Available at: http://www.ictp.it/~pub− off IC/2006/048

United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

STRANGE ATTRACTORS AND SYNCHRONIZATION DYNAMICS OF COUPLED VAN DER POL-DUFFING OSCILLATORS

Ren´eYamapi1 Department of Physics, Faculty of Sciences, University of Douala, P.O. Box 24157, Douala, Cameroon and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

and

Giovanni Filatrella2 Laboratorio Regionale SuperMat, INFM/CNR Salerno and Dipartimento di Scienze Biologiche ed Ambientali, Universit`adel Sannio, via Port’Arsa 11, I-82100 Benevento, Italy.

Abstract

We consider in this paper the dynamics and synchronization of coupled chaotic Van der Pol- Duffing systems. The stability of the synchronization process between two coupled autonomous Van der Pol model is first analyzed analytically and numerically, before following the problem of synchronizing chaos both on the same and different chaotic orbits of two coupled Van der Pol-Duffing systems. The stability boundaries of the synchronization process are derived and the effects of the amplitude of the periodic perturbation of the coupling parameter on these boundaries are analyzed. The results are provided on the stability map in the (q, K) plane.

MIRAMARE – TRIESTE July 2006

1Junior Associate of ICTP. Corresponding author: [email protected] 2fi[email protected] 1 Introduction

In recent years, extensive investigations have been carried out to analyze chaotic synchro- nization dynamics of coupled non-linear oscillators [1]. The idea of synchronizing two chaotic oscillators was presented by Pecora et al. [2] in 1990 by coupling both oscillators with a common drive signal. The idea has in fact been successfully tested in a variety of nonlinear dynamical systems, including Lorentz equation, the R¨osseler system and hysteresis circuits. The interest devoted to such topics is due to potential applications of synchronization in communication engineering (using chaos to mask the information signals) [2, 3, 4, 5], electrical and automation engineering, biology, and chemistry [6, 7, 8]. Chaotic orbits are known to be sensitive to initial conditions. It is in indeed surprising to find that two chaotic orbits which are initiated differently could be brought to synchronize with each other [2]. Recent studies on their synchronization process have been carried out and various coupling schemes have been employed [9, 10]. The synchronization can exhibit interesting phenomena such as phase-locking and cluster phase synchronization [11, 12, 13]. Considering the synchronization dynamics of coupled non-linear oscillators on the same and different chaotic orbits, Leung has recently analyzed in Ref. [14] the synchronization dynamics of coupled driven Van der Pol systems. The author shows that for a given set of parameters, a driven oscillator could possess two types of chaotic attractors. With different initial conditions, the results are the appearance of two trajectories which have inversion symmetry with respect to each other. The possibility to synchronize a chaotic trajectory with itself and with different ones have been probed. Here we aim to shed some light on the issue for oscillators in the chaotic states. A class of self-sustained oscillators, where different chaotic orbits appear and one can consider the synchronization problem, is the driven Van der Pol-Duffing oscillator described by the following nonlinear equation [15]

2 3 x¨ − µ(1 − x )x ˙ + x + αx = E0 cos wt, (1)

(dots denote differentiation with respect to time.) The quantities µ and α are two positive coefficients. E0 and w are respectively the amplitude and frequency of the external excitation. The interest in this model is due to the fact that it can be used to model several phenomena and has many applications in science and engineering, like to model an electronic circuit shown in Ref.[15], to name one. Particularly interesting is the case of several oscillators mutually con- nected to form a ring of coupled self-excited systems. For example, the mutually connected system can be used in electronic engineering as a model of parallel operating components in microwave oscillators, or to model the control pattern generator that rules the rhythmic ac- tivity in invertebrates. The Van der Pol-Duffing oscillator without external drive possesses a rich dynamical behavior [12, 13] and generates the sinusoidal limit cycle for small values of µ, developing into relaxation oscillations when µ becomes large. Szempli´nska-Stupmika et al. [12]

2 reported that the model described by Eq. (1) with an external excitation, exhibits various types of behaviors, namely, periodic and quasi-periodic oscillations, in the principal resonance region. In this paper, we consider three points. We first consider the problem of synchronizing two coupled autonomous Van der Pol models using analytical and numerical methods. Secondly, after finding interesting values in the parameter space where bifurcation structures and strange attractors appear, we consider the problem of synchronizing two driven chaotic Van der Pol- Duffing systems, either on the same or on different chaotic orbits. Our study uses a variation of the continuous feedback scheme of Ref. [18]. At variance with the Pyragas method where a chaotic system is coupled with itself (at a previous time), we coupled the system with an identical system that supplies unidirectional coupling [1, 2]. The latter is called the master system, while the driven system is called the slave. Previous studies on the same model have shown the possibility of chaos synchronization for a Van der Pol oscillator [14] and have derived the analytic conditions for the self sustained system [19]. We use numerical simulations to derive stability boundaries and the optimal coupling parameter of the synchronization process. In the last part, the problem of synchronization of two chaotic Van der Pol-Duffing systems with time periodic coupling is also considered. The organization of the paper is as follows. In the next section, the stability of the syn- chronization process of two coupled Van der Pol oscillators is considered. Bifurcation structures and transitions from the regular to chaotic states are investigated in section 3. Section 4 is devoted to the synchronization of two driven chaotic Van der Pol-Duffing systems. We find the stability boundaries of the synchronization process and derive the synchronization time. We find in Section 5 the effects of the amplitude of the parametric perturbation for the coupling element on the stability boundaries. Conclusions are given in the last section.

2 Synchronization dynamics of coupled systems

2.1 Statement of the problem

The final state of the VdPD oscillator without external excitation is a sinusoidal limit cycle for small values of the coefficient µ, developing into relaxation oscillations when µ becomes large. One particular characteristics in the VdPD model is that its phase depends on initial conditions. Therefore, if two VdPD oscillators are launched with different initial conditions, their trajectory will finally circulate on the same limit cycle, but with different phases ϕ1 and ϕ2. The objective of the synchronization in this case is to phase-lock the oscillators (phase synchronization) so that ϕ1 − ϕ2 = 0. As we have quoted in the introduction, one aim of this survey is to study the stability and derive the characteristics of the synchronization of two VdPD oscillators. The master system is described by the component x while the slave system has the corresponding component y. The

3 enslavement is carried out by coupling the slave to the master through the following scheme

2 3 x¨ − µ(1 − x )x ˙ + x + αx = 0, 2 3 y¨ − µ(1 − y )y ˙ + y + αy = −K(y − x)H(t − T0), (2) where K is the feedback coupling coefficient, t the time, T0 the onset time of the synchronization process and H(z) is the Heaviside function defined as

0 for z < 0, H(z)= ( 1 for z ≥ 0.

Practically, this type of unidirectionnal coupling between the master system and the slave system can be done through a linear resistor and a buffer. The buffer acts a signal-driving element that isolates the master system variable from the slave system variable, thereby providing a one-way coupling. In the absence of the buffer the system represents two identical Van der Pol-Duffing

Oscillators coupled by a common resistor Rc, when both the master and slave systems will mutually affect each other. It is worth to note that the sign of K has an intuitive meaning: K > 0 corresponds to the attractive case between the master and the slave, while K < 0 corresponds to repulsion. A natural expectation would then be: for some value of positive K the system achieves synchronization, while for negative values of K one would not expect synchronization. We shall demonstrate that the nonlinear character of the system defies intuition, and synchronization can occur also for some negative range of the coupling constant K.

2.2 Stability investigation of the synchronization process

When the synchronization process is launched, the slave system changes its configuration. We must assume that the process is stable to avoid irreversible damages to the system. It is then particularly important to develop criteria that guarantee the asymptotic stability of the process. To measure the closeness between the master and the slave at each time, let us introduce a new variable ε(t)= y(t) − x(t). (3)

The stability of the process is thus examined by the boundedness of ε(t) which obeys to the following equation

2 2 2 ε¨− µ(1 − x )ε ˙ + (2µxx˙ + 3αx +1+ K)ε + 2µxεε˙ + µε ε˙ 2 3 +(µx˙ + 3αx)ε + αε = 0. (4)

Saying that the process is stable here means that the synchronization is really effective. To achieve the synchronization process, we have to be sure that ε goes to zero as t increasing or is less than a given precision. The behavior of ε depends on K and on the form of the master x. The form of the master solution can been found following the Lindsted’s perturbation method

4 [] and can been approximated given by

α 3 1 2 x(t)= A cos ωt + cos 3ωt + µ( sin ωt − sin 3ωt)+ O(µ ), 4 4 4 (5) with

1 3 27 2 1 2 3 A = 2 − α, ω =1+ α − α − µ + O(µ ). (6) 2 2 16 16 Thus, the above variationnal equation (17) takes the form

2 2 3 ε¨+ [2λ + F (τ)]ε ˙ + G(τ)ε + Q(τ)ε + R(τ)εε˙ + a2εε˙ + a3ε = 0, (7) where µ A2 µ2 α2 µ α τ = ωt, λ = ( − 1+ + ), a2 = , a3 = 2ω 2 4 32 ω2 ω2

while the following quantities F (τ), G(τ), Q(τ), R(τ), g0, fis, fic and gis and gic (i = 1, 2, 3) are given in the appendix (A1). From the expression of G(τ), we find that if

2 2 3α 2 µ α K < Kb(α)= − (A + + ) − 1, (8) 2 2 16 ε(t) will grow indefinitely leading the slave to continuously drift away from its original limit cycle. In this case, the feedback coupling is dangerous since it continuously adds energy to the slave system. It’s found that there’s no synchronization between the master and the slave when

K decreasing from Kb to infinity since the deviation between the slave and the master becomes large or increases indefinitely. Since the analytical investigation of the variational equation (7) is very difficult due to its nature, we only consider the linear stability analysis. In this case, we suppose that the measure of the nearness ε of the slave to the master is very small, so that one can neglect both quadratic and cubic terms. Then in the linear regime, equation (7) is reduced to

ε¨+ [2λ + F (τ)]ε ˙ + G(τ)ε = 0. (9)

To examine the stability analysis of the synchronization process, let us transform equation (9) into the standard one by introducing a new variable η as τ 1 ′ ′ ε = η exp(−λτ)exp − F (τ )dτ . (10) 2 0  Z 

5 This yields the following Hill equation

η¨ + (ao + 2a1c cos 2τ + 2a1s sin 2τ + 2a2c cos 2τ + 2a2s sin 4τ

+2a3c cos 6τ + 2a3s sin 6τ + 2a4c cos 8τ + 2a4s sin 8τ

+2a5c cos 10τ + 2a6c cos 12τ + 2a6s sin 12τ)η = 0, (11) whose coefficients are given in the appendix (A2). Following the Floquet theory [16,17] the solution of equation (22) may be either stable or unstable and the stability boundaries of the synchronization process are to be found around the six main parametric resonances defined at 2 ao = n (with n = 1, 2, 3, 4, 5, 6). We have found that [19] the synchronization process is stable under the condition

n 2 2 2 2 4 2 H = (ao − n ) + 2(ao + n )λ + λ − an > 0, n = 1, 2, 3, 4, 5, 6. (12)

The above inequality will allow us to seek analytically the range of the coupling coefficient K where the process of synchronization is stable in the linear regime. The conditions (12) and (8) have been called to seek the range of the coupling parameter where the synchronization process is stable. It is important to note that for n = 2, 3, 4, 5, 6, the conditions (25) are satisfied for any value of K, and we only use this condition around the first main parametric resonance (n = 1). To identify different dynamical states which appears in the coupled VdPD oscillators, we varied the coefficient K to see if the conditions (8) and (12) are simultaneously verified or not. We find from these two conditions (8) and (12), that depending for the coupling coefficient K, there are some domains where the synchronization process could be stable or not. We fix µ = 0.1 and used two values of α. For instance when α = 0.01, the synchronization is unstable for K ∈] −∞, −1.0597] ∪ [−0.3130, 0[∪]0, 0.260] while for α = 0.05, the process is not achieved for K ∈] − ∞, −1.2929] ∪ [−0.4550, 0[∪]0, 0.1690]. Saying here that the synchronization is unstable means that ε(t) never goes to zero as the time is increased but has a bounded oscillatory behavior or goes to infinity. Three ranges of the coupling parameter are found. The first region is K ∈] −∞, −1.0597] for α = 0.01 and K ∈] −∞, −1.2928] for α = 0.05 where the synchronization process is unstable linearly and nonlinearly, called global unstable domain of the synchronization process. Generally in that first region, the amplitude of oscillations become very large and tend to infinity for certain values of the coupling parameter K (see Figure 1a with α = 0.01 and K = −30). Nevertheless according to α, the deviation x − y between the master and the slave can be bounded by high amplitude oscillations as shown in Figure 1b for α = 0.05 and K = −30. Such a behavior is the characteristic of global unstable phenomena of the synchronization process analyzed. In the second region, K ∈ [−0.3130; 0[∪]0; 0.260] for α = 0.01 and K ∈] − 0.4551, 0[∪]0; 0.1690] for α = 0.05, the synchronization is not achieved as we mentioned before since the deviation x − y is bounded (see Figure 2). Here, the amplitude of oscillations are very small compared to those

6 observed in the global unstable. At the last region, K ∈] − 1.0597, −0.3131[∪[0.260, +∞[ for α = 0.01 and K ∈] − 1.2929; −0.4551[∪[0.1690; +∞[ for α = 0.05, the synchronization process is stable and called for that global stability area. This means that the deviation between the master and the slave goes to zero as the time inreases as we shown in Figure 3(a) for α = 0.01 and in Figure 5(b) for α = 0.05.

3 Strange attractors

Bifurcation structures occur for some values of the parameters. To illustrate this, we nu- merically solve Eq. (1) and plot the resulting bifurcation diagram as a function of the driven amplitude E0. We have used a standard 4th order Runge-Kutta method with time step size around 0.01. Our investigations show that the driven Van der Pol-Duffing oscillator exhibits chaotic behavior as it appears in Fig. 1 with the parameters: µ = 5, α = 0.01, E0 = 5, w = 2.463. For a judicious choice of the initial conditions, we find that the system is able to exhibit two degenerated chaotic attractors. The first one shown in Fig. 1(i), obtained with the initial a conditions IC1 : (x(0), x˙(0)) = (2.0; 2.0), we shall call the upward attractor. The second one, a obtained with the initial conditions IC2 : (x(0), x˙(0)) = (0.1; 0.1), will be named the downward attractor. As it appears in Fig. 4(ii), this attractor is in inversion symmetry with the first one shown in Fig. 4(i). With the above two sets of initial conditions, we have drawn the bifurcation diagrams (see Fig. 5) when the amplitude E0 varies and the following results are observed.

As the amplitude E0 increases from zero, the chaotic orbit persists until E0 = 2.15 where the chaotic state stabilizes into a period-5 orbit. As E0 is further increased, the system passes to another chaotic state at E0 = 3.99. Then at E0 = 4.5, a tiny periodic orbit transition appears and the system passes into the chaotic state and bifurcates to a period-3 orbit at E0 = 5.1. At the other side, a reverse periodic-doubling sequences takes place leading to a period-1 or- bit (harmonic oscillations). The second indicator used to analyze the presence of chaos is the maximum Lyapunov exponent λ. Figure 6 presents the dependence of the Lyapunov exponent as a function of E0. As it is apparent, the Lyapunov exponent shows both positive (chaotic behaviors) and negative (periodic motion) signs, as the driving amplitude is changed.

4 Synchronization of two chaotic systems

In the chaotic state, the main feature is the high-sensitivity to initial conditions. This is the result of the combined effects of the cubic and intrinsic nonlinearities, and of the extrinsic periodic drive. Consequently, a very small difference in the initial conditions will lead to different time histories or orbits. If two systems are launched with two initial conditions IC1 and IC2, they will circulate on different degenerated chaotic orbits. The goal of the synchronization in this case is to call one of the system (slave) from its degenerated chaotic orbit to that of the

7 other system (master). For this aim, equations (2) become

2 3 x¨ − µ(1 − x )x ˙ + x + αx = E0 cos wt, 2 3 y¨ − µ(1 − y )y ˙ + y + αy = E0 cos wt − K(y − x)H(t − T0), (13)

4.1 Synchronizing chaos on the same orbit

In this subsection, we first investigate the possibility of synchronizing two driven chaotic Van der Pol-Duffing systems with their phase trajectories circulating along an identical upward chaotic orbit or downward chaotic orbit. To find the appropriate coupling coefficient K which enables the slave system to adjust its oscillations and to synchronize with the master one, we numerically compute the synchronization time defined as

Ts = ts − T0, ∀ts > T0, (14) where ts is the time instant at which the two trajectories are close enough to be considered as synchronized. Here, synchronization is achieved with the following synchronization criterion:

2 (y − x) + (y ˙ − x˙ ) < h, ∀t > T0, (15) q where h is the synchronization precision. We have chosen h consistently with the numerical precision, that has resulted in h ≃ 10−6. The coupling coefficient K is then varied, and we find the range of K in which the synchronization process is achieved. T0 = 400 is used in all the paper. It is important to note that, due to the finite precision of the numerical schemes, it does not suffices that the deviation ǫ(t) = y(t) − x(t) tends to zero (or, more accurately, that it is smaller than the given precision h) to achieve the synchronization, it is also necessary to be sure that the criterion (4) is satisfied successively for a large enough time interval ∆t [14] . In this work we used ∆t = 10000. When the two driven chaotic systems are circulated along an identical upward chaotic orbit, a b with the initial conditions IC1 : (x(0), x˙(0)) = (2.0; 2.0)) for the master system and IC1 : (x(0), x˙(0)) = (2.3; 2.3)) for the slave system, we find that the two systems are synchronized on an upward chaotic orbit in the range of K defined as K ∈]−1.02; −0.41[∪]5.29, 10]. For the case of the two systems circulating along an identical downward chaotic orbit with the initial conditions a b IC2 (x(0), x˙(0)) = (0.1; 0.1)) for the master system and IC2 (x(0), x˙(0)) = (0.3; 0.3)) for the slave system, it appears that the synchronization process requires that K ∈]−1.021; −0.45[∪]5.23; 10], thus confirming that the two attractors have similar properties. To illustrate this results, let us present graphically the time history of the deviation ǫ(t) between the master and slave systems for a coupling coefficient K chosen both in the stable and unstable regions. In the stable domain, we find in Fig. 7 that the deviation ǫ(t) goes to zero when the time goes up (see the correlated states between the master and slave variables).

8 In both case, the variation of the synchronization time Ts is plotted versus K in Fig. 8. It is found that in the stability boundaries, Ts is very large, But as K moves from the boundaries, Ts decreases quickly and for large K, it attains a limiting small value depending on the precision. Summarizing this part, we have found that for strong enough attraction the system gets synchronous, as expected. More surprisingly, we have also found that there is a window in the repulsive parameter region where the system achieves synchronization. A similar synchronization for repulsive coupling was analytically predicted and observed for the non-chaotic solution in Ref. [19].

4.2 Synchronizing chaos on different orbit

We study now the possibility of synchronizing chaos on two different chaotic attractors. The master and the slave are initially launched with the initial conditions IC1 and IC2 respectively. These sets of initial conditions lead respectively to upward and downward chaotic orbits. The master is circulated along an upward chaotic orbit while the slave is on a downward one. Here, the slave is forced to come from a downward chaotic orbit to follow the master to an upward chaotic orbit. This slave transition requires that K ∈] − 1.02; −0.43[∪]5.29; 10]. In this range of K, the slave system eventually synchronizes with the master, which circulates independently along this upward chaotic attractor. For the slave transition from a downward chaotic orbit to follow the master on an upward chaotic orbit, we just inverse the initial conditions. The slave transition requires that K ∈]−1.02; −0.45[∪]5.23; +10[. The time history ǫ(t) and the correlated state between the velocitiesx ˙ andy ˙ are also plotted in Fig. 9, and we find that ǫ(t) tends to zero in the stable domain of K. A straight line segment which has π/4 slope and shows no broadening is a guarantee for genuine synchronization. (We monitored the system for a longer time to check that it has entered into a steady synchronous motion). Numerical investigation shows that the dependence of the synchronization on K is similar to that in last section for the interaction of two identical chaotic attractors. Fig. 9 displays this feature. The result remain essentially the same for an upward chaotic attractor synchronized with a downward one. The downward chaos can be perturbed to bifurcate to an upward one by varying the initial conditions; put it differently an attractor is dragged out of its basin of attraction. Fig. 10 shows the synchronization time versus K for transitions of the slave system from both the upward and the downward attractors.

5 Effects of the parametric coupling on the stability boundaries

In section 4, we have found different bifurcation mechanisms which appear in the coupled systems when K varies. This section deals to the synchronization of two chaotic systems with time periodic coupling. For this aim, we assume that the coefficient K is varied as K = K(1 +

9 q sin 2Ωt), and equations (2) become

2 3 x¨ − µ(1 − x )x ˙ + x + αx = E0 cos wt, 2 3 y¨ − µ(1 − y )y ˙ + y + αy = E0 cos wt

−K(1 + q sin 2Ωt)(y − x)H(t − T0), (16)

The coupled systems (5) are practically realized by coupling the master and slave systems with the buffer and by means of a turning capacitor C (high-pass oscillations) with variable characteristics. The variation of the turning capacitor is induced by the external rotational motor and then the quantity 1/C changes with time by the torsional motor, as in [20] 1 1 = (1 + q sin 2Ω) C C where q and Ω are respectively the amplitude and frequency of the parametric modulation (0 < q ≤ 0.9). This type of time periodic coupling coefficient has been used in Ref.[21] to explore the synchronization of a ring of four mutually identical self-sustained electrical systems by means of the periodic parameter perturbations of the coupling element. We assume in the remaining of the paper that the external rotational motor has the same frequency as the external driven Ω = w, which is known as the resonant periodic parametric perturbation on the coupling element. With the periodic perturbation amplitude of the parametric coupling q, we have only con- sidered the possibility of synchronizing chaos on different orbits. We have chosen several values of q, and we have then computed the effect of q on the stability boundaries, as well as different bifurcation mechanisms which appear in the systems. For instance, with q = 0.1, our investiga- tions show that the synchronization process is achieved for K ∈ [−1.13; −0.51] ∪ [5.8; 10] (slave transition from an upward orbit to the downward orbit) and K ∈ [−1.13; −0.48] ∪ [5.87; 10] (for the reverse slave transition), while for q = 0.6, we find that K ∈ [−2.0; −0.88] slave transition from an upward orbit to the downward orbit) and K ∈ [−2.47; −1.06] (for the reverse slave transition). We note that in the stability domains, the deviation between the master and slave systems tends to zero as the time goes up. For other values of q, different bifurcation mechanisms which appear in the systems are provided in the stability map shows in Fig. 11. Analyzing the effects of the amplitude q on the stability boundaries, the following results are observed. We note from Fig. 8 that the larger the amplitude q, the larger the domain of instability of the synchronization process. However, the area of stability in the right on the chart disappears for q > 0.45, whereas the distance between the stability boundaries of right-hand side increases increasing q. Fig. 12 shows the effects of q on the synchronization time for the slave transition from the downward attractor to follow the master to an upward. Summarizing, we can say that inserting a time periodic perturbation on the coupling element does not contribute to the process of synchronization, because the increase of q leads to the reduction of the stability of

10 the synchronization process. However, the parametric coupling for small values of q does not significantly reduce the stability boundaries.

6 Conclusion

We have studied in this paper the possibility of synchronizing chaos in two coupled Van der Pol-Duffing systems. We have considered the transient characteristics of two attractors of a Van der Pol-Duffing oscillator, and we have found the corresponding bifurcation structures which appear in the system as the drive amplitude increases. It is interesting to find that two chaotic orbits, which are in inversion symmetry with each other, can coexist for a given set of parameters. These two degenerated attractors are the result of the combined effects of intrinsic nonlinearity and extrinsic periodic forcing. Stability boundaries of the synchronization process have been derived using numerical simulations. A generic feature is that there is a threshold for synchronization for the K > 0 attractive parameter of the master-slave interaction. For K < 0, the repulsive case, the system can be synchronized only for a special interval of the coupling. The effect of the amplitude of the parametric perturbations on these stability boundaries has been derived and provided in the stability chart in the (q, K) plane.

Acknowledgments

Part of this work was done within the framework of the Associateship Scheme of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. R. Yamapi expresses special thanks to the Swedish International Development Cooperation Agency (SIDA) for financial support.

11 Appendices

Appendix A-1

The functions F (τ), G(τ), R(τ) and R(τ) are defined as

F (τ) = f1c cos 2τ + f1s sin 2τ + f2c cos 4τ

+f2s sin 4τ + f3c cos 6τ + f3s sin 6τ,

G(τ) = g0 + g1c cos 2τ + g1s sin 2τ + g2c cos 4τ

+g2s sin 4τ + g3c cos 6τ + g3s sin 6τ,

Q(τ) = 2q1c cos τ + 2q1s sin τ + 2q3c cos 3τ + 2q3s sin 3τ,

R(τ) = 2r1c cos τ + 2r1s sin τ + 2r3c cos 3τ + 2r3s sin 3τ,

with 2 2 µ 2 αA 15µ µ 3α f1c = (A + − ), f1s = (A − ), 2ω 2 16 2ω 8 µ 3µ2 µ2 3α f2c = (αA + ), f2s = ( − A), 4ω 4 4ω 4 µ α2 − µ2 αµ2 f3c = ( ), f3s = − , ω 32 16ω 2 2 1 3α 2 µ α g0 = 2 1+ K + (A + + ) , ω " 2 2 16 # 3α αA 15µ2 µ2 3α g1c = (A + − )+ (A − ), 2ω2 2 16 ω 8 2 3µα 3α µ 2 15µ g1s = (A − )+ (−A − 2αA + ), 2ω2 8 ω 16 3α 3µ2 µ2 3α g2c = (αA + )+ (−A + ), 4ω2 4 ω 4 7A + 3µ2 3αµ 3α g2s = −µ( )+ ( − A), 4ω 4ω2 4 3α(α2 − µ2) 3µ2α 3α2µ 3µ(µ2 − α2) g3c = + , g3s = − + . 32ω2 8ω 16ω2 16ω

12 Appendix A-2

The quantities ao, ans and anc are defined as

2 1 2 2 2 2 2 2 a0 = g0 − λ − (f1c + f1s + f2c + f2s + f3c + f3s), 8 1 1 1 a1c = g1c + f1s − (4λf1c + f1cf2c + f1sf2s + f2cf3c + f2sf3s) 2 2 8 1 1 1 a1s = g1s − f1c − (4λf1s + f1cf2s + f2cf3s − f2sf3c) 2 2 8 1 1 1 2 1 a2c = g2c + f2s − ( f1c − f1s + 4λf2c + f1cf3c + f1sf3s), 2 8 2 2 1 1 a2s = g2s − f2c − (4λf2s + f1cf1s + f1cf3s − f1sf3c), 2 8 1 3 1 a3c = g3c + f3s − (4λf3c + f1cf2c − f1sf2s), 2 2 8 1 3 1 a3s = g3s − f3c − (4λf3s + f1cf2s − f1sf2c), 2 2 8 1 1 2 1 2 a4c = − ( f2c − f2s + f1cf3c − f1sf3s), 8 2 2 1 a4s = − (f1cf3s − f1sf2c + f1sf3c + f2sf2c), 8 1 1 a5c = − (f2cf3c − f2sf3s), a5s = − (f2cf3s + f2sf3c), 8 8 2 2 f3s − f3c 1 a6c = , a6s = − f3cf3s. 16 8

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[12] W Szempli´nska-Stupnicka and J Rudonski, The coexistence of periodic, almost-periodic and chaotic attractors in the chaotic attractors in the Van der Pol-Duffing oscillator, J. of Sound and Vib. 1997;199:165-.

[13] TE Vadivasora, GI Strelkova and VS Anishchenko, Phase-frequency synchronization in a chain of periodic oscillators in the presence of noise and harmonic forcings, Phys. Rev. E 2002;63:036225.

[14] HK Leung,Synchronization dynamics of coupled van der Pol systems, Physica A 2003;321:248-255.

[15] K Murali and M Lakshmanan,Transmission of signals by synchronization in a chaotic Van der PolDuffing oscillator, Phys. Rev. E 1993;48:R1624.

14 [16] C Hayashi, Nonlinear Oscillations in Physical Systems. New York: Mc-Graw-Hill; 1964.

[17] AH Nayfeh and DT Mook, Nonlinear Oscillations, New York: Wiley-Interscience;1979.

[18] K Pyragas, Phys. Lett. A 170, 421 (1992); Synchronization of coupled time-delay systems: Analytical estimations. Phys. Rev. E 1998;58: 3067-.

[19] R Yamapi, P Woafo, Dynamics and synchronization of coupled electromechanical devices, J. of Sound and Vib. 2005;285: 1151-.

[20] Y Rocard, Dynamique g´en´erale des vibrations, Paris: Masson et Cie: 1971.

[21] R Yamapi and S Bowong, Dynamical states in a ring of four mutually coupled self-sustained electrical systems with time periodic coupling, Int. J. of Bifurcation and Chaos 2006 (In press).

15 (a) 120

100

80

60 x-y

40

20

0

-20 580 585 590 595 600 605 610 615 620 625 TIME

(b) 30

25

20

15 x-y

10

5

0

-5 580 585 590 595 600 605 610 615 620 625 TIME

Figure 1: Time history of the deviation ε(t) displaying both linear and nonlinear unstable phenomena of the synchronization process. (a) α = 0.01, K = −30, (b) α = 0.05, K = −30.

16 (a) 0.4

0.3

0.2

0.1

0 x-y

-0.1

-0.2

-0.3

-0.4 500 550 600 650 700 750 TIME

(b) 3

2

1

0 x-y

-1

-2

-3 500 550 600 650 700 750 TIME

Figure 2: Time history of the deviation ε(t) displaying bounded oscillations in the process of synchronization. (a) α = 0.01, K = −0.04, (b) α = 0.05, K = −0.35.

17 (a) 0.3

0.2

0.1

0 x-y

-0.1

-0.2

-0.3 550 600 650 700 750 TIME

(b) 2

1.5

1

0.5

0 x-y

-0.5

-1

-1.5

-2 550 600 650 700 750 TIME

Figure 3: Time history of the deviation ε(t) displaying the stability of the synchronization process. (a) α = 0.01, K = −0.50, (b) α = 0.05, K = 0.70

18 6 (ii) 4

2

0

−2

−4 Velocity, dx/dt

−6

−8

−10 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Position, x 10 (ii) 8

6

4

2

0

Velocity, dx/dt −2

−4

−6

−8 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 Position, x

Figure 4: Chaotic phase portrait of the system prepared with different initial conditions: (i) IC1 = (2.0; 2.0) and (ii) IC2 = (0.1; 0.1). In both cases the parameters used are: µ = 5.0; α = 0.01; E0 = 5.0; w = 2.463.

19 2.5

2

1.5

1

0.5

0

−0.5 Position, x

−1

−1.5

−2 (i)

−2.5 0 5 10 15 Drive amplitude, E o 2.5

2

1.5

1

0.5

0

−0.5 Position, x

−1

−1.5

−2 (ii)

−2.5 0 5 10 15 Drive amplitude, E o

Figure 5: Bifurcation diagrams show the coordinate x versus E0. (i): IC1 = (2.0; 2.0) and (ii): IC2 = (0.1; 0.1). In both cases the parameters used are: µ = 5.0; α = 0.01; w = 2.463.

20 0.3

(i) 0.2

λ 0.1

0

−0.1

−0.2 Lyapunov exponent,

−0.3

−0.4 4 4.5 5 5.5 6 6.5 7 Driven amplitude E o 0.3

(ii) 0.2

λ 0.1

0

−0.1

−0.2 Lyapunov exponent,

−0.3

−0.4 4 4.5 5 5.5 6 6.5 7 Driven amplitude, E o

Figure 6: The Lyapunov exponent versus the drive amplitude E0. (i): IC1 = (2.0; 2.0) and (ii): IC2 = (0.1; 0.1). In both cases the parameters used are: µ = 5.0; α = 0.01; w = 2.463.

21 2

1.5

1

0.5

0 x −0.5

−1

−1.5

−2 K=−1.0

−2.5 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 y 2

1.5

1

0.5

0 x −0.5

−1

−1.5

−2 K=−0.6

−2.5 −2.5 −2 −1.5 −1 −0.5 y 0 0.5 1 1.5 2 2

1.5

1

0.5

0 x −0.5

−1

−1.5

−2 K=5.3

−2.5 −2.5 −2 −1.5 −1 −0.5 y 0 0.5 1 1.5 2 2

1.5

1

0.5

0 x −0.5

−1

−1.5

−2 K=6.0

−2.5 −2.5 −2 −1.5 −1 −0.5 y 0 0.5 1 1.5 2

Figure 7: Correlated states between the master and slave systems for synchronous chaos of the downward attractor. K is chosen in the stable domain. The parameters are the same as in Fig. 1.

22 3500 (i) 3000

s

2500

2000

1500

1000 Synchronization time, T

500

0 −2 0 2 4 6 8 10 Coupling strenght, K

(ii) 3000

s

2500

2000

1500

1000 Synchronization time, T

500

−1 0 1 2 3 4 5 6 7 8 Coupling strenght, K

Figure 8: The synchronization time Ts versus the coupling strength K for the master and the slave systems both prepared on the same orbit: (i) corresponds to the downward attractor and (ii) to an upward attractor.The parameters are the same as in Fig. 1.

23 6 (CS) 4

2

0

−2

−4

Master velocity dx/dt −6

−8 K=−0.5

−10 −10 −8 −6 −4 −2 0 2 4 6 Slave velocity dy/dt 6 (CS) 4

2

0

−2

−4

Master velocity dx/dt −6

−8 K=6

−10 −10 −8 −6 −4 −2 0 2 4 6 Slave velocity dy/dt 4 (ts) 3

2

1 ε

0

−1 Deviation,

−2

−3 K=−0.5

−4 300 320 340 360 380 400 420 440 460 480 500 Time, t 4 (ts) 3

2

ε 1

0

Deviation, −1

−2

−3 K=6

−4 300 320 340 360 380 400 420 440 460 480 500 Time, t

Figure 9: Correlation states (CS) between the velocities of the master and slave systems and time evolution of the deviation ǫ(t) (ts) for the two systems prepared on different orbits: slave system transition from an upward attractor to the downward attractor to follow the master prepared in the downward attractor. The coupling K is chosen in the stable domain The parameters are the same as in Fig. 1.

24 3000 (i)

2500s

2000

1500

1000 Synchronization time, T

500

0 −1 0 1 2 3 4 5 6 7 8 9 Coupling strenght, K

3000 (ii)

2500s

2000

1500

1000 Synchronization time, T

500

0 −1 0 1 2 3 4 5 6 7 8 9 Coupling strenght, K

Figure 10: Dependence of the synchronization time Ts versus the coupling strength K, corresponding to the synchronization of different orbits: (i) slave transition from to an upward attractor to follow the master to the downward; (ii) slave transition from to downward attractor to follow the master to the upward attractor. The parameters are the same as in Fig. 1.

25 0.8 (S) (i) 0.7

0.6

0.5

0.4 q (US)

0.3 (US) 0.2 (S) 0.1

0 −5 0 K 5 10 0.8 (S) (ii) 0.7

0.6

0.5 (US)

0.4 q

0.3

0.2 (US) (S) 0.1

0 −5 0 5 10 K

Figure 11: Stability map in the (q, K) plane showing the effects of the parametric coupling on the stability boundaries which appear in the systems. (i): slave transition from to downward attractor to follow the master to the upward; (ii): slave transition from to upward attractor to follow the master to the downward. (S) corresponds to the stability region while (US) corresponds to the unstable region. The parameters are the same as in Fig. 1.

26 q=0 3000

s 2500

2000

1500

1000 Synchronization time, T 500

−1 0 1 2 3 4 5 6 7 8 9 Coupling strenght, K

q=0.3 3000

s 2500

2000

1500

1000 Synchronization time, T 500

−1 0 1 2 3 4 5 6 7 8 9 Coupling strenght, K

q=0.5 3000

s 2500

2000

1500

1000 Synchronization time, T 500

0 −2 0 2 4 6 8 Coupling strenght, K

q=0.8 3000

s 2500

2000

1500

1000 Synchronization time, T 500

0 −4 −2 0 2 4 6 8 Coupling strenght, K

Figure 12: Effects of the amplitude q on the synchronization time Ts, from the slave system transition from the downward attractor to follow the master system on an upward attractor. The parameters are the same as in Fig. 1.

27