Time Delay in the Swing Equation: a Variety of Bifurcations Time Delay in the Swing Equation: a Variety of Bifurcations T

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Time Delay in the Swing Equation: A Variety of Bifurcations Time Delay in the Swing Equation: A Variety of Bifurcations T. H. Scholl,1, a) L. Gröll,1, b) and V. Hagenmeyer1, c) Institute for Automation and Applied Informatics, Karlsruhe Institute of Technology, 76344 Eggenstein-Leopoldshafen, Germany (Dated: 2019) The present paper addresses the swing equation with additional delayed damping as an example for pendulum-like systems. In this context, it is proved that recurring sub- and supercritical Hopf bifurcations occur if time delay is increased. To this end, a general formula for the ﬁrst Lyapunov coefﬁcient in second order systems with additional delayed damping and delay-free nonlinearity is given. In so far the paper extends results about stability switching of equilibria in linear time delay systems from Cooke and Grossman. In addition to the analytical results, periodic solutions are numerically dealt with. The numerical results demonstrate how a variety of qualitative behaviors is generated in the simple swing equation by only introducing time delay in a damping term.

The swing equation is, e.g., at the core of any power system model. By delayed frequency control it becomes a pendulum equation with delayed damping - a system which is shown to exhibit highly complex dynamics. Repeating Hopf bifurcations lead again and again to the emergence of limit cycles as delay is increased. These limit cycles un- dergo further bifurcations including period doubling cascades and the birth of invariant tori.

I. INTRODUCTION

Second order ordinary differential equations with periodic nonlinearity describe various non-related systems such as syn- chronous generators or single-machine-infinite-bus (SMIB) equivalents of power systems, Josephson junction circuits, phase-locked loops or mechanical pendulum mechanisms1. Consider the scalar equation FIG. 1. Phase portrait of the delay-free dynamics (parameterization (11), τ = 0), i.e. damping coefficient a + a˜. y¨+ ay˙+ ks sin(y) = u (1) with an external input u. Either in order to increase damping or to control the frequency (if y is expressed w.r.t. a rotat- al.2. Hence, a thorough analysis of these dynamics is an open ing reference frame), u might involve a derivative-dependent question. feedback u = w−a˜y˙. However, due to measurement, commu- nication, data processing, or response behavior of actuators, Therefore, the present paper examines how the delay this controller action is more or less delayed. Consequently, changes the overall system dynamics and gives the respective damping with delay τ > 0 arises in results (throughout the paper, the term increase denotes not a time-varying sense but addresses another autonomous system arXiv:1908.07996v3 [math.DS] 8 Dec 2019 y¨(t) + ay˙(t) + a˜y˙(t − τ) + ks sin(y(t)) = w, (2) with changed parametrization). Dynamics of the delay-free 2 swing equation, or equivalently the driven pendulum equation a,a˜,ks,w > 0. Schaefer et al. show that these dynamics with constant torque, is well known3–7 (Figure 1). All might become of high technical relevance in a future power additional complex phenomena which are made possible by system, where time delay is caused by frequency-dependent time delay must successively emerge with increasing delay. reactions of the demand side. In this smart grid, y(t) ∈ R Hence, we start with the delay-free system and tackle the represents an aggregated relative generator angle w.r.t. a 2 problem by a bifurcation analysis with time delay as the bi- rotating reference . However, the thereby generated system furcation parameter. Foundation is laid by bifurcation theory dynamics are only described in a first attempt in Schaefer et of retarded functional differential equations (RFDEs)8,9.

The linearization of (2) belongs to the general class of a)Electronic mail: [email protected] linear second order systems with delayed damping10–17. For b)Electronic mail: [email protected] these linear systems, Cooke and Grossman10 have already c)Electronic mail: [email protected] proved alternate switching between stability and instability Time Delay in the Swing Equation: A Variety of Bifurcations 2 with increasing delay. Such switching points in the lineariza- more general system class are retarded functional differential n tion are candidates for Hopf bifurcations in the nonlinear equations (RFDE) x˙(t) = F(xt ) , F : X → R . A Cauchy prob- system (2). The latter is a well known fact from ordinary lem requires an initial state x0 = φ ∈ X, i.e. an initial function differential equations (ODEs) and indeed also applies to the segment. Obviously, the latter has to stem from the infinite RFDE (2). To the authors’ best knowledge, systems with dimensional state space which is usually a predefined Banach delayed damping and delay-free nonlinearity h(y(t)) have space X. A very common state space for time delay systems n not yet been subject to bifurcation analysis in literature. is the space of continuous functions C([−τ,0],R ) endowed Besides of delayed dampinga ˜y˙(t − τ),a ˜ ∈ R>0 Cooke and with the uniform norm kφkC = maxθ∈[−τ,0](kφ(θ)k2). Grossman10 have proved similar stability switching behavior A zero equilibrium of an autonomous RFDE is locally for linearized systems with delayed restoring forcecy ˜ (t − τ), asymptotically stable (LAS) in X, if it is stable in X, c˜ ∈ R>0. For this related system class, results are already i.e. ∀ε > 0, ∃δ(ε) > 0 : kφkX <δ ⇒ kxt kX <ε, ∀t ≥ 0 18 given by Campbell at al. , revealing not only limit cycles and locally attractive in X, i.e. ∃δa > 0 : kφkX <δa ⇒ but also invariant tori. Similar observations are also valid limt→∞ kxt kX = 0. The domain of attraction DX of an for Minorsky’s equation17, which in contrast to (2) exhibits asymptotically stable zero equilibrium is the set of all initial a derivative-dependent and overall delayed nonlinear term functions which lead to a zero-convergent solution, i.e. which does not depend on y(t). The occurrence of oscillatory DX = {φ ∈ X : xt exists on t ∈ [0,∞) and limt→∞ kxt kX = 0}. problems due to limit cycles19, invariant tori20,21 and even 21 period doubling cascades which lead to chaos are also The Principle of Linearized Stability (Diekmann8, Chap- relevant in the context of power systems. However, their ter VII, Theorem 6.8) allows to conclude stability proper- observation requires higher order models. The present paper ties of equilibria based on a respective linearization about aims to show that only by introducing time delay, the simple them. For well-posed autonomous linear RFDEs in a suited swing equation alone is able to reveal a rich variety of such Banach space, the family {T (t)}t≥0 of solution operators phenomena. T (t) : X → X, x0 7→ xt forms a C0-semigroup. The spectrum of its infinitesimal generator A is a pure point spectrum Moreover, the presented result on the first Lyapunov coeffi- σ(A ). These eigenvalues λ ∈ σ(A ) decide about stability cient (Theorem III.1) is valid for more general systems of the linear system and are equal to the characteristic roots. Such characteristic roots can be derived analogously to the y¨(t) + ay˙(t) + a˜y˙(t − τ) + h(y(t)) = 0. (3) ODE case, i.e. either based on Laplace transformation of the linear RFDE, or on the ansatz x(t) = veλt , v ∈ n. Concerning This structure arises whenever a feedback law u for a plant R limit cycle stability, a linearization about the periodic solution results in a linear time-periodic RFDE with family of solution y¨+ ay˙+ cy + h0(y) = u (4) operators {T (t,t0)}t≥t0 as an analogue to the state transition uses a delayed measurement of the derivative. For instance, a matrix in ODEs. It should be noted that a complete Floquet 9 PD-like controller with static feedforward w theory does not exist for RFDEs , but space decompositions w.r.t generalized eigenspaces are very helpful8,9. Similarly to u(t) = w − kPy(t) − kDy˙(t − τ) (5) ODEs, the Floquet multipliers, i.e. eigenvalues of the mon- odromy operator T (T,0), are decisive for stability of linear (state feedback realization with delay in x2 = y˙) results in (3) time-periodic RFDEs with period T. Thereby, instability or witha ˜ = kD and h(y) = cy + h0(y) − w + kPy. asymptotic stability in the sense of Poincaré can be concluded for the limit cycle of the nonlinear system. The paper is organized as follows. Subsequently, the used notation and needed preliminaries are given. Section II pro- vides basic insights to the system dynamics by an analysis of the linearized system. Section III addresses analytical as well II. BASIC ANALYSIS as numerical results on equilibrium bifurcations and a numerical treatment of limit cycle bifurcations. In a first step, we will introduce a state space description of (2). W.l.o.g. we assume ks := 1 in (2), since any other swing kˆ > A. Notation and Preliminaries equation with arbitrary s 0

00 ˆ 0 ˆ ˆ 0 ˆ ˆ ˆ We are concerned with an autonomous difference differ- yˆ (t) + aˆyˆ (t) + a˜yˆ (t − τˆ) + ks sin(yˆ(t)) = wˆ, (6) n ential equation22,23 x˙(t) = f (x(t),x(t − τ)), with x(t) ∈ R 0 p and a single discrete delay τ > 0. Consequently, not only (·) = d/(dtˆ) can be transformed by tˆ = t/ kˆs and n p the instantaneous value x(t) ∈ R which is an element of the yˆ(t/ kˆ ) = y(t) to (2) with dimensionless parameters n-dimensional Euclidean space is decisive for the dynamics. s Instead the delay-wide last solution segment x(t˜), t˜ ∈ [t − τ,t] ˆ q has to be considered. Such delay-spanning solution segments aˆ a˜ wˆ ks = 1, a = , a˜ = , w = , τ = τˆ kˆs. (7) def p ˆ p ˆ kˆ are commonly denoted as xt (θ) = x(t + θ), θ ∈ [−τ,0]. The ks ks s Time Delay in the Swing Equation: A Variety of Bifurcations 3

A. State Space Description B. Linearization Based Stability Analysis

First, we introduce a relative coordinate x1 = y − ye with Linearizing the delayed swing equation gives a linear sec- respect to some reference value ye. Hence, if ye is chosen to be ond order differential equation with delayed damping. For an equilibrium of interest, the common assumption, that this such systems a stability analysis can already be found in equilibrium lies in the origin, is already met in x-coordinates. literature10. The following lemma summarizes results about > For x = [y − ye,y˙] , ye ∈ R, a state space representation of (2) asymptotic stability and instability as far as the equilibrium is with ks = 1 reads hyperbolic.

10 0 1 0 0 0 Lemma II.1 (Stability Switching ) x˙(t) = x(t) + x(t − τ)+ 0 −a 0 −a˜ w − sin(x1(t) + ye) Consider the linear RFDE x(θ) = φ(θ), θ ∈ [−τ,0] 0 1 0 0 x˙(t) = x(t) + x(t − τ) (12) (8) −c −a 0 −a˜

2 for ﬁxed a,a˜ ∈ R>0,c ∈ R and with a variable delay parameter x(t) ∈ R , t ∈ R≥(−τ), a,a˜,w ∈ R>0, τ ∈ R≥0. Thereby, 2 τ > 0. φ ∈ C([−τ,0],R ) describes the initial function. (i) If c > 0, the system is asymptotically stable for τ = 0. (i-a) Ifa ˜ < a, the system remains asymptotically stable for We are interested in w ∈ (0,1], since for |w| > 1 no equilib- τ ≥ 0. rium point exists. By assigning the principal value (i-b) If a < a˜, there are two sequences of delay values (τ1n)n∈N and (τ2n)n∈N ye := arcsin(w) (9) 1 a τ1n = arccos − + 2πn (13a) as reference, the equilibrium y(t) ≡ arcsin(w) in (2) is shifted ω1 a˜ > to x(t) ≡ [0,0] in (8). Besides of y(t) ≡ ye +2kπ, k ∈ Z there 1 a are also equilibrium points in y(t) ≡ ( − y ) + 2k , k ∈ τ2n = −arccos − + 2π(n + 1) (13b) π e π Z ω2 a˜ (Figure 1) - unless w = 1 where both sets coincide. Con- sequently, in the coordinates of (8), equilibria are x(t) ≡ xe, with corresponding xe ∈ Ex with r 1p 2 2 1 2 2 ω1,2 = ± a˜ − a + c + (a˜ − a ) (14) 2kπ π − 2y 2kπ 2 4 = ,k ∈ ∪ e + ,k ∈ , Ex 0 Z 0 0 Z such that the system switches to instability at τ1n and back (10) to asymptotic stability at τ2n with increasing delay, as long as τ2n < τ1(n+1). If τ ∈ (τ jn)n∈N, j ∈ {1,2} the equilibrium 2 if w ∈ (0,1]. By X = C([−τ,0],R ) the codomain of φ,xt ,x is non-hyperbolic and a simple complex conjugate eigenvalue 2 has been chosen to be the plane such that x(t) ∈ R . In the pair {iω j,−iω j} occurs. plane, equilibria which differ in the angle value x1 by 2π are (ii) If c = 0, the equilibrium is non-hyperbolic. considered as distinct equilibria. Hence, a slip over before (iii) If c < 0, the system is unstable for τ = 0 and remains settling down to [2π,0]> will not contribute to the domain unstable ∀τ ≥ 0. > of attraction of xe = [0,0] , albeit the states are physically indistinguishable afterwards. This distinction would not be The reader is referred to Cooke and Grossman10 for a de- true on the cylinder 1 × 3 [x (t),x (t)]>, which is an S R 1 2 tailed proof. We will shortly motivate equations (13), (14) alternative and quite natural choice due to periodicity in the since we will later need some intermediate results. above system. Sketch of Proof for (i-b). Eigenvalues are roots of the charac- Unless otherwise stated, we align the parametrization with teristic quasipolynomial det(∆(λ)) with Schaefer al al.2. According to (6), the parameter seta ˆ = −1 ˆ −1 −2 ˆ −2 λ −1 0.1s , a˜ = 0.25s ,w ˆ = 2s , ks = 16s becomes ∆ : → 2×2, λ 7→ ∆(λ) := . (15) C C c λ + a + a˜e−λτ k = 1, a = 0.025, a˜ = 0.0625, w = 0.125. (11) s By continuity of the spectrum w.r.t. parameter changes (Michiels and Niculescu28, Theorem 1.15) in RFDEs, a stabil- We are aware that considerations of small time scales serve ity switch necessarily goes along with an eigenvalue λ cross- as a legitimization for the strong simpliﬁcation (2) as a power i ing the imaginary axis. Since system model24–27. However, in accordance with Schaefer at 2 al. we will use (2) also on large time scales in order to cre- λi 6= 0, if c > 0 (16) ate awareness for qualitative effects of a large delay term. A physical time delay of τˆ = 10s (Schaefer et al.2 2015, Figure this can only be accompanied by a complex conjugated eigen- 5) results in τ = 40. value pair on the imaginary axis {λi,λ i} = {iω,−iω}, ω > 0. Time Delay in the Swing Equation: A Variety of Bifurcations 4

Hence, if the delay value τ should be a candidate point for Proposition II.1 (Stability of Equilibria) stability switching, the characteristic equation det(∆(λ)) = 0 Consider the delayed swing equation (8) with w ∈ (0,1). > > must be fulﬁlled for λ = iω, ω > 0. Thereby, det(∆(iω)) = 0 (i) The equilibria xe = [0,0] +[2kπ,0] , k ∈ Z are asymptot- is equivalent to ically stable in the delay-free case τ = 0. (i-a) If undelayed damping is prevalent, i.e.a ˜ < a, these equi- iω(a + a˜e−iωτ ) = ω2 − c. (17) libria remain delay-independently stable. (i-b) If delayed damping is prevalent, i.e. a < a˜, these equi- Comparing imaginary parts of (17) libria switch stability with increasing delay. They are non- hyperbolic for delay sequences (τ1n)n∈N, (τ2n)n∈N given by a + a˜cos(ωτ) = 0 (18) (13) with

( ) ( ) p leads to sequences τ1n n∈N, τ2n n∈N with c = 1 − w2 (25) ω τ =arccos− a + 2πn (19) 1 1n a˜ and hyperbolic otherwise. In the hyperbolic case, if τ ∈ Θ , q u a2 ⇒ sin(ω1τ1n) = 1 − , (20) a˜2 [ a Θu = τ1n,τ2n ∪ τ1(nmax+1),∞ (26) ω2τ2n = − arccos − a˜ + 2π(n + 1) (21) n∈N q 2 n≤nmax ⇒ ( ) = − − a , sin ω2τ2n 1 a˜2 (22) with while comparing real parts of (17) and using (20), (22) gives

nmax = max{n ∈ N : τ2n < τ1(n+1)} (27) q 2 2 ∓ a − a − c = . ω ˜ 1 a˜2 ω 0 (23) the equilibria are unstable and if τ ∈ R≥0 \ Θu asymptotically These two equations for ω1 (−-case) and ω2 (+-case) with stable, where Θu denotes the closure of (26). 10 > > the restriction to ω1,2 > 0 are solved by (14). Furthermore , (ii) The equilibria in xe = [π − 2ye,0] + [2kπ,0] are delay- roots λ = iω are simple and independently unstable.

d dτ (ℜeλ) > 0 ∀τ ∈ (τ1n)n∈N, (24a) d Proof. The results are direct consequences of the Principle ( e ) < ∀ ∈ ( ) . 8 dτ ℜ λ 0 τ τ2n n∈N (24b) of Linearized Stability (Diekmann et al. , Chapter VII, The- orem 6.8). The linearization of the delayed swing equation The latter inequalities determine, whether roots cross the c = / x (x + y )| (8) is given by (12) with√ ∂ ∂ 1 sin 1 e x=xe . This is imaginary axis from left to right or from right to left, respec- 2 c = cos(arcsin(w)) = 1 − w for√ the delay-free stable equi- tively, when delay is increased. Starting with stability of the 2 delay-free system (Lemma II.1 (i)), it can be concluded how libria as considered in (i) and c = − 1 − w for the delay-free unstable equilibria in (ii). Stability properties of the linearized the number of eigenvalues in the open right half-plane C+ develops with increasing τ. systems follow from Proposition II.1.

Remark II.1 (D-Subdivision Method and Stability Charts) Figure 2 confirms the above result (i-b) by the numeri- Substitution of λ = γ + iω in the characteristic equation cally determined spectrum for the linearization about the triv- det∆(λ) = 0, comparing real and imaginary parts, then set- ial equilibrium. Delay values τ ∈ \ Θ , with Θ given ting γ = 0 in order to get ω-parametrized curves that sepa- R≥0 u u in (26), correspond to stability of the equilibrium point and rate γ < 0 from γ > 0 in the parameter plane (so-called D- are depicted green. It should be noted that there is only a curves) and finally analyzing the ascent direction of γ at those finite number of stable intervals, the last one with boundary curves by its partial derivative w.r.t. a system parameter, is τ . This is due to a different rate of increase of the known as D-subdivision method29. It goes back to Neimark30. 1(nmax+1) sequences (τ ) and (τ ) in (13), such that accord- D-curves (also called exponent-crossing curves or transition 1n n∈N 2n n∈N ing to (27), τ < τ but τ ≤ τ . Al- curves) separate the parameter space in regions with a con- 2nmax 1(nmax+1) 1(nmax+2) 2(nmax+1) beit at τ an eigenvalue pair indeed moves back to the stant number of unstable eigenvalues. For the most funda- 2(nmax+1) left half plane, at τ a second eigenvalue pair has al- mental linear RFDEs, the analytic stability criteria and corre- 1(nmax+2) ready reached the right half plane. Thus, instead of a switch- sponding stability charts in parameter planes are available in ing between no and one unstable eigenvalue pair, henceforth a literature, see for instance Stépán29 or Insperger and Stépán31. switching between one and two unstable eigenvalue pairs occurs, which in a similar manner will only last for a finite delay Our main interest focuses on the case of prevalent delayed range. The effect becomes more obvious with a larger coeffi- dampinga ˜. Indeed, the above described changes between be- cient of delayed dampinga ˜: in Figure 2 (b) the second white ing stable and unstable also occur for the equilibrium point marker is still left to the third dark marker, i.e. τ22 < τ13, but > xe = [0,0] in the nonlinear system (8). τ14 < τ23 and thus nmax = 2. Time Delay in the Swing Equation: A Variety of Bifurcations 5

Indeed, to mark a Hopf bifurcation is even a generic property of stability switching points with ω 6= 0, since they have already proven to obtain a pair of complex conjugate roots λ = ±iω 6= 0 which crosses the imaginary axis with increasing delay (transversality condition). Assume there is no other pair of roots simultaneously on the imaginary axis. The only circumstance that could hinder these stability switching points from being Hopf points is a vanishing first Lyapunov coefficient L = 0. The latter is referred to as degenerate case32. The linearized equation is an example of degeneracy. It is the pro- (a) parametrization (11), i.e.a ˜ = 0.0625 totype system of stability switching. However, obviously it cannot give rise to a limit cycle (which requires the closed orbit to be isolated). However, not only whether the Lyapunov coefficient is zero is of interest. The sign of the Lyapunov coefficient is responsible for the type of the Hopf bifurcation.

Definition III.1 (Sub- and Supercritical Hopf Bifurcation32) Let L denote the first Lyapunov coefficient in a Hopf bifurcation point. The Hopf bifurcation is called supercritical, if L < 0 and subcritical, if L > 0. (b) parametrization (11), buta ˜ = 0.225

FIG. 2. Numerically derived real parts of the most critical eigen- The term Hopf bifurcation is sometimes restricted to values for different delay values: their maximum defines the spectral switches of the equilibrium between being stable and unsta- abscissa. Lemma II.1 gives two sequences of delay values (τ1n)n∈ N ble as described above (obviously, in a system with only two (dark vertical lines) and (τ2n)n∈N (white vertical lines) at which the trivial equilibrium of (8) is non-hyperbolic, i.e. ∃λi : ℜeλi = 0. Col- eigenvalues, e.g., in the restriction to the two-dimensional ors indicate the number of unstable eigenvalues nu, which changes at center manifold, an eigenvalue pair cannot cross the imagi- these points. nary axis without such a stability switch). Then, the sign of the first Lyapunov coefficient has the following consequences:

III. BIFURCATION ANALYSIS Lemma III.1 (Sub- and Supercritical Hopf Bifurcation32) Consider a Hopf bifurcation at a stability switching point. The following implications are valid in a neighborhood of the bi- According to Proposition II.1, delay values (τ1n)n∈ , and N furcation point. (τ2n)n∈N in (13) mark points at which an eigenvalue pair crosses the imaginary axis. These delay values are candidates (i) L < 0 (supercritical) ⇔ a stable limit cycle occurs for bifur- for Hopf bifurcations. A Hopf bifurcation goes along with cation parameter values at which the equilibrium is unstable. the emergence of a limit cycle, meaning that either for some (ii) L > 0 (subcritical) ⇔ an unstable limit cycle occurs for bi- smaller or for some larger delay values a limit cycle surrounds furcation parameter values at which the equilibrium is stable. the equilibrium point. Thereby, a stability analysis - be it analytically (Lemma II.1) or numerically (Figure 2) - already reveals whether A. Analytic Equilibrium Bifurcation Analysis stability of the equilibrium point is given on the side of smaller delay values or on the side of larger delay values. There are different indicators in literature which describe the character of a Hopf bifurcation. In the following we use However, unstable eigenvalues can already exist when an the value of the first Lyapunov coefficient in the bifurcation eigenvalue pair crosses the imaginary axis. Consequently, point L := l1(0). For precise Hopf theorems in time delay there might be, besides of the center and stable manifold, also systems we refer to Diekmann et al.8 and Hale and Verduyn an unstable manifold in the bifurcation point. The equilibrium Lunel9. changes from being of type k, i.e. having k eigenvalues on the right half-plane, to type (k + 2) or vice versa. Then, Remark III.1 (Relation between the First Lyapunov Coefficient in analogy to Lemma III.1, a Hopf bifurcation can lead to and other Indicators) the emergence of a limit cycle with k or (k + 1) unstable For the relation to the coefficient of the resonant cubic term in Floquet multipliers. While system (2) has, dependent on the the Poincaré normal form see Kuznetsov32. For the relation parametrization, an arbitrary but finite number10 of stability to the second order coefficient in a series expansion of the switching points τ1n, n ≤ nmax + 1, τ2n, n ≤ nmax, there is 33 34 Floquet exponent see Hassard et al. and Kazarinoff et al. . actually an infinite number of non-hyperbolic points τ1n, It should be noted that all three mentioned numbers obtain the τ2n, n ∈ N at which a pair of complex eigenvalues crosses same sign, which is actually the decisive value. the imaginary axis. All delay values in both sequences are Time Delay in the Swing Equation: A Variety of Bifurcations 6 candidates for Hopf bifurcations in this extended sense. Proposition III.1 Consider the delayed swing equation (2), (11). Let a < a˜ and ∈ ({ } ∪ { } ) \ ({ } ∩ { } ) In the following Theorem, we give a result for the sign τ τ1n n∈N τ2n n√∈N τ1n n∈N τ2n n∈N with τ jn of the first Lyapunov coefficient which is decisive according given by (13) and c := 1 − w2. There occur to Lemma III.1. We consider the general case of second order differential equations with delayed damping and arbitrary, • supercritical Hopf bifurcations (L < 0) at τ ∈ (τ1n)n∈N, delay-free nonlinearity. • subcritical Hopf bifurcations (L > 0) at τ ∈ (τ2n)n∈N. Theorem III.1 (First Lyapunov Coefficient for a General Class of Systems) Proof. The delayed swing equation (2) is equivalent to sys- Consider the RFDE with delayed damping and delay- and tem (28) with h(y) = sin(y) − w. The equilibrium of inter- derivative-free nonlinearity est is ye = arcsin√ (w) and derivatives in this point become c = 0 2 00 000 h = cosye = 1 − w , h = −sinye = −w, h = −cosye = y¨(t) + a˜y˙(t − τ) + ay˙(t) + h(y(t)) = 0, (28) e√ e e − 1 − w2. For the chosen parameters (11), the second sum- ω mand in (30) is dominant and thus (34) is decisive. To be more y(t) ∈ R,a ˜,a,τ ∈ R>0, a < a˜, h ∈ C (R,R). Let ye ∈ R such 0 ∂h precise, a numerical evaluation of (30) gives that h(ye) = 0 and he := ∂y |ye > 0 and ( (0.692n+0.260)+(−149.155n−47.057) , ∈ ( ) sgn n2+4.691n+27.137 τ τ1n n∈N τ ∈ ({τ1n}n∈ ∪ {τ2n}n∈ ) \ ({τ1n}n∈ ∩ {τ2n}n∈ ) (29) sgn L = N N N N (−0.899n−0.552)+(157.982n+108.140) , ∈ ( ) . sgn n2+5.429n+29.004 τ τ2n n∈N 0 with τ jn given by (13) and c := he. Then, the sign of the first Lyapunov coefficient is determined by " Hence, where elements in (τ ) alternate with elements 1 1n n∈N sgn L = sgn ℜe of (τ2n)n∈N, sub- and supercritical Hopf bifurcations alternate β det∆(2iω) as well. Consequently, all bifurcations are equally orientated 000 # in the following sense. 1 2 he + ℜe 0 − 00 2 (30) β he (he ) Corollary III.1 (Direction of Limit Cycle Continuation) For each Hopf bifurcation point τ jn in (2) with (11), existence with and increase of the emerging limit cycle is locally ensured on the side of larger delay values τ > τ jn. 2 0 0 2 β = − (ωτ)(ω − he) + i((ωτ)ωa + he + ω ) (31) and Proof. Due to (24), the eigenvalue pair crosses the imaginary axis from left to right if τ ∈ (τ1n)n∈ . Hence, a larger delay a N det∆(2iω) = − 3ω2 − 1 + 4 (ω2 − h0 ) value goes along with two more unstable eigenvalues. A Hopf a˜ e bifurcation with negative first Lyapunov coefficient L < 0 re- a2 sults in a limit cycle on this side (Lemma III.1 (i)). In contrast, + 2iω a − a˜ + 2 , (32) a˜ if τ ∈ (τ2n)n∈N, eigenvalues cross from right to left. Thus, the side of larger delays goes along with two more stable eigen- 0 00 000 where he,he ,he denote respective derivatives of h evaluated values. However, since the sign of the first Lyapunov coeffi- in the equilibrium point ye. cient is also reversed, i.e. L > 0 at τ ∈ (τ2n)n∈N, the emerging limit cycle occurs again for larger delay values (Lemma III.1 (ii)). Proof. The proof can be found in the Appendix. Figure 3 is based on the above analytic results for (2), (11). It should be noted, that, due to (31), in the second summand It considers the number of unstable eigenvalues not only de- of (30) the term pendent on τ but also on a variation ofa ˜. Positions of bifurcation delay values (13) change if another parameter likea ˜ is 1 ℜe(β) sgn ℜe = sgn = −sgn(ω2 − c) (33) varied. Thereby, possible points of codimension two bifur- β |β|2 cations become visible. However, there are some further non- degeneracy conditions for these types of bifurcations. Accord- 0 with c := he is decisive. The latter depends on whether the ing to the numerical bifurcation tool DDE-BIFTOOL35 these bifurcation point is an element of (τ1n)n∈N (hence ω = ω1) or conditions indeed hold in the points we have examined. collected in (τ2n)n∈N (hence ω = ω2), since according to (23) 2 2 ω1 − c > 0 and ω2 − c < 0. Consequently, • Hopf-Hopf bifurcations (also called two-pair bifurac- tion, double-Hopf bifurcation) in the (τ,a˜) plane or in 1 −1, τ ∈ (τ ) the (τ,w) plane become possible if a,a ˜ and w are cho- sgn ℜe = 1n n∈N (34) β +1, τ ∈ (τ2n)n∈N. sen such that a < a˜ and τ ∈ ({τ1n}n∈N ∩ {τ2n}n∈N). Time Delay in the Swing Equation: A Variety of Bifurcations 7

B. Numeric Approximations of Emerging Limit Cycles

A numerical analysis does not only determine normal form coefﬁcients and thus the type of bifurcation. It also allows to track limit cycles which emerge from Hopf bifurcations and to approximate the corresponding periodic solution. The MATLAB Toolbox DDE-BIFTOOL35 is a valuable tool for this purpose. Figure 4 visualizes our results.

The colored surface in Figure 4 describes the equilibrium (a) manifold in the (τ,w,x1) space. Assume parameter values τ and w are given, then the x1 coordinates of equilibria can be read off. Stability ranges, which are indicated by color, are due to analytic results (Proposition II.1). The upper equilibrium branch is delay independently unstable (one unstable eigenvalue), while the lower branch switches its stability property with increasing delay. For w > 1 no equilibrium exists. Not only in the delay-free system, but also for τ > 0, the upper and lower equilibrium branch collide in a (b) detail fold bifurcation at w = 1. We consider an equilibrium branch > > along [τ,w,x1] = [τ,0.125,0] with increasing τ-values. At FIG. 3. Colors indicate the analytically derived number n of un- u the Hopf bifurcation points τ ∈ (τin)n∈N, i ∈ {1,2} a limit stable equilibrium eigenvalues in the (τ,a˜) parameter plane (other cycle branches off. Maximum and minimum x1 values of the parameters according to (11)). The separating black and white lines corresponding periodic solutions are plotted. mark Hopf points in (τ1n)n∈N and (τ2n)n∈N, respectively (cmp. vertical lines in Figure 2). At their intersection points, Hopf-Hopf bifurcations occur. These Hopf-Hopf equilibrium bifurcation points form Thereby, the meaning of alternating super- and subcriti- starting points of Neimark-Sacker bifurcation curves of limit cycles, cal Hopf bifurcations (Proposition III.1) becomes clear. For > > at which invariant tori emerge. The numerically derived dashed lines τ < τ1,1 the equilibrium in [τ,w,x1] = [τ,0.125,0] is sta- in (b) refer to those Neimark-Sacker bifurcations which have been ble. For τ1,1 ∈ (τ1n)n∈N, n ≤ nmax + 1 stability is lost and a detected in Figure 4 witha ˜ = 0.0625 (non-transparent line in (a) and supercritical Hopf bifurcation is responsible for the occurence (b)). of a stable limit cycle at larger delay values. The limit cycle surrounds the henceforth unstable equilibrium. Of course, only locally, i.e. for delay values nearby the bifurcation point, Such intersection points of τ1n and τ2n curves in the stability and even existence of the limit cycle are granted. At (τ,a˜) plane are marked in Figure 3. At these param- parameter values τ2,1 ∈ (τ2n)n∈N, n ≤ nmax the equilibrium re- eter combinations, two pairs of purely imaginary eigen- gains stability. Again, a Hopf bifurcation causes the branch values ±iω1, ±iω2 cross simultaneously the imaginary off of a limit cycle. However, this time, it does not surround axis in opposite directions. the unstable equilibrium which would mean the limit cycle exists for delay values smaller than τ2,1. Instead, the limit • Bautin bifurcations (also called degenerate Hopf bifur- cycle occurs again for larger delay values (Corollary III.1) - cation, generalized Hopf bifurcation) in the (τ,a˜) plane now surrounding the stable equilibrium as result of a subcritical Hopf bifurcation. This pattern will repeat for larger delay are possible if a = a˜. Then, at τ = (1−w2)−1/2π(2n+ n values. The number of detected coexisting limit cycles rises 1), n ∈ an eigenvalue pair lies on the imaginary axis, N with increasing delay. but the Lyapunov coefﬁcient vanishes (Figure 3). Remark III.2 (Consequences for the Domain of Attraction) • Fold-Hopf bifurcations (also called zero-Hopf, saddle- Domains of attraction are essential in power system stability node Hopf, Gavrilov-Guckenheimer bifurcation) in the analysis. Results of Schaefer et al.2 suggest that for certain (τ,w) plane become possible if w = 1 (cmp. top view delay ranges an increasing time delay is able to enlarge (in the of Figure 4) anda ˜ > a. Due to (25) c = 0. Hence, λ = 0 sense of a generalized basin stability38,39) the domain of at- becomes an√ eigenvalue (cmp. (16)), while at {τ1n}n∈N traction of the delay-free stable equilibrium point. This seems 2 2 with ω1 = a˜ − a an eigenvalue pair is simultane- to be in accordance with the above detected growing limit ously on the imaginary axis. cycles. Indeed, in second order ODEs a limit cycle which stems from a Hopf bifurcation would clearly mark the bound- Furthermore, in Figure 3 Hopf bifurcation curves bound the ary of the domain of attraction for parameter values nearby. green marked area of stability. Such Christmas tree like sta- Hence, the detected growing limit cycles would go along with bility charts manifest the possibility of stability switching and a growing domain of attraction as the bifurcation parameter are well known in time delay systems36,37. increases. However, inﬁnite dimensionality of time delay sys- Time Delay in the Swing Equation: A Variety of Bifurcations 8

FIG. 4. The bended surface corresponds to y = ye + x1 coordinates of equilibria over the (τ,w) parameter plane. Thereby, colors indicate the number of unstable eigenvalues. Additionally, for ﬁxed parameter values w ∈ {0.125,0.8}, bold, striped horizontal lines are drawn along the lower equilibrium surface (the ﬁrst one corresponds to Figure 2 (a) and can also be found in Figure 3). They emphasize that the number of unstable eigenvalues changes with increasing delay. Where Hopf bifurcations occur along such lines, limit cycles emerge. On the vertical, white (τ,y) planes, maximal and minimal x1 coordinates of these limit cycle solutions are given. Line colors are chosen according to the number of unstable Floquet-Multipliers. This limit cycle representation equals the outline of a top view in the representation of Figure 5 with different τ-scaling. Letters mark detected limit cycle bifurcations like Neimark-Sacker (N), period doubling (D) or limit cycle fold (F).

tems makes meaningful conclusions of this manner hardly (x1,x2) plane. possible. There is no reason why solutions which are closer to the attractor should not diverge and cross the limit cycle in • A homoclinic bifurcation becomes visible as the grow- the (x1,x2) plane, i.e. in the (x1t (0),x2t (0)) projection of the ing limit cycle evolves to a homoclinic orbit of the ad- infinite dimensional state space. Nevertheless, the results en- jacent equilibrium, i.e. T → ∞ (Figure 5). able us to gain upper bounds on the radius of attraction. Such upper bounds are needed in order to prove that a minimum Besides the location, numeric approximations of the Floquet norm requirement on the domain of attraction is not fulfilled. multipliers allow to conclude stability of limit cycles. As We refer to Scholl et al.40, Example VI.1. in ODEs, there is always the trivial Floquet multiplier at µ = 1+0i, which does not affect stability of the limit cycle42. However, in the course of a bifurcation parameter variation, a further Floquet multiplier might come to be at the the stabil- C. Numeric Results on Bifurcations of Limit Cycles ity boundary, the unit cycle in the C plane. Again this cannot happen without a major qualitative change in the system The Hopf theorem does only guarantee stability properties behavior, i.e. a bifurcation. The following well known types and existence of a limit cycle locally in a small one-sided of limit cycle bifurcations have been detected in the delayed neighborhood of the bifurcation point. However, in limit swing equation (Figure 4): cycle bifurcations, stability and even existence of limit cycles can get lost for larger parameter variations. • In a limit cycle fold bifurcation (tangent bifurcation, saddle-node bifurcation, limit point) a stable and an un- Based on a two-point boundary value problem, with bound- stable limit cycle collide, i.e. at the bifurcation point, ary condition x0 = xT formulating periodicity, a numeric ap- there remains only one limit cycle. Thereby, a (further, proximation of the T-periodic solution as well as the unknown besides the trivial one) Floquet multiplier is on the unit 41 period T can be derived . Figure 5 visualizes how the limit cycle at µc = 1 + 0i. By larger parameter variations the cycle, which emerges in the first Hopf bifurcation of (8),(11) limit cycle disappears at all. On the back plane with at τ = τ11, develops with increasing delay τ: for each τ value, w = 0.8 in Figure 4, such a fold bifurcation is visible at the corresponding vertical slice shows the limit cycle in the the point where the first two limit cycles collide. Time Delay in the Swing Equation: A Variety of Bifurcations 9

counterpart to the Hopf bifurcation of an equilibrium. It occurs if a complex conjugate Floquet multiplier pair crosses the unit cycle at e±iϕ with eikϕ 6= 1 + 0i for k ∈ {1,2,3,4}. While in a Hopf bifurcation of an equilibrium a limit cycle evolves which encircles the equilibrium, by a Neimark-Sacker bifurcation an invariant torus around the limit cycle might result. How the position of Neimark-Sacker bifurcation points in Figure 4 changes ifa ˜ varies, is given in Figure 3 (b).

IV. CONCLUSIONS

When time delay is increased, linear second order systems with additional delayed damping or restoring force are able to show again and again characteristic roots crossing the imaginary axis. In nonlinearily perturbed systems each crossing is a candidate for a Hopf bifurcation point, which is further char- acterized by the sign of the first Lyapunov coefficient. The present paper gives a formula of the latter for general second order differential equations with additional delayed damping FIG. 5. For a given delay value τ, a vertical slice shows the ap- and arbitrary, delay-free nonlinearity. The result is applied to the delayed swing equation. It allows to prove, that sub- and pearance of the limit cycle in the (x1,x2) plane. Color indicates the elapsed time, if the limit cycle solution segment with φ1(0) = 0, supercritical Hopf bifurcations are combined in such a way φ2(0) ≥ 0 is chosen as initial function φ. At the bottom, the extreme that all emerging limit cycle branches head towards the direc- x1 values are sketched. These correspond to the first arc at w = 0.125 tion of larger delays. As a consequence, more and more limit in Figure 4. The limit cycle emerges at τ = 1.93 due to a super- cycles are likely to coexist with increasing delay. These limit critical Hopf bifurcation. At τ = 3.26 it looses stability in a period cycles are tracked by numeric means and respective graphical doubling bifurcation. A stable limit cycle branches off. The original representations are given. The latter show how due to further limit cycle as well as its period doubling branch off grow and evolve limit cycle bifurcations a surprisingly rich variety of compli- = . to homoclinic orbits of the type-one saddle equilibrium at τ 3 29 cated dynamics results - including invariant tori and period and τ = 3.83, respectively. doubling cascades. Summarizing, bifurcations of the following types are realized: Hopf, Bautin, Hopf-Hopf, fold, limit • Period doubling bifurcations (flip bifurcations) go along cycle fold, homoclinic, period doubling and Neimark-Sacker. with a crossing of the unit cycle at the opposite side: Hence, time delay already generates in the simple swing equation phenomena which are only known from higher order µc = −1 + 0i. A new limit cycle with double period branches off. Figure 6 shows a sequence of period dou- power system models. bling bifurcations. Delay values and limit cycle periods of consecutive period doubling bifurcations are marked by black squares in Figure 6c. The linear increase of ACKNOWLEDGMENTS these points in the double logarithmic diagram does not only show the doublings of limit cycle periods (i.e. The authors acknowledge support by the Helmholtz Asso- the solution over two periods does no longer consist of ciation under the Joint Initiative "Energy Systems Integration" two equal solution segments - thus twice as many en- (ZT-0002). circlings of the equilibrium are needed to describe the limit cycle). It also reveals the geometric progression of bifurcation points: an accelerating succession of bi- V. APPENDIX furcations can be observed as a certain delay value τ∞ is approached. The horizontal axis of Figure 6c describes time delay τ in terms of the difference to this reference In order to prove Theorem III.1, first the following Lemma value τ∞. Linearity in the logarithmic representation is recalled in the general case. uncovers the exponential behavior. It is well known, that period doubling cascades form a possible route to Lemma V.1 (First Lyapunov Coefficient43,44) chaos. Whether the cascade has indeed infinitely many Consider a nonlinear RFDE with m discrete delays elements and hence leads to a non-periodic (T = ∞) limit set at τ = τ∞, cannot be inferred from our results. x˙(t) = f (X), (35) n×(m+1) • A Neimark-Sacker bifurcation (torus bifurcation) is the X = [x(t),x(t − τ1),...,x(t − τm)] ∈ R , Time Delay in the Swing Equation: A Variety of Bifurcations 10

Assume the simple pair of purely imaginary eigenvalues is λ1,2 = ±iω, given as roots of the characteristic equation n det∆(λ) = 0. Let p,q ∈ R fulﬁll p>∆(iω) = 0, ∆(iω)q = 0 with p>D∆(iω)q = 1. (37) Assign

n n×(m+1) M : C([−τ,0],R ) → R , φ 7→ M(φ) := [φ(0),φ(−τ1),...,φ(−τm)],

i.e. X = M(xt ). Then, the ﬁrst Lyapunov coefﬁcient is given by 1 h L = ℜe p> D2 f (0 )(M(φ),M(h )) ω n×(m+1) 11 1 + D2 f (0 )(M(φ),M(h )) 2 n×(m+1) 20 1 i + D3 f (0 )(M(φ),M(φ),M(φ)) , 2 n×(m+1) (38)

(a) Poincaré section where φ(θ) = eiωθ q −1 2 h11(θ) = ∆ (0) D f (0n×(m+1))(M(φ),M(φ)) 2iωθ −1 2 h20(θ) = e ∆ (2iω) D f (0n×(m+1))(M(φ),M(φ)). (39)

Based on Lemma V.1, we are able to state the proof of The- orem III.1. Proof of Theorem III.1. Consider the state space representa- > tion of (28) in x(t) = [x1(t),x2(t)] (b) Limit cycles in the (x1,x2) plane (c) Delay values τ of period for τ = 12.25 and position of the doubling bifurcations (black x˙ (t) = x (t) (40a) Poincaré section in (a). Points mark squares) and periods T of the 1 2 the two equilibria. emerging limit cycles as delay x˙2(t) = −ax2(t) − ax˜ 2(t − τ) − h(x1(t)) (40b) increases. The horizontal axis > describes the delay range with Taylor expansion about xe = [0,0] τ ∈ [5,12.21306] in a logarithmic scaling w.r.t τ∞ := 12.21308. 0 1 0 0 x˙(t) = x(t) + x(t − τ) −h0 −a 0 −a˜ FIG. 6. A ﬁnite dimensional Poincaré section (as black marked cut- e ting edges) of a period doubling cascade is shown in (a). Depicted is 0 a horizontal cut through limit cycles from 6 subsequent period dou- + 1 00 2 1 000 3 4 . (41) − 2 he x1(t) − 3! he x1(t) + O(x1) bling bifurcations. In contrast to simulatively derived Poincaré sec- tions, the unstable limit cycles remain visible. The perspective in (a) (k) where he denotes the k-th derivative of h(x ) in x = 0. In is similar to Figure 5 albeit ∈ [11.7,12.25], x < 0, x > 2.1. This 1 1 τ 2 1 order to evaluate (37), the characteristic matrix given by (15) time, extreme x1 values of the full limit cycles refer to the second 0 w = . with c := he is considered. Due to (17), an evaluation at λ = green arc at 0 125 in Figure 4. −iωτ c iω can be simpliﬁed by a + a˜e = −iω + i ω . It follows iω −1 n ω n×(m+1) n ( ) = x(t) ∈ R , f ∈ C (R ,R ) , f (0n×(m+1)) = 0n in a ∆ iω c (42) c i ω Hopf bifurcation point with Taylor expansion about Xe = 0 > > > n×(m+1) with ker∆ (iω) = ic ω C and ker∆(iω) = 1 iω C. According to (37) normalization factors αp,αq ∈ C, have to 1 2 be chosen, such that x˙(t) =D f (0n×(m+1))(X) + D f (0n×(m+1))(X,X) 2 1 > > 1 + D3 f (0 )(X,X,X) + O(X4). (36) p := αp ic ω , q := αq (43) 3! n×(m+1) iω Time Delay in the Swing Equation: A Variety of Bifurcations 11

! fulﬁll a relative normalization p>D∆(iω)q = 1, where the by derivative of the characteristic matrix is given by det∆(2iω) = − 4ω2 + 2iωa + 2iωa˜(e−iωτ )2 + c (50) 2i = − 4ω2 + 2iωa − (ω2 − c − iωa)2 + c (51) 1 0 ωa˜ D∆(λ) = − . (44) a 0 1 − τa˜e λτ = − 3ω2 − (ω2 − c) − 4 (ω2 − c) a˜ a (ω2 − c)2 + 2i ωa 1 + − (52) Hence, the normalization condition can be written with (43) a˜ ωa˜ as a = − 3ω2 − 1 + 4 (ω2 − c) a˜ a2 a2 1 ! 1 0 1 + 2i ω a + − ωa˜ 1 − (53) = ic ω −iωτ . (45) a a2 αqαp 0 1 − τa˜e iω ˜ ˜ a = − 3ω2 − 1 + 4 (ω2 − c) a˜ Again, relation (17) yields a simpliﬁcation iω(a˜e−iωτ ) = a2 2 + 2iω a − a˜ + 2 . (54) ω − c − iωa, such that a˜

With p2 = αpω, ω ∈ R and q1 = αq, (47) results in 1 ! 2 2 = = c + − ( − c − a) h 00 1 00 β : i iω ωτ ω iω L =ℜe α − h φ (0)h (0) − h φ (0)h (0) αqαp p e 1 11,1 2 e 1 20,1 = − (ωτ)(ω2 − c) + i((ωτ)ωa + c + ω2). (46) 1 i − h000φ (0)2φ (0) 2 e 1 1 " 00 1 00 1 00 1 00 2 Since the nonlinearity h in (40) and thus also the higher order =ℜe αp h αq h αqαq + h αq h α e c e 2 e det∆(2iω) e q terms in (41) depend only on x1(t), i.e. on X11 in (35) and (36), equation (38) results in #! 1 − h000α2α . (55) 2 e q q " 1 > 0 1 0 According to (46), α α = β −1, and with c = h0 the ﬁrst Lya- L = ℜe p 00 + 00 p q e ω −he φ1(0)h11,1(0) 2 −he φ 1(0)h20,1(0) punov coefﬁcient can be evaluated as

#! " 000 #! 1 0 1 −1 2 00 2 2 1 he + 000 . (47) L =ℜe β |α | (h ) + − . −h φ (0)2φ (0) q e 0 00 2 2 e 1 1 2 he det∆(2iω) (he ) (56)

> > The decisive terms for the sign of L are stated in (30). Let e1 = [1,0] , e2 = [0,1] , then according to (39) Remark V.1 (Noninvariance of L) L 0 The value of the first Lyapunov coefficient in (56) depends φ (0) = q , h (0) = e>∆−1(0) , 1 1 11,1 1 −h00q q on |αq|, i.e. on the arbitrary normalization of q. However, this e 1 1 ambiguity does not affect the sign of L, which alone is decisive 32 > −1 0 in bifurcation analysis (cmp. Kuznetsov , page 99). h20,1(0) = e1 ∆ (2iω) 00 , (48) −he q1q1 Remark V.2 (Transformed Equation) There are some mathematical issues discussed in Hale and where q q = q q = α2 and 1 1 1 1 q Verduyn Lunel9 if the delay value is considered as bifurcation parameter. However, by a time scale transformation t = tˆτ in combination with y(tˆτ) = yˆ(tˆ), it can be achieved that the bi- 0 −1 2iω −1 ∆(0) = , ∆(2iω) = furcation parameter τ contributes only to coefficients, while c a + a˜ c + a + a( −iωτ )2 2iω ˜ e the delay value is fixed at one in > −1 1 > −1 1 00 0 0 2 2 e ∆ (0)e2 = , e ∆ (2iω)e2 = . (49) yˆ (tˆ) + aτyˆ (tˆ) + a˜τyˆ (tˆ− 1) + τ sin(yˆ(tˆ))) = τ w. (57) 1 c 1 det∆(2iω) Analogously to (17), the characteristic equation at purely imaginary roots λˆ = iωˆ is a( −iωτ ) = 2 − c − a Solving (17) for iω ˜ e ω iω and (23) for −iω 2 2 (ω2 − c)2 = a˜(1 − (a/a˜)2), the needed determinant is given iωτˆ (a + a˜e ) = ω − τ c. (58) Time Delay in the Swing Equation: A Variety of Bifurcations 12

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