Regular oscillations, chaos, and multistability in a system of two coupled van der Pol oscillators: numerical and experimental studies J. Kengne, J. C. Chedjou, M. Kom, K. Kyamakya & V. Kamdoum Tamba
Nonlinear Dynamics An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems ISSN 0924-090X Nonlinear Dyn DOI 10.1007/s11071-013-1195-y
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Author's personal copy Nonlinear Dyn DOI 10.1007/s11071-013-1195-y
ORIGINAL PAPER
Regular oscillations, chaos, and multistability in a system of two coupled van der Pol oscillators: numerical and experimental studies J. Kengne · J. C. Chedjou · M. Kom · K. Kyamakya · V. Kamdoum Tamba
Received: 26 August 2013 / Accepted: 13 December 2013 © Springer Science+Business Media Dordrecht 2014
Abstract In this paper, the dynamics of a system of two coupled van der Pol oscillators is investigated. The coupling between the two oscillators consists of adding to each one’s amplitude a perturbation proportional to the other one. The coupling between two laser oscillators and the coupling between two vacuum tube oscillators are examples of physical/experimental systems related to the model considered in this paper. The stability of fixed points and the symmetries of the model equations are discussed. The bifurcations structures of the system are analyzed with particular attention on the effects of frequency detuning between the two oscillators. It is found that the system exhibits a variety of bifurcations including symmetry breaking, period doubling, and crises when monitoring the frequency detuning parameter in tiny steps. The multistability property of the system for special sets of its parameters is also analyzed. An experimental study of the coupled system is carried out in this work. An appropriate elecJ. Kengne · V. K. Tamba Laboratoire d’Automatique et Informatique Apliquée (LAIA), IUT-FV Bandjoun, University of Dschang, Dschang, Cameroon J. Kengne (B) · J. C. Chedjou · K. Kyamakya Institute for Smart-Systems Technologies, University of Klagenfurt, Klagenfurt, Austria e-mail:
[email protected] M. Kom Ecole Nationale Superieure Polytechnique (ENSP), University of Younde-1, Younde, Cameroon
tronic simulator is proposed for the investigations of the dynamic behavior of the system. Correspondences are established between the coefficients of the system model and the components of the electronic circuit. A comparison of experimental and numerical results yields a very good agreement. Keywords Coupled van der Pol oscillators · Bifurcation analysis · Multistability · Crisis · Analog circuit implementation 1 Introduction In recent years, coupled limit cycle oscillator models have drained tremendous research efforts due to the useful insight they provide into the collective behavior of many physical, chemical, and biological systems [1–11]. The system of coupled van der Pol oscillator is one of the simplest such models. Briefly recall that the van der Pol equation is originally an example of an oscillator with nonlinear damping that arises in the modeling of electrical circuits with a triode valve [12–14]. A system of coupled van der Pol oscillators is highly documented. Depending on the coupling scheme and the values of parameters, such a coupled system can demonstrate some striking dynamic behavior including oscillation death, phase locking of oscillators with different ratios of frequencies, two frequency quasi-periodic regimes, multistability, and chaos [3]. Some interesting factors such as nonlinear coupling [15], delay coupling [16,17], and ‘coupling via a bath’
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[18] on the dynamics of the coupled system have been considered. A highly symmetric coupling of two coupled van der Pol oscillators was considered in [8]. The authors analyzed the structure of the equilibrium points and the discrete symmetries of the model. A complete characterization of the dynamics was also performed on three specific cases, as a function of the coupling parameters. It was found that several attractors coexist in phase space, either having the symmetry of the model equations or appearing in pairs that restore such symmetry. The possibility that the asymptotic dynamics is different in the coexisting symmetric and asymmetric attractors that were investigated along with their creation or destruction, splitting, and merging, when a control parameter is monitored. When only one attractor is explored, the authors revealed that the system followed a perioddoubling route to chaos. In Pastor and Lopez-Fraguas [9], an extensive analysis of the multistability properties of the model proposed in [8] was carried out. Using special tools such as first return map, Poincare section, and probability distribution function, they showed that the complex dynamics found in the system could be understood in terms of simple discrete transformations related to the logistic map. For some specific values of the control parameters, the authors also showed that a combined master–slave system based on the coupled oscillators of [8] exhibits a chaotic synchronization regime. Correspondingly, the practical implications of the observed phenomena were underlined. However, the analysis performed so far [8,9] is restricted to the case of coupled isochronous oscillators. Furthermore, according to the best of our knowledge, no experimental verification of the results presented in [8,9] is reported in the relevant state of the art. Our goal in this paper is to make a contribution in the study of the dynamic behavior of the coupled system described in [8,9] by, (a) extending the analysis to the general (and more realistic) case of coupled nonisochronous oscillators; (b) carrying out an experimental study of the system to validate/verify the theoretical results; and (c) pointing out some of the unknown and striking behavior of the coupled van der Pol oscillators. One of the interest of this work is to prove that the analog computation is suitable than that of its numerical counterpart for understanding the real behavior of nonlinear and chaotic systems. The rest of the paper is structured as follows. Section 2 describes the mathematical model of the system
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under investigation and underlines possible symmetries. The stability of the equilibrium points is also discussed. In Sect. 3, the bifurcation structures of the system are investigated numerically with particular attention on the effects of frequency detuning between the two oscillators. The possibility of the coexisting solutions for the same parameters’ setting is pointed out. In Sect. 4, an appropriate analog computer is proposed for the experimental study of the dynamic behavior of the system. Correspondences are established between the coefficients of the system model and the components of the analog simulator. A comparison between experimental and numerical results is presented. In Sect. 5, the conclusions and perspectives for future work are outlined.
2 Description and analysis of the model 2.1 System model The dynamics of a system consisting of two mutually coupled van der Pol oscillators considered in this paper are described by the following set of equations: (1a) x¨ − ε1 − (x + βy)2 x˙ + (x + βy) = 0, y¨ − ε2 − (y + αx)2 y˙ + (1 + δ) (y + αx) = 0, (1b) where δ (introduced in this work) is the frequency detuning between the second and the first oscillators and ε j ( j = 1, 2) is a parameter responsible for the Andronov–Hopf bifurcation in the uncoupled oscillators [5], while α and β represent the coupling strengths [8]. The dot denotes differentiation with respect to time. Setting x˙ = u and y˙ = v, system (1a, 1b) can be rewritten as a set of four first-order differential equations in the form: x˙ = u, u˙ = ε1 − (x + βy)2 u − (x + βy) , y˙ = v, v˙ = ε2 − (y + αx)2 v − (1 + δ) (y + αx) .
(2a) (2b) (2c) (2d)
For α = β = 0, both oscillators are uncoupled and each of them exhibit a limit cycle attractor with amplitude and periods determined by parameters ε1 , ε2 , and δ. Obviously, the coupling between the two oscillators
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consists of adding to each one’s amplitude a perturbation proportional to the other one. We note that system (2) is invariant under the transformation: (x(t), u(t), y(t), v(t)) ⇔ (−x(t),−u(t),−y(t),−v(t)) . Therefore, if (x(t), u(t), y(t), v(t)) is a solution of system (2) for a specific set of parameters, ε1 , ε2 , δ, α, and β, then, (−x(t), −u(t), −y(t), −v(t)) is also a solution for the same parameters’ set. The fixed point O(0, 0, 0, 0) is a trivial symmetric solution. Consequently, attractors in state space have to be symmetric with respect to the origin; otherwise they must appear in pairs, to restore the exact symmetry of the model
After some algebraic manipulations, we obtained (1 − αβ)x = 0 and (1 − αβ)y = 0. Therefore, the origin O = (0, 0, 0, 0)T is the only equilibrium point except for the case αβ = 1. If αβ = 1, then the manifold of the equilibrium points is a line in R4 ; namely u = v = 0, x = −βy. For this special combination of coupling parameters, infinitely many new fixed points appear, and that represents a nonstandard feature of our model equation. This special case is not considered in this work and we refer the reader to [8] for further details. We now examine the stability of this unique equilibrium point. In this regard, we compute the Jacobian matrix at any given point (x, u, y, v)T as follows:
⎤ 0 1 0 0 ⎥ ⎢ 0 −1 − 2u (x + βy) ε1 − (x + βy)2 −β − 2β (x + βy) ⎥. MJ = ⎢ ⎦ ⎣ 0 0 0 1 2 −2αv (y + αx) − α (1 + δ) 0 −1 − δ − 2v (y + αx) ε2 − (y + αx) ⎡
equations. This exact symmetry may serve to explain the existence of several attractors in state space. Interested readers are referred to the literature [8] for further discussions about other symmetries of the model with respect to the parameters α and β. It is important to stress that from an experimental point of view, the coupling between two vacuum tube oscillators may be performed using the scheme described above. Furthermore, owing to the fact that the laser field obeys a differential equation with a van der pol type nonlinearity [8], the coupling between two laser oscillators is another physical system related to the mathematical model considered in this paper. Also, the motivation behind this work is to understand what might happen when two self-excited oscillators are coupled in a simple way, so that a physical system of evolution equations and thereafter the results obtained are exploited as predictions on a real physical system.
2.2 Equilibrium points and stability The fixed points of Eqs. (2a–2d) are determined by setting the right-hand side to zero. Thus, the equilibrium points are the solutions of the following linear sets of equations: u = 0, x + βy = 0, v = 0, (1 + δ) (y + αx) = 0. (3)
(4)
Thus, the Jacobian matrix evaluated at the equilibrium point O(0, 0, 0, 0)T satisfies the following characteristic equation (det (M J − λId ) = 0): λ4 + c3 λ3 + c2 λ2 + c1 λ + c0 = 0,
(5a)
where Id is the 4 × 4 identity matrix and the coefficients ci (i = 0, 1, 2, 3) are defined as c0 = (1 + δ)(1 − αβ), c1 = −ε2 − (1 + δ)ε1 , c2 = 2 + δ + ε1 ε2 , c3 = −ε1 − ε2 .
(5b)
A set of necessary and sufficient conditions for all the roots of Eq. (4) to have negative real parts is given by the well-known Routh–Hurwitz criterion expressed in the form: ci > 0 (i = 0, 1, 2, 3) ,
(6a)
c3 c2 − c1 > 0,
(6b)
c3 (c1 c2 − c0 c3 ) − c12
> 0.
(6c)
Obviously, the origin is always unstable, since the characteristic polynomial has coefficients with different signs (e.g., c2 > 0 and c3 ≤ 0) provided that ε1 , ε2 , and δ are positive parameters. Figure 1 shows the representation of the eigenvalues (roots of the characteristic equation) in the complex plane. These roots are computed using the Newton–Raphson algorithm for −20.00 ≤ α ≤ 20.00 and 0 ≤ δ ≤ 5.00; the rest of system parameters being ε1 = ε2 = 1.00 and β = −1.50. From Fig. 1, the presence of eigenvalues with
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Fig. 1 Representation of the eigenvalues solutions of Eq. (5) in the complex plane (Re (λ) , Im (λ)): a δ = 0, −20.00 ≤ a ≤ 20.00); b −20 ≤ a ≤ 20, 0 ≤ δ ≤ 5.00. Given that M J is a real
matrix, complex eigenvalues occur in complex conjugate pairs responsible of the observed symmetry along the real axis
positive real part and thus the unstable nature of the trivial equilibrium point are numerically evidenced [19].
states, LLE < 0 for regular states, and LLE = 0 for torus states) and draw the corresponding bifurcation diagram. The graph of Lyapunov exponent is obtained by simultaneously integrating system (2a–2d) and corresponding variational equations with the help of Wolf algorithm presented in [20]. Sample results are depicted in Fig. 2 where we show the bifurcation diagrams for x and y and corresponding graph of LLE. Two diagrams are superimposed. For the diagrams shown in red color, the final state at each iteration of the control parameter serves as the initial state for the next iteration while the blue ones are obtained with the same initial state [i.e., (−0.2, −0.5, −0.3, 0.2)]. From Fig. 2, a very rich and striking dynamic behavior is observed when the control parameter is monitored in tiny steps. The bifurcation structures include period doubling, symmetry breaking, crisis, and narrow windows of periodicity in chaotic domains. Those bifurcation structures are perfect in accordance with the Lyapunov spectra. From Fig. 2c, one can note that the system is weakly chaotic due to the smaller values of LLE (LLE ≤ 0.20). With the same parameters’ settings in Fig. 2, the various numerical computations of the phase portraits of the system with their corresponding power spectra were obtained confirming different scenarios to chaos reported previously. Briefly recall that for a periodic motion, all spikes in the power spectrum are harmonically related to the fundamental
3 Numerical simulations 3.1 Bifurcation and onset of chaos Following the results presented in the pioneering work [8,9], the dynamics of the coupled system with respect to the four parameters ε1 , ε2 , α, and β is well understood (see Sect. 1). Now, we concentrate on the effects of frequency detuning (i.e., parameter δ) on the dynamics of the coupled system. For this aim, Eqs. (2a–2d) are solved numerically to define routes to chaos, in our model, using the fourth-order Runge–Kutta integration formulas. For each value of δ, the system is integrated for a sufficiently long time and transient is discarded. The rest of the parameters are assigned the following values: ε1 = ε2 = 1.00, α = 2.00, and β = −1.50. The bifurcation diagrams result from the plot of maxima of x(t) and y(t) versus the control parameter δ that is varied in tiny steps. During the numerical experiment, the time step is always kept at t ≤ 0.005 and the computations are performed using variables and constants parameters in extended mode. We classify the type of motion by computing the largest numerical 1D Lyapunov exponent (LLE), (LLE > 0 for chaotic
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shown in Fig. 3 where both attractors (left column) and associated power spectrum (right column) are depicted for some discrete values of parameter δ. Obviously, four chaotic attractors can be distinguished in Fig. 3: two asymmetric solutions (Fig. 3c) and two symmetric attractors (Fig. 3d–e). The presence of different asymptotic behaviors for the same set of parameters is discussed in the next section. A zoom of the bifurcation diagram of Fig. 2 in the region of period-3 window and the corresponding graph of largest 1D LLE illustrating saddle-node (SN) bifurcation, period-doubling (PD), and interior crisis (IC) phenomenon is depicted in Fig. 4.
3.2 Coexistence of solutions
Fig. 2 Bifurcation diagram of the system showing maximal values of the coordinates X (a) and Y (b) and corresponding graph of Lyapunov exponent (c) versus the (increasing) control parameter δ. The rest of parameters are: ε1 = ε2 = 1.00; α = 2.00; andβ = −1.50
whereas a broadband noise-like power spectrum corresponds to a chaotic steady state. The periodicity of the attractor (i.e., total number of frequency components in a wave) is obtained by counting the number of spikes located at the left-hand side of the highest spike (the latter is included) of the spectrum. The results are
The coexistence of multiple attractors is one of the most exciting phenomena in nonlinear dynamics. Such a phenomenon has been observed in various systems including laser [21], biological system [22], and electronic circuits [23,24], to name a few. Concerning our model defined in Eqs. (2a–2d) (with δ = 0), the existence of multiple attractors for the same parameters’ settings has been discussed extensively by Pastor and co-workers in [9]. During our numerical experiment, the effects of the initial conditions on the behavior of our coupled system (with δ = 0) were also observed (see Fig. 2a, b). It was found that the behavior of our coupled system is sensitive to changes in the initial conditions. For instance, the phase portrait of Fig. 5a can be obtained under the initial conditions x0 = −0.2, u 0 = −0.50, y0 = −0.3, and v0 = 0.20; using the initial conditions x0 = −2.0, u 0 = −5.0, y0 = 3.0, and v0 = 2.0, a completely different solution (i.e., period3 attractor) is obtained in Fig. 5b. Therefore, considering the set of parameters in Fig. 5 and performing a scan of initial conditions (in the (u 0 , v0 ) plane (see Fig. 6), we have defined the domain of initial conditions in which the chaotic solutions can be found. In Fig. 6, we present the basin of attraction for the chaotic solution (black regions). These regions represent initial conditions that lead to chaotic trajectories. The white regions correspond to the regular period3 orbit. From a practical view point such behavior is not desirable and justifies the need for control. Detail study on this direction is beyond the scope of this paper.
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Fig. 3 Numerically computed phase portraits of the coupled system (left) and corresponding power spectra (right) showing various dynamic regimes: a Period-1 for δ = 3.50; b Period-2
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for δ = 3.00; c spiraling asymmetric chaos for δ = 2.75; d symmetric chaos for δ = 1.50; e star like chaos for δ = 0.50. The rest of parameters are those in Fig. 2
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Fig. 3 continued
4 Experimental study The aim of this section is to design and implement an appropriate analog circuit/simulator [6,25– 30] for the analysis of the model defined by the set of Eqs. (1a–1b) to verify the theoretical results obtained previously. The routes to chaos in the system are searched experimentally. Numerical and experimental phase portraits are compared.
4.1 Design of the analog simulator The circuit diagram of the proposed electronic simulator is provided in Fig. 7. In the diagram of Fig. 7, the upper network consisting of capacitors (C1 j ), resistors
(R1 j ), operational amplifiers (U1 , U2 A), and related analog multipliers (U3 , U4 ) implements oscillator X. The lower network models oscillator Y. Electronic multipliers, Mk (k = 1, 2, 3, 4), are the analog devices AD633JN versions of the AD633 four-quadrant voltage multiplier chips used to implement the nonlinear terms of our model. The operational amplifiers (TL084CN) and associated circuitry implement the basic operations of addition, subtraction, and integration. Adopting an appropriate time scaling, the simulator outputs can directly be displayed on an oscilloscope by connecting the output voltage of U1 C to the X input and the output voltage of U3 C to the Y input. It can be shown that the voltage at x (output of U1 C) and y (output of U3 C) satisfy the set of coupled nonlinear differential Equation (1a, 1b):
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Fig. 6 Structure of the section of the basin of attraction with (x0 , y0 ) = (−0.2, 0.3) and u 0 , v0 in the range [−20, 20]. White regions correspond to the three tone attractor, while black ones are associated to the chaotic attractor. The parameters are those in Fig. 5
Fig. 4 Blow-up of the bifurcation diagram of Fig. 2 in the region of period-3 window and corresponding graph of largest Lyapunov exponent b showing period doubling (PD), saddle-node bifurcation (SN) and interior crisis (IC)
1 R16 2 1 x− x¨ = − y x˙ R12 C12 100R14 C12 R15 R16 1 x− y , (7a) − R11 C11 R13 C12 R15
1 R26 2 1 y+ − x y¨ = y˙ R22 C22 100R24 C22 R25 1 R26 x . (7b) − y+ R21 C21 R23 C22 R25
Fig. 5 Coexistence of (solutions) a chaotic attractor a with a period-3 attractor b illustrating the multistability of the coupled system in the range: 0.210 ≤ δ ≤ 0.375 (see Fig. 2). The parameters are: ε1 = ε2 = 1.00, δ = 0.25, α = 2.00, β = −1.50
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Fig. 7 Schematic diagram of the complete analog simulator. The circuit is highly symmetric; the upper and lower networks implement oscillators X and Y respectively
Adopting a time unit of 33 × 10−4 s, the parameters of Eqs. (1a–1b) are expressed in terms of the values of capacitors and resistors values as follows: 33 33 R16 ; ε2 = 4 ; β=− ; ε1 = 4 10 R12 C12 10 R22 C22 R15 332 R26 ; δ+1= 8 . (8) α= R25 10 R23 R21 C21 C22 These mathematical definitions are valid if the following critical relationships between the values of resistors and capacitors are fulfilled:
106 R14 C12 = 106 R24 C22 = 33,
(9a)
10 R16 R11 C11 C12 = 33 .
(9b)
8
2
Mention that the time scaling process offers to the analog devices (operational amplifiers and analog multipliers) the possibility to operate under their bandwidth. Furthermore, the time scaling offers the possibility to simulate the behavior of the system at any given frequency depending on the values of resistors and capacitors used in the analog computer. Interested reader can consult the profitable literature [6] for the complete derivation procedure.
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Fig. 8 Experimental results [and corresponding numerical ones (right)] showing typical phase portraits of the coupled system: a period-1 for R23 = 2, 220 ; b period-2 for R23 = 2, 500 ; c spiraling asymmetric chaos for R23 = 2, 665 ; d symmet-
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ric chaos for R23 = 4, 000 , e star-like chaotic attractor for R23 = 6, 660 . For all pictures displayed, oscilloscope scales are: x = 200mV/div and y = 2V /div except for e where x = y = 1V /div
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Fig. 8 continued
4.2 Experimental results Now we concentrate on the experimental study of our model with the help of the electronic simulator. The effects of frequency detuning (i.e., parameter δ) on the behavior of the coupled system are investigated by monitoring a single resistor (R23 ), while keeping the rest of electronic components values constant. The following values of circuits components are selected: C j ( j = 11, 12, 21, 22) = 33n F, Rk (k = 11, 12, 13, 16, 17, 18, 21, 22, 26, 27, 28) = 10k , R15 = 5k R j ( j = 14, 24) = 100 , and R25 =
6, 667 . The choice of these values is justified by our intention to use the same sets of parameters for both numerical and experimental analyzes (i.e., ε1 = ε2 = 1.00, α = 2.00, β = −1.50). A real physical implementation of Fig. 7 is carried out using operational amplifiers (TL084CN), analog multipliers (AD633JN), high precision resistors, and capacitors with corresponding values fixed as above. The bias is provided by a 15V DC symmetric source. When monitoring the control resistors R23 , it is found that the electronic simulator experiences a rich and striking dynamical behavior. Some sample phase portraits obtained experimentally
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Fig. 9 Coexistence of solutions for R23 = 8k (i.e., δ = 0.25). Both the star like a and the period-3 attractor b appear randomly in experiment when switching on and off the power supply. The scales are: x = y = 1v/div
are shown in Fig. 8. Note the similarity between the numerically computed phase portraits and the experimental ones. Furthermore, from Fig. 8, one can note that the experimental circuit experiences the same bifurcation scenarios as those obtained numerically. In order to experimentally demonstrate/evidence the coexistence of solutions, the control resistor value is set to R23 = 8k (i.e., δ = 0.25). When switching on and off the power supply (and thereby randomly selecting initial conditions), both regular and star-like chaotic attractor were obtained. A comparison between experimental and numerical results is shown in Fig. 9. Once more a good similarity between numerical and experi-
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mental phase portraits is observed. The results obtained in this work show that the analog simulator is a powerful tool for the investigation of complex nonlinear models.
5 Conclusion In summary, this paper has investigated the dynamics of two mutually coupled van der Pol oscillators with particular attention on the effects of frequency detuning. By using classical nonlinear dynamic tools including phase portraits, frequency spectra, Lyapunov exponent,
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and bifurcation diagrams, we have shown that the coupled system can experience very complex behaviors depending on the choice of its parameters. We have found that the system can exhibit a variety of bifurcations namely symmetry breaking, period doubling, and crises when monitoring the frequency detuning parameter in tiny ranges. Furthermore, the multistability phenomenon marked by the coexistence of various attractors in phase space for some special parameters’ settings was also discussed. An appropriate analog simulator was designed and used for the investigations. The results from the analog simulator were compared to that of the numerical ones, and we obtained a very good agreement between the two approaches. The results obtained in this work show that the coupled van der Pol oscillators can demonstrate very complex behaviors in case of nonisochronism as well as in the critical case of isochronism (i.e., δ = 0) as earlier reported by Pastor and co-workers [8,9]. In contrast to [9] where only asymmetric coexisting solutions were reported, we have shown, in this work, that the model can exhibit symmetric coexisting attractors in addition to asymmetric ones. An interesting issue under consideration is the analysis of the effects of time delay on the dynamics of the coupled van der Pol oscillators (described in this paper) in spite of practical coupling conditions and possible applications. Another challenging task is the extension of the present study to the case of an array of coupled van der Pol oscillators as well as the synchronization of such arrays (as earlier initiated in [9]) in view of potential applications in chaos-based secure communication.
References 1. Aronson, D.G., Doedel, E.J., Othmer, H.G.: An analytical and numerical study of the bifurcations in a system of linearly-coupled oscillators. Physica D 25(1), 320–104 (1987) 2. Balanov, A., Janson, N., Postnov, D., Sosnovtseva, O.: Synchronization: From Simple to Complex. Springer, Berlin (2009) 3. Yu, P.E., Kuznetsov, A.P., Sataev, I.R., Turukina, L.V.: Synchronization and multi-frequency oscillations in a lowdimensional chain of the self-oscillators. Physica D 244(1), 36–49 (2013) 4. Chakraborty, T., Rand, R.H.: The transition from phase locking to drift in a system of two weakly coupled van der Pol oscillator. Int. J. Non-Linear Mech. 23(5–6), 369–376 (1988)
5. Kuznetsov, A.P., Seleznev, E.P., Stankevich, N.V.: Nonautonomous dynamics of coupled van der pol oscillators in the regime of amplitude death. Commun. Nonlinear Sci. Numer. Simul. 17(9), 3740–3746 (2012) 6. Chedjou, J.C., Fotsin, H.B., Woafo, P., Domngang, S.: Analog simulation of the dynamics of a van der Pol oscillator coupled to a Duffing oscillator. IEEE Trans. Circuits Syst. I 48, 748–756 (2001) 7. Gonzalez-Miranda, J.M.: Synchronization and Control of Chaos: An Introduction for Scientists and Engineers. Imperial College Press, London (2004) 8. Pastor, I., Perez-Garcia, V.M., Encinas-Sanz, F., Gerra, J.M.: Ordered and chaotic behavior of two coupled van der pol oscillators. Phys. Rev. E 48, 171–182 (1983) 9. Pastor-Diaz, I., Lopez-Fragas, A.: Dynamics of two coupled van der Pol oscillators. Phys. Rev. E 52(2), 1480–1489 (1995) 10. Poliasshenko, M., Mckay, S.R., Smith, C.W.: Hysteresis of synchronous–asynchronous regimes in a system of two coupled oscillators. Phys. Rev. A 43, 5638–5641 (1991) 11. Storti, D.W., Rand, R.H.: Dynamics of two strongly coupled van der Pol oscillators. Int. J. Non-Linear Mech. 17(3), 143–152 (1982) 12. Van der Pol, B.: On ‘relaxation-oscillation’. Philos. Mag. 7, 978–992 (1926) 13. Pinto, C.M., Machado, J.T.: Complex-order forced van der Pol. J. Vib. Control 18(14), 2201–2209 (2012) 14. Barbosa, R.S., Tenreiro Machado, J.A., Vinagre, B.M., Calderon, A.J.: Analysis of the van der Pol oscillator containing derivatives of fractional order. J. Vib. Control 13(9– 10), 1291–1301 (2007) 15. Pikovsky, A., Rosenblum, M.: Self-organized partially synchronous dynamics in populations of nonlinearly coupled oscillators. Physica D 238, 27–37 (2009) 16. Wirkus, S.: The dynamics of two coupled van der Pol oscillators with delay coupling. Nonlinear Dyn. 3(3), 205–221 (2002) 17. Jianming, Z., Xinsheng, G.: Stability and bifurcation analysis in the delay-coupled van der Pol oscillators. Appl. Math. Model. 34, 2291–2299 (2010) 18. Camacho, E., Rand, R., Howland, H.: Dynamics of two coupled van der Pol oscillators via a bath. Int. J. Solids Struct. 41(8), 1233–2143 (2004) 19. Nayfeh, A.H., Balakumar, B.: Applied Monlinear Dynamics: Analytical, Computational and Experimental Methods. Wiley, New York (1995) 20. Wolf, A., Swift, J.B., Swinney, H.L., Wastano, J.A.: Determining Lyapunov exponents from time series. Physica D 16, 285–317 (1985) 21. Masoller, C.: Coexistence of attractors in a laser diode with optical feedback from a large external cavity. Phys. Rev. A 50, 2569–2578 (1994) 22. Cushing, J.M., Henson, S.M., Blackburn, : Multiple mixed attractors in a competition model. J. Biol. Dyn. 1(4), 347– 362 (2007) 23. Vaithianathan, V., Veijun, J.: Coexistence of four different attractors in a fundamental power system model. IEEE Trans. Circuits Syst. I 46(3), 405–409 (1999) 24. Maggio, G.M., De Feo, O., Kennedy, M.P.: Nonlinear analysis of the Colpitts oscillator and application to design. IEEE Trans. Circuits Syst. 46, 1118–1130 (1999)
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Author's personal copy J. Kengne et al. 25. Itoh, M.: Synthesis of electronic circuits for simulating nonlinear dynamics. Int. J. Bifurcation Chaos 11(3), 605–653 (2001) 26. Hamil, D.C.: Learning about chaotic circuits with SPICE. IEEE Trans. Educ. 36(1), 28–35 (1991) 27. Yujun, N., Xingyuan, W., Mingjun, W., Huaguang, Z.: A new hyperchaotic system and its circuit implementation. Commun. Nonlinear Sci. Numer. Simul. 15, 3518–3524 (2010) 28. Chen, D.Y., Wu, C., Liu, C.F., Ma, X.Y., You, Y.J., Zhan, R.F.: Synchronization and circuit simulation of a new double-wing chaos. Nonlinear Dyn. 67, 1481–1504 (2012)
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29. Liu, Y.: Circuit implementation and finite-time synchronization of the 4D Rabinovich hyperchaotic system. Nonlinear Dyn. 67, 89–96 (2012) 30. Jia, H.Y., Chen, Z.Q., Qi, G.Y.: Topological horseshoes analysis and circuit implementation for a four-wing chaotic attractor. Nonlinear Dyn. 65(1–2), 131–140 (2011)
