cos(πω2/ω1) ω1 ω2 − 432cos(πω2/ω1) ω1 ω2

4 4 3 4 2 5 2 6 2 4 3 4 3 + 480cos(πω2/ω1) ω1 ω2 − 48cos(πω2/ω1) ω1 ω2 + 144cos(πω2/ω1) ω1 ω2 − 16cos(πω2/ω1) ω1 ω2 − 144ω1 ω2

2 5 2 8 7 8 5 2 8 3 4 8 6 + 16ω1 ω2 v1v2 + − 144cos(πω2/ω1) ω1 + 1456cos(πω2/ω1) ω1 ω2 − 1456cos(πω2/ω1) ω1 ω2 + 144cos(πω2/ω1) ω1ω2

6 7 6 5 2 6 3 4 6 6 4 7 + 288cos(πω2/ω1) ω1 − 2912cos(πω2/ω1) ω1 ω2 + 2912cos(πω2/ω1) ω1 ω2 − 288cos(πω2/ω1) ω1ω2 − 144cos(πω2/ω1) ω1

4 5 2 4 3 4 4 6 3 + 1456cos(πω2/ω1) ω1 ω2 − 1456cos(πω2/ω1) ω1 ω2 + 144cos(πω2/ω1) ω1ω2 v2 72

R T 2π 4. INTEGRAL: f11cos(ω2t), T = 0 ω2

R T 0 f11cos(ω2t)dt =

1 2 7 2 5 2 2 3 4 7 5 2 − 7 5 2 3 4 6 (24cos( πω2/ω1) ω1 − 224cos( πω2/ω1) ω1 ω2 + 72cos(πω2/ω1) ω1 ω2 − 24ω1 + 224ω1 ω2 4(9ω1 −91ω1 ω2 +91ω1 ω2 −9ω1ω2 )

3 4 3 4 7 4 5 2 4 3 4 4 6 − 72ω1 ω2 u1 + 108cos(πω2/ω1) ω1 − 1056cos(πω2/ω1) ω1 ω2 + 764cos(πω2/ω1) ω1 ω2 − 72cos(πω2/ω1) ω1ω2

2 7 2 5 2 2 3 4 2 6 2 − 108cos(πω2/ω1) ω1 + 1056cos(πω2/ω1) ω1 ω2 − 764cos(πω2/ω1) ω1 ω2 + 72cos(πω2/ω1) ω1ω2 u1u2

6 4 3 2 5 2 + 24cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 − 224cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 72cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 u1v1

3 7 3 5 2 3 3 4 + − 108cos(πω2/ω1) sin(πω2/ω1)ω1 + 1056cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 764cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

3 6 7 5 2 + 72cos(πω2/ω1) sin(πω2/ω1)ω1ω2 + 54cos(πω2/ω1)sin(πω2/ω1)ω1 − 528cos(πω2/ω1)sin(πω2/ω1)ω1 ω2

3 4 6 6 4 3 2 5 7 2 + 382cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 − 36cos(πω2/ω1)sin(πω2/ω1)ω1ω2 − 18πω1 ω2 + 182πω1 ω2 − 182πω1 ω2 + 18πω2 u1v2

6 7 6 5 2 6 3 4 6 6 + 288cos(πω2/ω1) ω1 − 2048cos(πω2/ω1) ω1 ω2 + 1952cos(πω2/ω1) ω1 ω2 − 192cos(πω2/ω1) ω1ω2

4 7 4 5 2 4 3 4 4 6 2 7 − 432cos(πω2/ω1) ω1 + 3072cos(πω2/ω1) ω1 ω2 − 2928cos(πω2/ω1) ω1 ω2 + 288cos(πω2/ω1) ω1ω2 + 216cos(πω2/ω1) ω1

2 5 2 2 3 4 2 6 7 5 2 3 4 6 2 − 1680cos(πω2/ω1) ω1 ω2 + 1480cos(πω2/ω1) ω1 ω2 − 144cos(πω2/ω1) ω1ω2 − 72ω1 + 656ω1 ω2 − 504ω1 ω2 + 48ω1ω2 u1u2

3 6 3 4 3 3 2 5 + 72cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 656cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 + 72cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

6 4 3 2 5 − 36cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 328cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 − 36cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 u1u2v1

5 7 5 5 2 5 3 4 + − 576cos(πω2/ω1) sin(πω2/ω1)ω1 + 4096cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 3904cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

5 6 3 7 3 5 2 + 384cos(πω2/ω1) sin(πω2/ω1)ω1ω2 + 576cos(πω2/ω1) sin(πω2/ω1)ω1 − 4096cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

3 3 4 3 6 7 + 3904cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 384cos(πω2/ω1) sin(πω2/ω1)ω1ω2 − 144cos(πω2/ω1)sin(πω2/ω1)ω1

5 2 3 4 6 + 1168cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 − 1424cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 144cos(πω2/ω1)sin(πω2/ω1)ω1ω2 u1u2v2 73

2 5 2 2 3 4 5 2 3 4 2 4 6 + 16cos(πω2/ω1) ω1 ω2 − 144cos(πω2/ω1) ω1 ω2 − 16ω1 ω2 + 144ω1 ω2 )u1v1 + (72cos(πω2/ω1) ω1 ω2

4 4 3 4 2 5 2 6 2 4 3 − 656cos(πω2/ω1) ω1 ω2 + 72cos(πω2/ω1) ω1 ω2 − 72cos(πω2/ω1) ω1 ω2 + 656cos(πω2/ω1) ω1 ω2

2 2 5 6 7 6 5 2 6 3 4 − 72cos(πω2/ω1) ω1 ω2 u1v1v2 + − 288cos(πω2/ω1) ω1 + 2048cos(πω2/ω1) ω1 ω2 − 1952cos(πω2/ω1) ω1 ω2

6 6 4 7 4 5 2 4 3 4 + 192cos(πω2/ω1) ω1ω2 + 432cos(πω2/ω1) ω1 − 3072cos(πω2/ω1) ω1 ω2 + 2928cos(πω2/ω1) ω1 ω2

4 6 2 7 2 5 2 2 3 4 − 288cos(πω2/ω1) ω1ω2 − 144cos(πω2/ω1) ω1 + 1024cos(πω2/ω1) ω1 ω2 − 1408cos(πω2/ω1) ω1 ω2

2 6 3 4 6 2 + 144cos(πω2/ω1) ω1ω2 + 432ω1 ω2 − 48ω1ω2 u1v2

8 7 8 5 2 8 3 4 8 6 6 7 + 144cos(πω2/ω1) ω1 − 1456cos(πω2/ω1) ω1 ω2 + 1456cos(πω2/ω1) ω1 ω2 − 144cos(πω2/ω1) ω1ω2 − 288cos(πω2/ω1) ω1

6 5 2 6 3 4 6 6 4 7 + 2912cos(πω2/ω1) ω1 ω2 − 2912cos(πω2/ω1) ω1 ω2 + 288cos(πω2/ω1) ω1ω2 + 216cos(πω2/ω1) ω1

4 5 2 4 3 4 4 6 2 7 2 5 2 − 2184cos(πω2/ω1) ω1 ω2 + 2184cos(πω2/ω1) ω1 ω2 − 216cos(πω2/ω1) ω1ω2 − 72cos(πω2/ω1) ω1 + 728cos(πω2/ω1) ω1 ω2

2 3 4 2 6 3 − 728cos(πω2/ω1) ω1 ω2 + 72cos(πω2/ω1) ω1ω2 u2

5 6 5 4 3 5 2 5 + 288cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 320sin(πω2/ω1)cos(πω2/ω1) ω1 ω2 + 32sin(πω2/ω1)cos(πω2/ω1) ω1 ω2

3 6 3 4 3 3 2 5 − 288cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 + 320cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 32cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

6 4 3 2 5 2 + 72cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 − 224cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 24cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 u2v1

7 7 7 5 2 7 3 4 + − 432cos(πω2/ω1) sin(πω2/ω1)ω1 + 4368cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 4368cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

7 6 5 7 5 5 2 + 432cos(πω2/ω1) sin(πω2/ω1)ω1ω2 + 648cos(πω2/ω1) sin(πω2/ω1)ω1 − 6552cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

5 3 4 5 6 3 7 + 6552cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 648cos(πω2/ω1) sin(πω2/ω1)ω1ω2 − 342cos(πω2/ω1) sin(πω2/ω1)ω1

3 5 2 3 3 4 3 6 + 3458cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 3458cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 + 342cos(πω2/ω1) sin(πω2/ω1)ω1ω2

7 5 2 3 4 + 63cos(πω2/ω1)sin(πω2/ω1)ω1 − 637cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 637cos(πω2/ω1)sin(πω2/ω1)ω1 ω2

6 6 4 3 2 5 7 2 − 63cos(πω2/ω1)sin(πω2/ω1)ω1ω2 − 9πω1 ω2 + 91πω1 ω2 − 91πω1 ω2 + 9πω2 u2v2

4 7 4 5 2 4 3 4 2 7 2 5 2 + − 36cos(πω2/ω1) ω1 + 328cos(πω2/ω1) ω1 ω2 − 36cos(πω2/ω1) ω1 ω2 + 36cos(πω2/ω1) ω1 − 328cos(πω2/ω1) ω1 ω2 74

2 3 4 2 6 6 6 4 3 6 2 5 + 36cos(πω2/ω1) ω1 ω2 u2v1 + 576cos(πω2/ω1) ω1 ω2 − 640cos(πω2/ω1) ω1 ω2 + 64cos(πω2/ω1) ω1 ω2

4 6 4 4 3 4 2 5 2 6 2 4 3 − 864cos(πω2/ω1) ω1 ω2 + 960cos(πω2/ω1) ω1 ω2 − 96cos(πω2/ω1) ω1 ω2 + 288cos(πω2/ω1) ω1 ω2 − 32cos(πω2/ω1) ω1 ω2

4 3 2 5 8 7 8 5 2 8 3 4 − 288ω1 ω2 + 32ω1 ω2 u2v1v2 + − 432cos(πω2/ω1) ω1 + 4368cos(πω2/ω1) ω1 ω2 − 4368cos(πω2/ω1) ω1 ω2

8 6 6 7 6 5 2 6 3 4 + 432cos(πω2/ω1) ω1ω2 + 864cos(πω2/ω1) ω1 − 8736cos(πω2/ω1) ω1 ω2 + 8736cos(πω2/ω1) ω1 ω2

6 6 4 7 4 5 2 4 3 4 − 864cos(πω2/ω1) ω1ω2 − 576cos(πω2/ω1) ω1 + 5824cos(πω2/ω1) ω1 ω2 − 5824cos(πω2/ω1) ω1 ω2

4 6 2 7 2 5 2 2 3 4 + 576cos(πω2/ω1) ω1ω2 + 144cos(πω2/ω1) ω1 − 1456cos(πω2/ω1) ω1 ω2 + 1456cos(πω2/ω1) ω1 ω2

2 6 2 6 4 3 3 − 144cos(πω2/ω1) ω1ω2 u2v2 + 16cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 − 144cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 )v1

3 7 3 5 2 3 3 4 + (36cos(πω2/ω1) sin(πω2/ω1)ω1 − 328cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 + 36cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

7 5 2 − 18cos(πω2/ω1)sin(πω2/ω1)ω1 + 164cos(πω2/ω1)sin(πω2/ω1)ω1 ω2

3 4 6 4 3 2 5 7 2 − 18cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 − 18πω1 ω2 + 182πω1 ω2 − 182πω1 ω2 + 18πω2 v1 v2

5 6 5 4 3 5 2 5 + − 288cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 + 320sin(πω2/ω1)cos(πω2/ω1) ω1 ω2 − 32sin(πω2/ω1)cos(πω2/ω1) ω1 ω2

3 6 3 4 3 3 2 5 + 288cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 320cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 + 32cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

4 3 2 5 2 − 432cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 + 48cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 v1v2

7 7 7 5 2 7 3 4 + 144cos(πω2/ω1) sin(πω2/ω1)ω1 − 1456cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 + 1456cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

7 6 5 7 5 5 2 − 144cos(πω2/ω1) sin(πω2/ω1)ω1ω2 − 216cos(πω2/ω1) sin(πω2/ω1)ω1 + 2184cos(πω2/ω1) sin(πω2/ω1)ω1 ω2

5 3 4 5 6 3 7 − 2184cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 + 216cos(πω2/ω1) sin(πω2/ω1)ω1ω2 + 90cos(πω2/ω1) sin(πω2/ω1)ω1

3 5 2 3 3 4 3 6 − 910cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 + 910cos(πω2/ω1) sin(πω2/ω1)ω1 ω2 − 90cos(πω2/ω1) sin(πω2/ω1)ω1ω2

7 5 2 3 4 − 9cos(πω2/ω1)sin(πω2/ω1)ω1 + 91cos(πω2/ω1)sin(πω2/ω1)ω1 ω2 − 91cos(πω2/ω1)sin(πω2/ω1)ω1 ω2

6 6 4 3 2 5 7 3 + 9cos(πω2/ω1)sin(πω2/ω1)ω1ω2 − 9πω1 ω2 + 91πω1 ω2 − 91πω1 ω2 + 9πω2 v2 75

SECOND METHOD

Setting the transformation

  x1 = ρ1sin(ω1γ1)       x2 = ρ1ω1cos(ω1γ1) 

  x = ρ sin(ω γ )  3 2 2 2      x4 = ρ2ω2cos(ω2γ2) we have

  x˙ 1 =ρ ˙1sin(ω1γ1) + ρ1ω1cos(ω1γ1)γ ˙1      2  x˙ 2 =ρ ˙1ω1cos(ω1γ1) − ρ1ω1 sin(ω1γ1)γ ˙1     x˙ 3 =ρ ˙2sin(ω2γ2) + ρ2ω2cos(ω2γ2)γ ˙2      2 x˙ 4 =ρ ˙2ω2cos(ω2γ2) − ρ2ω2 sin(ω2γ2)γ ˙2.

Replacing in system (A.2) we get

  ρ˙1sin(ω1γ1) + ρ1ω1cos(ω1γ1)γ ˙1 = ρ1ω1cos(ω1γ1)      2 2 a−R1 a  ρ˙1ω1cos(ω1γ1) − ρ1ω1 sin(ω1γ1)γ ˙1 = −ρ1ω1 sin(ω1γ1) + ρ1ω1cos(ω1γ1) + ρ2ω2cos(ω2γ2) + f1(ρ1, ρ2, γ1, γ2)  L1 L1 (A.14)    ρ˙2sin(ω2γ2) + ρ2ω2cos(ω2γ2)γ ˙2 = ρ2ω2cos(ω2γ2)      2 2 a−R2 a  ρ˙2ω2cos(ω2γ2) − ρ2ω sin(ω2γ2)γ ˙2 = −ρ2ω sin(ω2γ2) + ρ2ω2cos(ω2γ2) + ρ1ω1cos(ω1γ1) + f2(ρ1, ρ2, γ1, γ2) 2 2 L2 L2

where 3b 2 f1(ρ1, ρ2, γ1, γ2) = − ρ1sin(ω1γ1) + ρ2sin(ω2γ2) ρ1ω1cos(ω1γ1) + ρ2ω2cos(ω2γ2) L1 and 3b 2 f2(ρ1, ρ2, γ1, γ2) = − ρ1sin(ω1γ1) + ρ2sin(ω2γ2) ρ1ω1cos(ω1γ1) + ρ2ω2cos(ω2γ2) . L2

Multiplying the ﬁrst equation in system (A.14) by ω1sin(ω1γ1), the second one by cos(ω1γ1) and adding the resulting equations we obtain

a−R1 2 a ω2 1 ρ˙1 = ρ1cos (ω1γ1) + ρ2 cos(ω1γ1)cos(ω2γ2) + f1(ρ1, ρ2, γ1, γ2) L1 L1 ω1 ω1

An analogous procedure with the equations in system (A.14) gives the new system 76

 a−R1 2 a ω2 1  ρ˙1 = ρ1cos (ω1γ1) + ρ2 cos(ω1γ1)cos(ω2γ2) + f1(ρ1, ρ2, γ1, γ2)  L1 L1 ω1 ω1     a−R2 2 a ω1 1  ρ˙2 = ρ2cos (ω2γ2) + ρ1 cos(ω1γ1)cos(ω2γ2) + f2(ρ1, ρ2, γ1, γ2)  L2 L2 ω2 ω2  (A.15)  a−R1 1 a ω2 ρ2 1  γ˙1 = 1 − sin(ω1γ1)cos(ω1γ1) − 2 sin(ω1γ1)cos(ω2γ2) − 2 sin(ω1γ1)f1(ρ1, ρ2, γ1, γ2)  L1 ω1 L1 ω ρ1 ρ1ω  1 1     a−R2 1 a ω1 ρ1 1  γ˙2 = 1 − L ω sin(ω2γ2)cos(ω2γ2) − L 2 ρ sin(ω1γ1)cos(ω2γ2) − 2 sin(ω2γ2)f2(ρ1, ρ2, γ1, γ2) 2 1 2 ω2 2 ρ2ω2 or equivalently

 ρ˙ = cos(ω γ ) L1 1 aρ ω cos(ω γ ) + ρ ω cos(ω γ ) − R ρ ω cos(ω γ ) + L f (ρ , ρ , γ , γ )  1 1 1 aω L 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 2 1 2  1 1      L1 1  ρ˙2 = cos(ω2γ2) a ρ1ω1cos(ω1γ1) + ρ2ω2cos(ω2γ2) − R2ρ2ω2cos(ω2γ2) + L2f2(ρ1, ρ2, γ1, γ2)  aω2 L2  (A.16)   L1 1  γ˙1 = 1 − sin(ω1γ1) 2 a ρ1ω1cos(ω1γ1) + ρ2ω2cos(ω2γ2) − R1ρ1ω1cos(ω1γ1) + L1f1(ρ1, ρ2, γ1, γ2)  aρ ω L1  1 1      L1 1  γ˙2 = 1 − sin(ω2γ2) 2 L a ρ1ω1cos(ω1γ1) + ρ2ω2cos(ω2γ2) − R2ρ2ω2cos(ω2γ2) + L2f2(ρ1, ρ2, γ1, γ2) aρ2ω2 2

Where = a . We can write system (A.16) in the form L1

  ρ˙1 = P1(ρ, γ)       ρ˙2 = P2(ρ, γ)  (A.17)   γ˙ = 1 − Γ (ρ, γ)  1 1      γ˙2 = 1 − Γ2(ρ, γ)

where ρ = (ρ1, ρ2) and γ = (γ1, γ2). Along with the system (A.17) we consider the averaged system

  ρ˙1 = P10(ρ)       ρ˙2 = P20(ρ)  (A.18)   γ˙ = 1 − Γ (γ)  1 10      γ˙2 = 1 − Γ20(γ2) where 77

1 R T a−R1 2 2 P10(ρ) = limT →∞ T 0 P1(ρ, γ + τ)dτ = −4 3b ρ1 + ρ1 ρ1 + 2ρ2

1 R T a−R2 2 2 P20(ρ) = limT →∞ T 0 P2(ρ, γ + τ)dτ = −4 3b ρ2 + ρ2 ρ2 + 2ρ1 (A.19) 1 R T Γ10(γ) = limT →∞ T 0 Γ1(ρ, γ) = 0

1 R T Γ20(γ2) = limT →∞ T 0 Γ2(ρ, γ) = 0

From the two last equations in (A.19) we get

γ10(t) = t + t1 γ20(t) = t + t2

where t1 and t2 are constant depending on the initial conditions. So, the solutions of system (A.17) to a zeroth-order in is given by

ρ1 = ρ10, ρ2 = ρ20, γ1 = γ10, γ2 = γ20 where ρ10 and ρ20 are solution of

a−R1 2 2 −4 3b ρ1 + ρ1 ρ1 + 2ρ2 = 0

a−R2 2 2 −4 3b ρ2 + ρ2 ρ2 + 2ρ1 = 0.

EQUIVALENCE BETWEEN THE AVERAGED SYSTEMS

The systems (A.13) and (A.18) are equivalent under the transformation

q 2 2 ρ1 = u1 + v1

q 2 2 ρ2 = u2 + v2 (A.20)

2ω1 v1 γ1 = arctan Ω1 u1

2ω1 v2 γ2 = arctan Ω2 u2

VAN DER POL TRANSFORMATION: Unidirectional coupling

Consider the oscillators

2 d ik,j dik,j ik,j 2 dik,1 dik,2 dik+1,1 dik+1,2 Lk,j 2 + Rk,j + = a − 3b ik,1 + ik,2 − λ ik+1,1 + ik+1,2 + − λ( + ) . (A.21) dt dt Ck,j dt dt dt dt

Setting k = 1, 2, ..., N, j = 1, 2, Lk,1 = L1, Lk,2 = L2, Rk,1 = R1, Rk,2 = R2, Ck,1 = C1 and Ck,2 = C2, then system (A.21) takes the form 78

 2 d ik,1 dik,1 ik,1 2 dik,1 dik,2 dik+1,1 dik+1,2  L1 2 + R1 + = a − 3b ik,1 + ik,2 − λ ik+1,1 + ik+1,2 + − λ( + )  dt dt C1 dt dt dt dt (A.22)  2  d ik,2 dik,2 ik,2 2 dik,1 dik,2 dik+1,1 dik+1,2  L2 2 + R2 + = a − 3b ik,1 + ik,2 − λ ik+1,1 + ik+1,2 + − λ( + ) . dt dt C2 dt dt dt dt

Consider the invertible Van der Pol transformation

ωk,1 lk,1 ωk,1 0 uk,1 = cos( t)ik,1 − sin( t)i lk,1 ωk,1 lk,1 k,1

ωk,1 lk,1 ωk,1 0 vk,1 = −sin( t)ik,1 − cos( t)i lk,1 ωk,1 lk,1 k,1 (A.23) ωk,2 lk,2 ωk,2 0 uk,2 = cos( t)ik,2 − sin( t)i lk,2 ωk,2 lk,2 k,2

ωk,2 lk,2 ωk,2 0 vk,2 = −sin( t)ik,2 − cos( t)i . lk,2 ωk,2 lk,2 k,2 for k = 1, 2, ..., N. Thus,

 l ω2 ω  u˙ = − k,1 k,1 i + i00 sin( k,1 t)  k,1 ω l2 k,1 k,1 l  k,1 k,1 k,1     2  lk,1 ωk,1 00 ωk,1  v˙k,1 = − 2 ik,1 + i cos( t)  ωk,1 l k,1 lk,1  k,1 (A.24)  l ω2 ω  u˙ = − k,2 k,2 i + i00 sin( k,2 t)  k,2 ω l2 k,2 k,2 l  k,2 k,2 k,2     2  lk,2 ωk,2 00 ωk,2  v˙k,2 = − ω 2 ik,2 + ik,2 cos( l t). k,2 lk,2 k,2

Also we have,

ωk,1 ωk,1 0 ωk,1 ωk,1 ωk,1 ωk,1 ik,1 = cos( t)uk,1 − sin( t)vk,1 i = − sin( t)uk,1 − cos( t)vk,1 lk,1 lk,1 k,1 lk,1 lk,1 lk,1 lk,1

ωk,2 ωk,2 0 ωk,2 ωk,2 ωk,2 ωk,2 ik,2 = cos( t)uk,2 − sin( t)vk,2 i = − sin( t)uk,2 − cos( t)vk,2 lk,2 lk,2 k,2 lk,2 lk,2 lk,2 lk,2 (A.25) ωk+1,1 ωk+1,1 0 ωk+1,1 ωk+1,1 ωk+1,1 ωk+1,1 ik+1,1 = cos( t)uk+1,1 − sin( t)vk+1,1 i = − sin( t)uk+1,1 − cos( t)vk+1,1 lk+1,1 lk+1,1 k+1,1 lk+1,1 lk+1,1 lk+1,1 lk+1,1

ωk+1,2 ωk+1,2 0 ωk+1,2 ωk+1,2 ωk+1,2 ωk+1,2 ik+1,2 = cos( t)uk+1,2 − sin( t)vk+1,2 i = − sin( t)uk+1,2 − cos( t)vk+1,2. lk+1,2 lk+1,2 k+1,2 lk+1,2 lk+1,2 lk+1,2 lk+1,2

We consider the following perturbation of system (A.22)

 2 d ik,1 1 R1 dik,1 2 dik,1 dik,2 dik+1,1 dik+1,2  2 = − ik,1 + − + a − 3b ik,1 + ik,2 − λ ik+1,1 + ik+1,2 + − λ( + )  dt L1C1 L1 dt dt dt dt dt  (A.26)  2  d ik,2 1 R2 dik,2 2 dik,1 dik,2 dik+1,1 dik+1,2  2 = − ik,2 + − + a − 3b ik,1 + ik,2 − λ ik+1,1 + ik+1,2 + − λ( + ) ,  dt L2C2 L2 dt dt dt dt dt or equivalently 79

 2 d ik,1 1 0 0 0 0  2 = − ik,1 + f1(ik,1, ik+1,1, ik,2, ik+1,2, ik,1, ik+1,1, ik,2, ik+1,2, t)  dt L1C1 (A.27)  2  d ik,2 1 0 0 0 0  2 = − ik,2 + f2(ik,1, ik+1,1, ik,2, ik+1,2, i , i , i , i , t) dt L2C2 k,1 k+1,1 k,2 k+1,2 where 0 0 0 0 f1(ik,1, ik+1,1, ik,2, ik+1,2, ik,1, ik+1,1, ik,2, ik+1,2, t) = di 2 di di di di − R1 k,1 + a − 3bi + i − λi + i k,1 + k,2 − λ( k+1,1 + k+1,2 ) L1 dt k,1 k,2 k+1,1 k+1,2 dt dt dt dt and 0 0 0 0 f2(ik,1, ik+1,1, ik,2, ik+1,2, ik,1, ik+1,1, ik,2, ik+1,2, t) = di 2 di di di di − R2 k,2 + a − 3bi + i − λi + i k,1 + k,2 − λ( k+1,1 + k+1,2 ). L2 dt k,1 k,2 k+1,1 k+1,2 dt dt dt dt

2 1 2 1 2 2 2 2 Setting lk,1 = lk,2 = lk+1,1 = lk+1,2 = 1, ω = , ω = , ω − ω = Ωk,1, ω − ω = Ωk,2, 01 L1C1 02 L2C2 01 k,1 02 k,2 ωk,j = ωk+1,j , j = 1, 2 taking into account (A.27), and then applying the transformation (A.23) and (A.25) in the system (A.24) we get

 R1  u˙ k,1 = ω Ωk,1 cos(ωk,1t)uk,1 − sin(ωk,1t)vk,1 − − L − ωk,1sin(ωk,1t)uk,1 − ωk,1cos(ωk,1t)vk,1  k,1 1      +a − ω sin(ω t)u − ω cos(ω t)v − ω sin(ω t)u − ω cos(ω t)v  k,1 k,1 k,1 k,1 k,1 k,1 k,2 k,2 k,2 k,2 k,2 k,2      −λ − ω sin(ω t)u − ω cos(ω t)v − ω sin(ω t)u − ω cos(ω t)v  k,1 k,1 k+1,1 k,1 k,1 k+1,1 k,2 k,2 k+1,2 k,2 k,2 k+1,2      −3b cos(ωk,1t)uk,1 − sin(ωk,1t)vk,1 + cos(ωk,2t)uk,2 − sin(ωk,2t)vk,2 − λ cos(ωk,1t)uk+1,1 − sin(ωk,1t)vk+1,1      2  +cos(ωk,2t)uk+1,2 − sin(ωk,2t)vk+1,2 − ωk,1sin(ωk,1t)uk,1 − ωk,1cos(ωk,1t)vk,1 − ωk,2sin(ωk,2t)uk,2       −ωk,2cos(ωk,2t)vk,2 − λ − ωk,1sin(ωk,1t)uk+1,1 − ωk,1cos(ωk,1t)vk+1,1 − ωk,2sin(ωk,2t)uk+1,2       −ω cos(ω t)v sin(ω t)  k,2 k,2 k+1,2 k,1      v˙k,1 = ω Ωk,1 cos(ωk,1t)uk,1 − sin(ωk,1t)vk,1 (A.28)  k,1      R1  − − − ωk,1sin(ωk,1t)uk,1 − ωk,1cos(ωk,1t)vk,1  L1       +a − ωk,1sin(ωk,1t)uk,1 − ωk,1cos(ωk,1t)vk,1 − ωk,2sin(ωk,2t)uk,2 − ωk,2cos(ωk,2t)vk,2       −λ − ωk,1sin(ωk,1t)uk+1,1 − ωk,1cos(ωk,1t)vk+1,1 − ωk,2sin(ωk,2t)uk+1,2 − ωk,2cos(ωk,2t)vk+1,2       −3b cos(ωk,1t)uk,1 − sin(ωk,1t)vk,1 + cos(ωk,2t)uk,2 − sin(ωk,2t)vk,2 − λ cos(ωk,1t)uk+1,1 − sin(ωk,1t)vk+1,1      2  +cos(ωk,2t)uk+1,2 − sin(ωk,2t)vk+1,2 − ωk,1sin(ωk,1t)uk,1 − ωk,1cos(ωk,1t)vk,1 − ωk,2sin(ωk,2t)uk,2       −ωk,2cos(ωk,2t)vk,2 − λ − ωk,1sin(ωk,1t)uk+1,1 − ωk,1cos(ωk,1t)vk+1,1 − ωk,2sin(ωk,2t)uk+1,2       −ωk,2cos(ωk,2t)vk+1,2 cos(ωk,1t) 80

 R2  u˙ k,2 = ω Ωk,2 cos(ωk,2t)uk,2 − sin(ωk,2t)vk,2 − − L − ωk,2sinωk,2t)uk,2 − ωk,2cos(ωk,2t)vk,2  k,2 2      +a − ω sin(ω t)u − ω cos(ω t)v − ω sin(ω t)u − ω cos(ω t)v  k,1 k,1 k,1 k,1 k,1 k,1 k,2 k,2 k,2 k,2 k,2 k,2      −λ − ω sin(ω t)u − ω cos(ω t)v − ω sin(ω t)u − ω cos(ω t)v  k,1 k,1 k+1,1 k,1 k,1 k+1,1 k,2 k,2 k+1,2 k,2 k,2 k+1,2      −3b cos(ωk,1t)uk,1 − sin(ωk,1t)vk,1 + cos(ωk,2t)uk,2 − sin(ωk,2t)vk,2 − λ cos(ωk,1t)uk+1,1 − sin(ωk,1t)vk+1,1      2  +cos(ωk,2t)uk+1,2 − sin(ωk,2t)vk+1,2 − ωk,1sin(ωk,1t)uk,1 − ωk,1cos(ωk,1t)vk,1 − ωk,2sin(ωk,2t)uk,2       −ωk,2cos(ωk,2t)vk,2 − λ − ωk,1sin(ωk,1t)uk+1,1 − ωk,1cos(ωk,1t)vk+1,1 − ωk,2sin(ωk,2t)uk+1,2       −ωk,2cos(ωk,2t)vk+1,2 sin(ωk,2t)  

  R2  v˙k,2 = Ωk,2 cos(ωk,2t)uk,2 − sin(ωk,2t)vk,2 − − − ωk,2sinωk,2t)uk,2 − ωk,2cos(ωk,2t)vk,2  ω L2  k,2      +a − ω sin(ω t)u − ω cos(ω t)v − ω sin(ω t)u − ω cos(ω t)v  k,1 k,1 k,1 k,1 k,1 k,1 k,2 k,2 k,2 k,2 k,2 k,2      −λ − ωk,1sin(ωk,1t)uk+1,1 − ωk,1cos(ωk,1t)vk+1,1 − ωk,2sin(ωk,2t)uk+1,2 − ωk,2cos(ωk,2t)vk+1,2       −3b cos(ωk,1t)uk,1 − sin(ωk,1t)vk,1 + cos(ωk,2t)uk,2 − sin(ωk,2t)vk,2 − λ cos(ωk,1t)uk+1,1 − sin(ωk,1t)vk+1,1      2  +cos(ωk,2t)uk+1,2 − sin(ωk,2t)vk+1,2 − ωk,1sin(ωk,1t)uk,1 − ωk,1cos(ωk,1t)vk,1 − ωk,2sin(ωk,2t)uk,2       −ωk,2cos(ωk,2t)vk,2 − λ − ωk,1sin(ωk,1t)uk+1,1 − ωk,1cos(ωk,1t)vk+1,1 − ωk,2sin(ωk,2t)uk+1,2       −ωk,2cos(ωk,2t)vk+1,2 cos(ωk,2t).

R T Averaging the system (A.28), that is setting limT →∞ 0 on the right hand side of system (A.28), we obtain the averaged system given by 81

Ωk,1 (a−R1) a 3b 2 2 2 2 3b 2 2 u˙ k,1 = − vk,1 + uk,1 − uk+1,1λ − u + v + 2(u + v ) uk,1 + λ (3u + v )uk+1,1 2ωk,1 2L1 2L1 8L1 k,1 k,1 k,2 k,2 8L1 k,1 k,1

2 2 +2(uk,2 + vk,2)uk+1,1 + (4uk,2uk+1,2 + 2vk,1vk+1,1 + 4vk,2vk+1,2)uk,1

3bλ2 2 2 2 2 − 2u + 2v + 3u + v uk,1 + 4uk,2uk+1,2 + 2vk,1vk+1,1 + 4vk,2vk+1,2 uk+1,1 8L1 k+1,2 k+1,2 k+1,1 k+1,1

3bλ3 2 2 2 2 + 2u + 2v + u + v uk+1,1 8L1 k+1,2 k+1,2 k+1,1 k+1,1

Ωk,1 (a−R1) a 3b 2 2 2 2 3b 2 2 v˙k,1 = uk,1 + vk,1 − vk+1,1λ − u + v + 2(u + v ) vk,1 + λ (3v + u )vk+1,1 2ωk,1 2L1 2L1 8L1 k,1 k,1 k,2 k,2 8L1 k,1 k,1

2 2 +(2uk,2 + 2vk,2)vk+1,1 + (4uk,2uk+1,2 + 2uk,1uk+1,1 + 4vk,2vk+1,2)vk,1

3bλ2 2 2 2 2 − (u + 2u + 3v + 2v )vk,1 + (4uk,2uk+1,2 + 2uk,1uk+1,1 + 4vk,2vk+1,2)vk+1,1 8L1 k+1,1 k+1,2 k+1,1 k+1,2

3bλ3 2 2 2 2 + u + 2u + v + 2v vk+1,1 8L1 k+1,1 k+1,2 k+1,1 k+1,2 (A.29) Ωk,2 (a−R2) a 3b 2 2 2 2 3b 2 2 u˙ k,2 = − vk,2 + uk,2 − uk+1,2λ − u + v + 2(u + v ) uk,2 + λ (3u + v )uk+1,2 2ωk,2 2L2 2L2 8L2 k,2 k,2 k,1 k,1 8L2 k,2 k,2

2 2 +2(uk,1 + vk,1)uk+1,2 + (4uk,1uk+1,1 + 2vk,2vk+1,2 + 4vk,1vk+1,1)uk,2

3bλ2 2 2 2 2 − 2u + 2v + 3u + v uk,2 + 4uk,1uk+1,1 + 2vk,2vk+1,2 + 4vk,1vk+1,1 uk+1,2 8L2 k+1,1 k+1,1 k+1,2 k+1,2

3bλ3 2 2 2 2 + (2u + u + 2v + v )uk+1,2 8L2 k+1,1 k+1,2 k+1,1 k+1,2

Ωk,2 (a−R2) a 3b 2 2 2 2 3b 2 2 v˙k,2 = uk,2 + vk,2 − vk+1,2λ − u + v + 2(u + v ) vk,2 + λ (3v + u )vk+1,2 2ωk,2 2L2 2L2 8L2 k,2 k,2 k,1 k,1 8L2 k,2 k,2

2 2 +2(uk,1 + vk,1)vk+1,2 + (4uk,1uk+1,1 + 2uk,2uk+1,2 + 4vk,1vk+1,1)vk,2

3bλ2 2 2 2 2 − 2u + 2v + 3v + u vk,2 + 4uk,1uk+1,1 + 2uk,2uk+1,2 + 4vk,1vk+1,1 vk+1,2 8L2 k+1,1 k+1,1 k+1,2 k+1,2

3bλ3 2 2 2 2 + (2u + u + 2v + v )vk+1,2 8L2 k+1,1 k+1,2 k+1,1 k+1,2 82

VAN DER POL TRANSFORMATION: Bidirectional coupling

Consider the oscillators

2 d ik,j dik,j ik,j 2 dik,1 dik,2 Lk,j 2 + Rk,j + = a − 3b ik,1 + ik,2 − λ ik−1,1 + ik−1,2 + ik+1,1 + ik+1,2 + dt dt Ck,j dt dt (A.30)

dik−1,1 dik−1,2 dik+1,1 dik+1,2 −λ( dt + dt + dt + dt ) .

Setting k = 1, 2, ..., N, j = 1, 2, Lk,1 = L1, Lk,2 = L2, Rk,1 = R1, Rk,2 = R2, Ck,1 = C1 and Ck,2 = C2, then system (A.30) takes the form

2  d ik,1 dik,1 ik,1 2 dik,1 dik,2  L1 2 + R1 + = a − 3b ik,1 + ik,2 − λ ik−1,1 + ik−1,2 + ik+1,1 + ik+1,2 +  dt dt C1 dt dt     dik−1,1 dik−1,2 dik+1,1 dik+1,2  −λ( + + + )  dt dt dt dt (A.31)  2  d ik,2 dik,2 ik,2 2 dik,1 dik,2  L2 2 + R2 + = a − 3b ik,1 + ik,2 − λ ik−1,1 + ik−1,2 + ik+1,1 + ik+1,2 +  dt dt C2 dt dt     dik−1,1 dik−1,2 dik+1,1 dik+1,2 −λ( dt + dt + dt + dt ) .

Consider the invertible Van der Pol transformation

ωk−1,1 lk−1,1 ωk−1,1 0 uk−1,1 = cos( t)ik−1,1 − sin( t)i lk−1,1 ωk−1,1 lk−1,1 k−1,1

ωk,1 lk,1 ωk,1 0 uk,1 = cos( t)ik,1 − sin( t)i lk,1 ωk,1 lk,1 k,1

ωk−1,1 lk−1,1 ωk−1,1 0 vk−1,1 = −sin( t)ik−1,1 − cos( t)i lk−1,1 ωk−1,1 lk−1,1 k−1,1

ωk,1 lk,1 ωk,1 0 vk,1 = −sin( t)ik,1 − cos( t)i lk,1 ωk,1 lk,1 k,1 (A.32) ωk−1,2 lk−1,2 ωk−1,2 0 uk−1,2 = cos( t)ik−1,2 − sin( t)i lk−1,2 ωk−1,2 lk−1,2 k−1,2

ωk,2 lk,2 ωk,2 0 uk,2 = cos( t)ik,2 − sin( t)i lk,2 ωk,2 lk,2 k,2

ωk−1,2 lk−1,2 ωk,2 0 vk−1,2 = −sin( t)ik−1,2 − cos( t)i lk−1,2 ωk−1,2 lk−1,2 k−1,2

ωk,2 lk,2 ω2 0 vk,2 = −sin( t)ik,2 − cos( t)i . lk,2 ωk,2 lk,2 k,2 83 for k = 1, 2, ..., N. Thus,

 l ω2 ω  u˙ = − k,1 k,1 i + i00 sin( k,1 t)  k,1 ω l2 k,1 k,1 l  k,1 k,1 k,1     2  lk,1 ωk,1 00 ωk,1  v˙k,1 = − 2 ik,1 + i cos( t)  ωk,1 l k,1 lk,1  k,1 (A.33)  l ω2 ω  u˙ = − k,2 k,2 i + i00 sin( k,2 t)  k,2 ω l2 k,2 k,2 l  k,2 k,2 k,2     2  lk,2 ωk,2 00 ωk,2  v˙k,2 = − ω 2 ik,2 + ik,2 cos( l t). k,2 lk,2 k,2

Also we have,

ωk−1,1 ωk−1,1 0 ωk−1,1 ωk−1,1 ωk−1,1 ωk−1,1 ik−1,1 = cos( t)uk−1,1 − sin( t)vk−1,1 i = − sin( t)uk−1,1 − cos( t)vk−1,1 lk−1,1 lk−1,1 k−1,1 lk−1,1 lk−1,1 lk−1,1 lk−1,1

ωk,1 ωk,1 0 ωk,1 ωk,1 ωk,1 ωk,1 ik,1 = cos( t)uk,1 − sin( t)vk,1 i = − sin( t)uk,1 − cos( t)vk,1 lk,1 lk,1 k,1 lk,1 lk,1 lk,1 lk,1

ωk+1,1 ωk+1,1 0 ωk+1,1 ωk+1,1 ωk+1,1 ωk+1,1 ik+1,1 = cos( t)uk+1,1 − sin( t)vk+1,1 i = − sin( t)uk+1,1 − cos( t)vk+1,1 lk+1,1 lk+1,1 k+1,1 lk+1,1 lk+1,1 lk+1,1 lk+1,1 (A.34) ωk−1,2 ωk−1,2 0 ωk−1,2 ωk−1,2 ωk−1,2 ωk−1,2 ik−1,2 = cos( t)uk−1,2 − sin( t)vk−1,2 i = − sin( t)uk−1,2 − cos( t)vk−1,2 lk−1,2 lk−1,2 k−1,2 lk−1,2 lk−1,2 lk−1,2 lk−1,2

ωk,2 ωk,2 0 ωk,2 ωk,2 ωk,2 ωk,2 ik,2 = cos( t)uk,2 − sin( t)vk,2 i = − sin( t)uk,2 − cos( t)vk,2 lk,2 lk,2 k,2 lk,2 lk,2 lk,2 lk,2

ωk+1,2 ωk+1,2 0 ωk+1,2 ωk+1,2 ωk+1,2 ωk+1,2 ik+1,2 = cos( t)uk+1,2 − sin( t)vk+1,2 i = − sin( t)uk+1,2 − cos( t)vk+1,2. lk+1,2 lk+1,2 k+1,2 lk+1,2 lk+1,2 lk+1,2 lk+1,2

We consider the following perturbation of system (A.31)

 2 d ik,1 1 R1 dik,1 2 dik,1 dik,2  2 = − ik,1 + − + a − 3b ik,1 + ik,2 − λ ik−1,1 + ik−1,2 + ik+1,1 + ik+1,2 +  dt L1C1 L1 dt dt dt      dik−1,1 dik−1,2 dik+1,1 dik+1,2  −λ( + + + )  dt dt dt dt  (A.35)  2  d ik,2 1 R2 dik,2 2 dik,1 dik,2  2 = − ik,2 + − + a − 3b ik,1 + ik,2 − λ ik−1,1 + ik−1,2 + ik+1,1 + ik+1,2 +  dt L2C2 L2 dt dt dt       dik−1,1 dik−1,2 dik+1,1 dik+1,2  −λ( dt + dt + dt + dt ) ,

or equivalently

 2 d ik,1 1 0 0 0 0 0 0  2 = − ik,1 + f1(ik−1,1, ik,1, ik+1,1, ik−1,2, ik,2, ik+1,2, ik−1,1, ik,1, ik+1,1, ik−1,2, ik,2, ik+1,2, t)  dt L1C1 (A.36)  2  d ik,2 1 0 0 0 0 0 0  2 = − ik,2 + f2(ik−1,1, ik,1, ik+1,1, ik−1,2, ik,2, ik+1,2, i , i , i , i , i , i , t) dt L2C2 k−1,1 k,1 k+1,1 k−1,2 k,2 k+1,2

where di 2 di di f = − R1 k,1 + a − 3bi + i − λi + i + i + i k,1 + k,2 − 1 L1 dt k,1 k,2 k−1,1 k−1,2 k+1,1 k+1,2 dt dt dik−1,1 dik−1,2 dik+1,1 dik+1,2 λ( dt + dt + dt + dt ) 84 and di 2 di di f = − R2 k,2 + a − 3bi + i − λi + i + i + i k,1 + k,2 − 2 L2 dt k,1 k,2 k−1,1 k−1,2 k+1,1 k+1,2 dt dt dik−1,1 dik−1,2 dik+1,1 dik+1,2 λ( dt + dt + dt + dt ) .

2 1 2 1 2 2 2 2 Setting lk,1 = lk,2 = lk+1,1 = lk+1,2 = 1, ω = , ω = , ω − ω = Ωk,1, ω − ω = Ωk,2, 01 L1C1 02 L2C2 01 k,1 02 k,2 ωk,j = ωk+1,j , j = 1, 2 taking into account (A.36), then applying the transformation (A.32) and (A.34) in the system (A.33) and ﬁnally averaging the R T resulting system, that is setting limT →∞ 0 on the right hand side of the resulting system , we obtain the averaged system given by

Ωk,1 (a−R1) 3b 2 2 2 2 a u˙ k,1 = − vk,1 + uk,1 − u + v + 2 u + v uk,1 − λ uk−1,1 + uk+1,1 2ωk,1 2L1 8L1 k,1 k,1 k,2 k,2 2L1

3b + λ 2 uk,1uk−1,1 + uk,1uk+1,1 + 2uk,2uk−1,2 + 2uk,2uk+1,2 + vk,1vk−1,1 + vk,1vk+1,1 8L1

2 2 2 2 +2vk,2vk−1,2 + 2vk,2vk+1,2 uk,1 + uk,1 + 2uk,2 + vk,1 + 2vk,2 uk−1,1 + uk+1,1

3b 2 2 2 2 2 2 2 − λ u + 2uk−1,1uk+1,1 + 2u + 4uk−1,2uk+1,2 + u + 2u + v + 2vk−1,1vk+1,1 + 2v 8L1 k−1,1 k−1,2 k+1,1 k+1,2 k−1,1 k−1,2

2 2 +4vk−1,2vk+1,2 + vk+1,1 + 2vk+1,2 uk,1 + 2 uk,1uk−1,1 + uk,1uk+1,1 + 2uk,2uk−1,2 + 2uk,2uk+1,2 + vk,1vk−1,1

3b 3 2 2 +vk,1vk+1,1 + 2vk,2vk−1,2 + 2vk,2vk+1,2 uk−1,1 + uk+1,1 + λ u + 2uk−1,1uk+1,1 + 2u + 4uk−1,2uk+1,2 8L1 k−1,1 k−1,2

2 2 2 2 2 2 +uk+1,1 + 2uk+1,2 + vk−1,1 + 2vk−1,1vk+1,1 + 2vk−1,2 + 4vk−1,2vk+1,2 + vk+1,1 + 2vk+1,2 uk−1,1 + uk+1,1

(A.37)

Ωk,1 (a−R1) 3b 2 2 2 2 a v˙k,1 = uk,1 + vk,1 − u + v + 2 u + v vk,1 − λ vk−1,1 + vk+1,1 2ωk,1 2L1 8L1 k,1 k,1 k,2 k,2 2L1

3b + λ 2 uk,1uk−1,1 + uk,1uk+1,1 + 2uk,2uk−1,2 + 2uk,2uk+1,2 + vk,1vk−1,1 + vk,1vk+1,1 8L1

2 2 2 2 +2vk,2vk−1,2 + 2vk,2vk+1,2 vk,1 + uk,1 + 2uk,2 + vk,1 + 2vk,2 vk−1,1 + vk+1,1

3b 2 2 2 2 2 2 2 − λ u + 2uk−1,1uk+1,1 + 2u + 4uk−1,2uk+1,2 + u + 2u + v + 2vk−1,1vk+1,1 + 2v 8L1 k−1,1 k−1,2 k+1,1 k+1,2 k−1,1 k−1,2

2 2 +4vk−1,2vk+1,2 + vk+1,1 + 2vk+1,2 vk,1 + 2 uk,1uk−1,1 + uk,1uk+1,1 + 2uk,2uk−1,2 + 2uk,2uk+1,2 + vk,1vk−1,1

3b 3 2 2 +vk,1vk+1,1 + 2vk,2vk−1,2 + 2vk,2vk+1,2 vk−1,1 + vk+1,1 + λ u + 2uk−1,1uk+1,1 + 2u + 4uk−1,2uk+1,2 8L1 k−1,1 k−1,2

2 2 2 2 2 2 +uk+1,1 + 2uk+1,2 + vk−1,1 + 2vk−1,1vk+1,1 + 2vk−1,2 + 4vk−1,2vk+1,2 + vk+1,1 + 2vk+1,2 vk−1,1 + vk+1,1 85

Ωk,2 (a−R2) 3b 2 2 2 2 a u˙ k,2 = − vk,2 + uk,2 − u + v + 2 u + v uk,2 − λ uk−1,2 + uk+1,2 2ωk,2 2L2 8L2 k,2 k,2 k,1 k,1 2L2

3b + λ 2 2uk,1uk−1,1 + 2uk,1uk+1,1 + uk,2uk−1,2 + uk,2uk+1,2 + 2vk,1vk−1,1 + 2vk,1vk+1,1 8L2

2 2 2 2 +vk,2vk−1,2 + vk,2vk+1,2 uk,2 + 2uk,1 + uk,2 + 2vk,1 + vk,2 uk−1,2 + uk+1,2

3b 2 2 2 2 2 2 2 − λ 2u + 4uk−1,1uk+1,1 + u + 2uk−1,2uk+1,2 + 2u + u + 2v + 4vk−1,1vk+1,1 + v 8L2 k−1,1 k−1,2 k+1,1 k+1,2 k−1,1 k−1,2

2 2 +2vk−1,2vk+1,2 + 2vk+1,1 + vk+1,2 uk,2 + 2 2uk,1uk−1,1 + 2uk,1uk+1,1 + uk,2uk−1,2 + uk,2uk+1,2 + 2vk,1vk−1,1

3b 3 2 2 +2vk,1vk+1,1 + vk,2vk−1,2 + vk,2vk+1,2 uk−1,2 + uk+1,2 + λ 2u + 4uk−1,1uk+1,1 + u + 2uk−1,2uk+1,2 8L2 k−1,1 k−1,2

2 2 2 2 2 2 +2uk+1,1 + uk+1,2 + 2vk−1,1 + 4vk−1,1vk+1,1 + vk−1,2 + 2vk−1,2vk+1,2 + 2vk+1,1 + vk+1,2 uk−1,2 + uk+1,2 (A.38)

Ωk,2 (a−R2) 3b 2 2 2 2 a v˙k,2 = uk,2 + vk,2 − u + v + 2 u + v vk,2 − λ vk−1,2 + vk+1,2 + 2ωk,2 2L2 8L2 k,2 k,2 k,1 k,1 2L2

3b + λ 2 2uk,1uk−1,1 + 2uk,1uk+1,1 + uk,2uk−1,2 + uk,2uk+1,2 + 2vk,1vk−1,1 + 2vk,1vk+1,1 8L2

2 2 2 2 +vk,2vk−1,2 + vk,2vk+1,2 vk,2 + 2uk,1 + uk,2 + 2vk,1 + vk,2 vk−1,2 + vk+1,2

3b 2 2 2 2 2 2 2 − λ 2u + 4uk−1,1uk+1,1 + u + 2uk−1,2uk+1,2 + 2u + u + 2v + 4vk−1,1vk+1,1 + v 8L2 k−1,1 k−1,2 k+1,1 k+1,2 k−1,1 k−1,2

2 2 +2vk−1,2vk+1,2 + 2vk+1,1 + vk+1,2 vk,2 + 4uk,1uk−1,1 + 4uk,1uk+1,1 + 2uk,2uk−1,2 + 2uk,2uk+1,2 + 4vk,1vk−1,1

3b 3 2 2 +4vk,1vk+1,1 + 2vk,2vk−1,2 + 2vk,2vk+1,2 vk−1,2 + vk+1,2 + λ 2u + 4uk−1,1uk+1,1 + u + 2uk−1,2uk+1,2 8L2 k−1,1 k−1,2

2 2 2 2 2 2 +2uk+1,1 + uk+1,2 + 2vk−1,1 + 4vk−1,1vk+1,1 + vk−1,2 + 2vk−1,2vk+1,2 + 2vk+1,1 + vk+1,2 vk−1,2 + vk+1,2