Intermittent lag synchronization in a nonautonomous system of coupled oscillators

Physics Letters A 338 (2005) 141 149 www.elsevier.com/locate/pla Intermittent lag synchronization in a nonautonomous system of coupled oscillators Alexander N. Pisarchik a,, Rider Jaimes-Reategui b a Centro de Investigaciones en Optica, Loma del Bosque, 115, Col. Lomas del Campestre, 37150 Leon, Guanajuato, Mexico b Universidad de Guadalajara, Campus Universitario Los Lagos, Enrique Díaz de León, Paseo de Las Montañas, Lagos del Moreno 47460, Jalisco, Mexico Received 7 April 2004; received in revised form 10 February 2005; accepted 16 February 2005 Communicated by A.R. Bishop Abstract Synchronization properties of two identical mutually coupled Duffing oscillators with parametric modulation in one of them are studied. Intermittent lag synchronization is observed in the vicinity of saddle-node bifurcations where the system changes its dynamical state. This phenomenon is seen as intermittent jumps from phase to lag synchronization, during which the chaotic trajectory visits closely a periodic orbit. Different types of intermittent lag synchronization are demonstrated and the simplest case of period-one lag synchronization is analyzed. 2005 Elsevier B.V. All rights reserved. PACS: 02.60.Cb; 05.45.Pq; 05.45.Xt; 05.45.Jn Keywords: Synchronization; Duffing oscillator; Intermittency In the last years, synchronization of coupled oscillatory systems has attracted a great attention in almost all areas of natural sciences, engineering and social life. The main reason of such interest is important practical applications which include communications, modelling brain and cardiac rhythm activity, earthquake dynamics, etc. (see, e.g., [1,2] and references therein). Different types of synchronization, complete [3], phase [4], lag[5], generalized [6], and almost synchronization [7] have been identified. Many of these theoretical finding have been experimentally verified in real systems, including biological and medical systems [8] and chaotic lasers [9]. However, the most of theoretical and experimental works have been carried out on synchronization of autonomous (self-oscillatory) systems of coupled anharmonic oscillators. Significantly less attention has been given to a study of synchronization * Corresponding author. E-mail address: apisarch@cio.mx (A.N. Pisarchik). 0375-9601/$ see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.02.025

142 A.N. Pisarchik, R. Jaimes-Reategui / Physics Letters A 338 (2005) 141 149 effects in nonautonomous coupled systems, in which oscillations and synchronization effects can be observed under periodical modulation of a system parameter or a state variable. Among nonautonomous systems, the most extensively investigated model is the Duffing oscillators with external forcing. The Duffing oscillator is one of the prototype systems of nonlinear dynamics and was successfully explored to model a variety of physical processes such as stiffening strings, beam buckling, nonlinear electronic circuits, superconducting Josephson parametric amplifiers, ionization waves in plasmas, and even biological and medical processes (see [10] and references therein). For example, Kapitaniak [11] studied the transition to hyperchaos for a system of coupled Duffing oscillators and Landa and Rosenblum [12] investigated synchronization phenomena for different types of coupled systems. The bifurcation structure of two coupled periodically driven Duffing oscillators in space of the parameters of modulation was studied by Kozłowski et al. [13] for the case of single-well potentials and by Kenfack [14] for double-well potentials. The effect of phase difference in mutually coupled chaotic oscillators was considered by Yin et al. [15]. Recently, Raj et al. [16] investigated coexisting attractors and synchronization of chaos in two coupled Duffing oscillators with two driving forces. In the all abovementioned works the external driving force was applied to a variable of one or both coupled oscillators. However, in many real experiments it is more convenient to modulate a system parameter rather than a state variable. Moreover, the parametric modulation is commonly used in electromechanical, electronic, and laser systems and, in particular, in communications [10]. Nevertheless, only few works were devoted to a study of coupled oscillators with parametric modulation [17,18]. It is known, that in chaotic autonomous nonidentical oscillators, a symmetric coupling can lead to phase synchronization [4]. This regime is characterized by a perfect locking of the phases of the two signals, whereas the two chaotic amplitudes remain uncorrelated. Lag synchronization consists of hooking one system to the output of the other shifted in time of a lag time τ lag [s 1 (t) = s 2 (t τ lag )] [5]. Recently, Boccaletti and Valladares [19] characterized intermittent lag synchronization of two nonidentical symmetrically coupled Rössler systems. They observed intermittent bursts away from the lag synchronization and described this phenomenon in terms of the existence of a set of lag times τlag n (n = 1, 2,...), such that the system always verifies s 1(t) s 2 (t τlag n ) for a given n. Inthis work we study a similar phenomenon in a nonautonomous system with parametric modulation. As distinct from a self-oscillatory system, all oscillations in a parametrically modulated system are the forced oscillations induced and driven by an external periodic forcing and hence they are always phase-locked with the forcing, even when the oscillators are identical (see, for example, [1]). In this Letter we will show that at certain conditions the regime of intermittent lag synchronization appears in the nonautonomous system of the coupled oscillators. Similarly to [19], this regime can be considered as the intermittent behavior between phase synchronization and lag synchronization. Generally, dynamics of two identical nonlinear oscillators is governed by the equation, ẍ + γ ẋ = dv(x) (1) dx, where x (x, y), γ is a damping factor, and V(x) is a two-dimensional anharmonic potential function of the coupled oscillators. The potential functions for two symmetric Duffing oscillators can be expressed as follows: V(x,y) = a 2 x2 + b 4 x4 + c 2 x2 y 2, (2) V(y,x) = a 0 (3) 2 y2 + b 4 y4 + c 2 x2 y 2, where a, a 0 and b are parameters and c is a coupling coefficient. In the following, we discuss some bifurcation properties of two mutually coupled, identical Duffing oscillators subjected to parametric modulation in one of them (master oscillator) a = a 0 [1 m sin ωt], (4)

A.N. Pisarchik, R. Jaimes-Reategui / Physics Letters A 338 (2005) 141 149 143 Fig. 1. Double-well potential function (Eq. (2)). where m and ω are the modulation depth and frequency. Here we consider only the case of a double-well potential, i.e., a 0 < 0, with positive b (b >0). This case is more interesting for modelling a real experiment for signal transmission, because the oscillators have nonzero stable equilibrium points as distinct from a single-well case. Without modulation (m = 0 and a = a 0 ) the system (1) (3) has nine possible steady states Q i = ( x 1, x 2, x 3, x 4 ) (i = 1,...,9): ( ) Q 1 (0, 0, 0, 0), Q 2,3 (0, 0, ± ), ab, 0 Q 4,5 ± ab, 0, 0, 0, Q 6 9 (± a ) b + c, 0, ± a b + c, 0. For simplicity we consider the case of γ = 0.4, a 0 = 0.25, and b = 0.5. The three-dimensional potential function Eq. (2) at fixed coupling strength c = 0.5 isshowninfig. 1. There exist a saddle equilibrium point x u = 0 and a conjugate pair of stable equilibrium points at x s =± 0.5 cy 2. The positions of stable points x s depend on the coupling strength as shown in Fig. 2. For the potential function Eq. (3) of another subsystem, the equilibrium points are at y u = 0 and y s =± 0.5 cx 2. For our parameters, point Q 1 is a saddle and points Q 2 5 are sinks for any value of c 0, while the stability of the other four solutions, Q 6 9, depends on c: they are sinks for c<0.5 and saddles for c 0.5. For stability analysis and numerical simulation, it is convenient to transform the second order differential Eq. (1) into a system of first-order differential equations of the following form: ẋ 2 = γx 2 a 0 [1 m sin x 5 ]x 1 bx1 3 cx 1x3 2, ẋ 1 = x 2, (5) (6) ẋ 3 = x 4, (7) ẋ 4 = γx 4 a 0 x 3 bx3 3 cx2 1 x 3, (8) ẋ 5 = ω. (9)

144 A.N. Pisarchik, R. Jaimes-Reategui / Physics Letters A 338 (2005) 141 149 Fig. 2. Position of stable equilibrium points x s versus coupling strength c and variable y. The system (5) (9) in vector notation can be written in the form Ẋ = Ψ(X; p), (10) where X =[x 1,x 2,x 3,x 4,x 5 ] t is the state-space vector ([...] t being transpose), Ψ =[ψ 1,ψ 2,ψ 3,ψ 4,ψ 5 ] t is the function space, and p = (γ, a 0,b,c,m,ω) is an element of the parameter space. This system generates a flow Φ ={Φ T } on the phase space R 4 S 1 and S 1 = R/T is the circle of length T = 2π/ω. The frequency f = ω/2π and the amplitude m of the parametric modulation are used as control parameters in the bifurcation diagrams that will be presented in the following. Due to the symmetric potential of both oscillators, the coupled system (5) (9) possesses the same symmetry properties as a single Duffing oscillator [20], i.e., it is symmetric since the transformation S : (x 1,x 2,x 3,x 4,x 5 ) ( x 1, x 2, x 3, x 4,x 5 + π) leaves (5) (9) invariant. Therefore, this system can support symmetric orbits and also asymmetric ones which are not invariant with the transformation S. Saddle-node, period-doubling, Hopf, and symmetry-breaking bifurcations occur in the coupled system (5) (9). We use a perturbation analysis to analyze solution stability in the Poincaré section defined by Σ ={(x 1,x 2,x 3,x 4,x 5 ) R 4 S 1 : x 5 = const}. The equations for small deviation δx from the trajectory X(t) are ( ) δx = L ij X(t) δx, i,j = 1, 2,...,5, (11) where L ij = Ψ i / x j is the Jacobian 5 5 matrix of derivatives. Eq. (11) for system (5) (9) becomes: δẋ 1 0 1 0 0 0 δx 1 δẋ 2 a δẋ 3 = 0 + a 0 m sin x 5 3bx1 2 cx2 3 γ 2cx 1 x 3 0 a 0 mx 1 cos x 5 δx 2 0 0 0 1 0 δẋ 4 2cx 1 x 3 0 a 0 3bx3 2 cx2 δx 3 1 γ 0 δx. 4 δẋ 5 0 0 0 0 0 δx 5 (12) When using the unit matrix I as the initial condition X 0 = I, the resulting solution X(T ) after one period T of the oscillation represents the linearized Poincaré map P. The solution of Eq. (11) can be found in the form X(t) = X 0 exp [ ( )] tl ij X(0), (13) where the time-independent matrix L ij (X(0)) and the matrix X 0 contains the initial conditions. Let µ k = λ k + iω k (k = 1, 2, 3, 4) be the eigenvalues of the matrix L ij (X(0)). Then the eigenvalues of the linearized Poincaré map P may be written as ρ k = e Tµ k = e Tλ k [ cos(t Ω k ) + i sin(t Ω k ) ]. (14)

A.N. Pisarchik, R. Jaimes-Reategui / Physics Letters A 338 (2005) 141 149 145 Globally, if all λ k < 0, then these eigenvalues spiral into the origin of the complex plane when the modulation frequency ω is decreased, i.e., all sufficiently small perturbations tend towards zero as t and the steady state (nodes, saddle nodes, spiral) is stable. For larger values of the modulation amplitude m, the influence of the nonlinearity becomes more important and the eigenvalues move towards the critical value +1 (saddle-node bifurcation) or 1 (period-doubling bifurcation). In the case of the coupled Duffing oscillators (5) (9) the symmetry of their potentials (Fig. 1) implies the existence of a root P = H of the Poincaré map P = P 0 P = ( H) 2, where the map H is obtained by integrating the variables of the Poincaré map over half a period of the oscillation [13]. The system (5) (9) exhibits a wide range of dynamical regimes ranging from steady states to periodic and chaotic oscillations and represents different types of synchronization. The general information analysis of the dynamical regimes which can be expected from the system (5) (9) allows us to reveal the following possible situations [18]. (i) When both the coupling strength and modulation amplitude are sufficiently small (c 0.1 and m 0.1), the mismatch of resonance natural frequencies of the two oscillators is also small, and bifurcation diagrams of the variables for each subsystem (x 1 and x 3 ) have a standard shape of a linear resonance response, i.e., the two oscillators are completely synchronized. (ii) With increasing c, the response becomes nonlinear and the resonances are shifted to the high-frequency region, while the mismatch of the resonant frequencies increases. Thus, the oscillations occur to be shifted in time, i.e., lag synchronization takes place. At c<0.5 and m<0.5, both subsystems oscillate in a periodic regime with ω (period-1 regime) over the all frequency range of parametric modulation. Finally, when c 0.5 the nonlinear resonances disappear in the system response. (iii) A further increase in coupling c leads to the appearance of coexisting multiple periodic attractors and steady state solutions [18]. Although the two oscillators are almost identical, the origin of the lag in their oscillations is the same as in the case of nonidentical autonomous oscillators [5,19], namely, a mismatch of their nonlinear resonance frequencies, that appears due to the nonlinear coupling and because the modulation is applied only to one of the oscillators. In this Letter we are interested in chaotic regimes. Chaotic oscillations are observed for relatively high modulation amplitudes (m>0.75) and low couplings (c < 0.25). As distinct from autonomous systems where the coupling strength is an unique control parameter, in our system three parameters (c, m, and f ) can be controllable. Therefore, dynamics of the system (5) (9) can be analyzed in the space of these three parameters. In Fig. 3 we present the codimensional-two bifurcation diagrams in the (c,m) (Fig. 3(a)) and (m, f )(Fig. 3(b)) parameter spaces. In the figures we plot the saddle-node bifurcation lines which bound different dynamical regimes: periodic orbits (PO), one-well chaos (OWC), cross-well chaos (CWC), and hopping oscillations (HO) (periodic windows). In our system all oscillations are excited by external periodic modulation and hence they are always phase-locked with the forcing. When the system oscillates in a periodic regime, the state variables of the two subsystems are shifted in time, i.e., lag synchronization takes place. Within very narrow parameter range, close to the saddle-node bifurcations, short periodic windows are observed in time, where system jumps from chaos to local periodicity (Figs. 4 and 5). During these jumps, the chaotic trajectory visits closely a periodic orbit. We identify this phenomenon with intermittent lag synchronization (ILS). Two kinds of ILS are seen in Figs. 4 and 5: one-state period-1 (P1) ILS (Fig. 4) and cross-state period-2 (P2) ILS (Fig. 5). In the former case, the x 1 -trajectory jumps intermittently from cross-well chaos to the small P1 orbit around each of the potential wells and back, whereas in the latter case, the trajectory jumps from cross-well chaos to the large P2 orbit oscillating between the two wells. Figs. 4(b) and 5(b) display the enlarged parts of the time series where lag synchronization is observed. The modulation parameters for the regimes shown in Figs. 4 and 5 are indicated in Fig. 3(b) by the dots. These dots lie on the saddle-node bifurcation lines which bound, respectively, the one-well and cross-well chaotic regimes and the regimes of hopping oscillations and cross-well chaos. Rosenblum et al. [5] proposed to describe the occurrence of ILS as a situation where during some periods of time the system verifies x 3 (t) x 1 (t τ) 1(τ being a lag time), but where bursts of local nonsynchronous behavior may occur. This phenomenon was identified with on-off intermittency [21] and the bursts from lag synchronization was found to result from the small, but negative value of the second global Lyapunov exponent of the system, so that the trajectory visits attractor regions where the local Lyapunov exponent is still positive. In the

146 A.N. Pisarchik, R. Jaimes-Reategui / Physics Letters A 338 (2005) 141 149 Fig. 3. Codimensional-two bifurcation diagrams in parameter spaces of (a) coupling strength c and modulation depth m for f = 0.1 and(b) modulation frequency f and depth m for c = 0.1. Intermittent lag synchronization occurs in the vicinity of the saddle-node bifurcation lines which bound different dynamical regimes: one-well chaos (OWC), cross-well chaos (CWC), hopping oscillations (HO), and periodic orbits (PO). The dots indicate the parameters for which the regimes of period-1 ILS (P1) and period-2 ILS (P2) are observed (see Figs. 4 and 5). case of periodically driven systems, this condition should be modified to be x 3 (t) x 1 (t τ)/η+ x 1 (t) 1, (15) where η is a proportional coefficient between the alternative amplitudes of the variables in the synchronous regime, η = (x1 max x1 min )/(x3 max x1 min ). As distinct from autonomous (self-oscillatory) coupled oscillators, in our system chaotic oscillations of the two variables, x 1 (t) and x 3 (t), are always phase synchronized. Therefore, the intermittent jumps from chaos to the windows of periodicity can be considered as the intermittent behavior from phase to lag synchronous regimes. Of course, the criterion Eq. (15) can be used only for characterization of the simplest case of P1 ILS (Fig. 4). For higher periodic regimes (P2, P3,...)ofILS,theshapes of oscillations in the periodic windows are different for two oscillators, and hence more complex relation would be required. The temporal behavior of 0 for the case of P1 ILS for τ 0 = 116 is shown in Fig. 6(a). Similarly to Rosenblum et al. [5], we may characterize lag synchronization by the similarity function S(τ), defined as the time-averaged difference, conveniently normalized to the geometrical average of the two mean

A.N. Pisarchik, R. Jaimes-Reategui / Physics Letters A 338 (2005) 141 149 147 Fig. 4. One-state period-1 intermittent lag synchronization of cross-well chaos (small orbit synchronization). (a) Time series of active (x 1 )and passive (x 3 ) oscillators, (b) enlarged part of (a) demonstrating synchronous windows of periodicity in one of the states. c = 0.1, f = 0.107, m = 0.8. Fig. 5. Two-state period-2 of intermittent lag synchronization of cross-well chaos (large orbit synchronization). (a) Time series of active (x 1 ) and passive (x 3 ) oscillators, (b) enlarged part of (a) demonstrating synchronous windows of periodicity between two states. c = 0.1, f = 0.087, m = 0.8.

148 A.N. Pisarchik, R. Jaimes-Reategui / Physics Letters A 338 (2005) 141 149 Fig. 6. (a) Time series of 0 in period-1 intermittent lag synchronization regime for τ 0 = 116. The windows with 0 can be viewed as the low-dimensional lag synchronous attractor. (b) Similarity function S (τ) vs lag time τ. There are exist a global minimum σ 0atτ 0 = 116 and local minima for smaller and larger times τ n (n = 1, 2, 3,...). signals S 2 (τ) = 2 [ x 2 1 (t) x2 3 (t) ]1/2, and search for its global minimum σ = min τ S(τ), forτ 0 0. The dependence of the similarity function on the lag time shown in Fig. 6(b) resembles the similar dependence reported previously by Boccaletti and Valladares [19] for Rössler systems. Looking at Fig. 6(b), one can see that, besides a global minimum σ 0atτ 0 = 116, S(τ) displays many other local minima at smaller and larger lag times τ n (n = 1, 2, 3,...). This means that the system (5) (8), besides being lag synchronized during some periods of time with respect to the global minimum τ 0, occasionally visits closely other lag configurations corresponding to the condition Eq. (15). The depth of the nth local minimum is closely related to the fraction of time that the corresponding lag configuration is closely visited by the system. The different lag times τ n can be expressed by the relation τ n τ 0 + nt, where T is the period of external modulation or the return time of the limit cycle onto the Poincaré section. The anharmonicity in function S(τ) results from the anharmonicity of the periodic oscillations due to the high nonlinearity of the system. In summary, we have studied the synchronization properties of two mutually coupled oscillators with parametric modulation in one of them and have found synchronous states, which we identify with intermittent lag synchronization. In the intermittent states, the system during its temporal evolution occasionally changes the behavior (16)

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