—journal of April 2005
physics pp. 503–511
Intermittent lag synchronization in a driven system of coupled oscillators
ALEXANDER N PISARCHIK1 and RIDER JAIMES-RE ´ATEGUI2
1Centro de Investigaciones en Optica, Loma del Bosque 115, Lomas del Campestre, Leon 37150, Guanajuato, Mexico
2Universidad de Guadalajara, Campus Universitario Los Lagos, Enrique D´ıaz de Le´on, Paseo de Las Monta˜nas, Lagos del Moreno 47460, Jalisco, Mexico
E-mail: apisarch@cio.mx
Abstract. We study intermittent lag synchronization in a system of two identical mu- tually coupled Duffing oscillators with parametric modulation in one of them. This phe- nomenon in a periodically forced system can be seen as intermittent jump from phase to lag synchronization, during which the chaotic trajectory visits a periodic orbit closely.
We demonstrate different types of intermittent lag synchronizations, that occur in the vicinity of saddle-node bifurcations where the system changes its dynamical state, and characterize the simplest case of period-one intermittent lag synchronization.
Keywords. Duffing oscillator; synchronization; chaos.
PACS Nos 02.60.Cb; 05.45.Pq; 05.45.Xt
Synchronization of coupled oscillatory systems has attracted great interest in al- most all areas of natural sciences, engineering and social life in the past few years because of its important practical applications which include communications, mod- elling brain and cardiac rhythm activity, earthquake dynamics, etc. (see, e.g., [1,2]
and references therein). Different types of synchronizations such as complete syn- chronization [3], generalized synchronization [4], phase synchronization [5], lag syn- chronization [6], measure synchronization [7], almost synchronization [8], and antic- ipated synchronization [9] have been identified. Many of these theoretical findings have been experimentally verified in real systems, such as biological and medical [10]
systems and chaotic lasers [11]. However, most of the theoretical and experimen- tal works are devoted to investigation of synchronization effects in self-oscillatory autonomous systems. Significantly less attention has been given to a study of syn- chronization of driven coupled oscillators, in which either a system parameter or a state variable is periodically modulated.
One of the most extensively investigated models of nonlinear systems with external forcing is the Duffing oscillator. The Duffing oscillator was success- fully explored to model a variety of physical processes such as stiffening strings, beam buckling, nonlinear electronic circuits, superconducting Josephson parametric
amplifiers, and ionization waves in plasmas, as well as biological and medical pro- cesses. For example, the transition to hyperchaos and synchronization phenomena in a system of coupled Duffing oscillators were investigated respectively by Kapita- niak [12] and Landa and Rosenblum [13]. The bifurcation structure of two coupled periodically driven Duffing oscillators in space of modulation parameters was stud- ied by KozÃlowskiet al [14] for the case of single-well potentials and by Kenfack [15]
for the case of double-well potentials. The effect of phase difference in mutually coupled chaotic oscillators was considered by Yin et al [16]. Recently, Raj et al [17] investigated coexisting attractors and synchronization of chaos in two coupled Duffing oscillators with two driving forces. Usually, an external driving force is ap- plied to a state variable in one of the oscillators or to variables of both oscillators.
However, in a real experimental practice it is more convenient to modulate a system parameter rather than a state variable. The parametric modulation is commonly used in electromechanical and electronic systems, in particular, for communication purpose. Nevertheless, only few works were devoted to a study of synchronization of parametrically modulated systems [18,19].
Recently, phase synchronization of chaotic oscillations has been found by Rosen- blum et al [5] in autonomous non-identical oscillators with symmetric coupling.
This regime is characterized by a perfect locking of the phases of the two signals, while the two chaotic amplitudes remain uncorrelated. Later, the same authors observed lag synchronization, that consists of hooking one system to the output of the other shifted in time of a lag time τlag[s1(t) =s2(t−τlag)] [6]. The latter phe- nomenon has been observed experimentally in two unidirectionally coupled Chua’s circuits [20].
The present research has been stimulated by the work of Boccaletti and Val- ladares [21] who characterized intermittent lag synchronization (ILS) of two non- identical symmetrically coupled R¨ossler systems. They observed intermittent bursts away from the lag synchronization to asynchronous regime and described this phe- nomenon in terms of the existence of a set of lag times τlagn (n = 1,2, ...), such that the system always obeys s1(t)' s2(t−τlagn ) for a given n. In this work we demonstrate a similar behavior in a driven system of coupled oscillators. Unlike a self-oscillatory system, all oscillations in our system are the forced oscillations induced and driven by the external periodic forcing and hence they are always phase-locked with the forcing, and so, in a driven system ILS manifests itself as an intermittent behavior between phase and lag synchronization.
Generally, we consider a system of two coupled subsystems: ˙x=g(x,y;A) and
˙
y = h(x,y;B), where x and y are phase-space variables, A and B are sets of parameters, andg andhare the corresponding nonlinear velocity fields. If one of the parameters, saya(a∈A), is a function of time, i.e., a=ϕ(t) while the other parameters are constants, the two subsystems are not completely identical, i.e.
g6=h. Nevertheless, we may consider the subsystems to be almost identical when averaged in time parameters are the same, i.e., hAi = B(h· · ·i denotes temporal average).
Dynamics of two identical nonlinear oscillators can be governed by the equation
¨
x+γx˙ =−dV(x)
dx , (1)
wherex≡(x, y),γis a damping factor, andV(x) is a two-dimensional anharmonic
potential function of the coupled oscillators. The potential functions for symmetric Duffing oscillators can be expressed as follows:
V(x, y) = a 2x2+b
4x4+δ
2x2y2, (2)
V(y, x) =a0
2 y2+b 4y4+δ
2x2y2, (3)
where a, a0 andbare parameters and δis a coupling coefficient. Without external modulation, eqs (1)–(3) do not have periodic solutions. The external modulation is added to one of the oscillators in the form of parametric modulation
a=a0[1−msin(2πf t)], (4)
wheremandf are the modulation depth and frequency. Here we consider only the case of a double-well potential, i.e., whena0<0, with positiveb(b >0). This case is more useful for modelling a real experiment for signal transmission, because the oscillators have non-zero stable equilibrium points unlike a single-well case.
Equations (1)–(4) can be written as
˙
x1=x2, (5)
˙
x2=−γx2−a0[1−msin(2πf t)]x1−bx31−δx1x23, (6)
˙
x3=x4, (7)
˙
x4=−γx4−a0x3−bx33−δx21x3. (8) The general information analysis of dynamical regimes, which can be expected from eqs (5)–(8) with parametersγ= 0.4,a0=−0.25, andb= 0.5, reveals the following possible situations: (i) When both coupling strength and modulation amplitude are sufficiently small (δ.0.1 andm.0.1), the mismatch of natural frequencies of the two oscillators is also small and the bifurcation diagrams of the variables of each subsystem (x1andx3) have the standard shape of a linear resonance response (fig- ure 1a), i.e., the two oscillators are completely synchronized. (ii) Whenδincreases, the response becomes nonlinear and both resonance frequencies move to a higher frequency region and the mismatch also increases (figure 1b). Thus, the oscillations occur to be shifted in time, i.e., lag synchronization takes place. For δ <0.5 and m <0.5, both subsystems oscillate in a periodic regime withf (period 1) over all frequency range. Finally, when δ→0.5 the nonlinear resonances disappear in the system response (figure 1c). (iii) A further increase inδleads to the appearance of coexisting multiple periodic attractors and steady-state solutions (figure 1d) [19].
Although the two oscillators are almost identical, the origin of the lag in their oscillations is the same as in the case of non-identical autonomous oscillators [6,21], 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 paper we are interested mainly in a chaotic region. Chaotic oscillations in the system (eqs (5)–(8)) are observed at relatively high modulation amplitudes (m >0.75) and low couplings (δ <0.25).In parametrically modulated coupled sys- tems, three parameters (δ,m, andf) can be used as control parameters, and hence
Figure 1. Bifurcation diagrams for driven (x1) and passive (x3) oscilla- tors with respect to modulation frequency form= 0.1 for different coupling strengths (a)δ= 0.1, (b) 0.4, (c) 0.5, and (d) 0.51.
the dynamics may be analysed in the space of these three parameters. In figure 2 we present the co-dimensional-two bifurcation diagrams in the (δ, m) (figure 2a) and (m, f) (figure 2b) parameter spaces. The saddle-node bifurcation (SNB) lines bound different dynamical regimes: periodic orbits (PO), one-well chaos (OWC), cross-well chaos (CWC), and hopping oscillations (HO) (periodic windows).
As mentioned above, all oscillations in our system are excited by the external periodic modulation and therefore they are always phase-locked with the forcing.
In a periodic regime, the state variables of the two subsystems are shifted in time, i.e., they are lag synchronized. Within very narrow parameter range, close to SNBs, short windows of periodicity appear intermittently in chaotic time series. In such an intermittent regime, the system occasionally jumps from chaos to local periodicity (figures 3 and 4). During these jumps, the chaotic trajectory visits a periodic orbit closely, i.e., we have a sort of intermittent lag synchronization observed in the driven system. The regime of ILS appears in regions of the parameter space in the vicinity of SNBs and is associated with on–off intermittency [22]. This type of intermittency (in other words, modulational intermittency) appears only in a driven system in which the external forcing is applied in the form of either noise, or chaos, or periodic modulation. On–off intermittency has been also detected experimentally in different dynamical systems [23–27].
Figure 2. Co-dimensional-two bifurcation diagrams in parameter spaces of (a) coupling strengthδ and modulation depthmforf = 0.1 and (b) modu- lation frequencyfand depthmforδ= 0.1. Intermittent lag synchronization occurs in the vicinity of the saddle-node bifurcation lines which bound dif- ferent 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) shown in figures 3 and 4 are observed.
In figures 3 and 4 we demonstrate two kinds of ILS in the driven system: one- state period-1 (P1) ILS (figure 3) and cross-state period-2 (P2) ILS (figure 4). In the former case, thex1-trajectory jumps intermittently from cross-well chaos to the small P1 orbit around each of the potential well and back. In the latter case, the trajectory jumps from cross-well chaos to the large P2 orbit oscillating between the two wells. Figures 3b and 4b display the enlarged parts of the time series where lag synchronization is observed. The regimes shown in figures 3 and 4 are observed for the parameters marked in figure 2 by the dots. These dots lie on the SNB lines which bound respectively the one-well and cross-well chaotic regions and the regions of hopping oscillations and cross-well chaos. Similar regimes of ILS are found near the other SNB lines.
Recently, Rosenblumet al [6] proposed to describe the occurrence of ILS as a situ- ation where during some periods of time the system verifies ∆≡ |x3(t)−x1(t−τ)|
¿ 1 (τ being a lag time), but where bursts of a local non-synchronous behavior
Figure 3. One-state period-1 intermittent lag synchronization of one-well chaos (small-orbit synchronization). (a) Time series of driven (x1) and passive (x3) oscillators, (b) enlarged part of (a) demonstrating synchronous unstable period-1 orbits around the potential wells. δ= 0.1,f= 0.107,m= 0.8.
Figure 4. Two-state period-2 intermittent lag synchronization of cross-well chaos (large-orbit synchronization) inx direction. (a) Time series of driven (x1) and passive (x3) oscillators, (b) enlarged part of (a) demonstrating syn- chronous unstable period-2 orbits. δ= 0.1,f= 0.087,m= 0.8.
may occur. They identified this phenomenon with on–off intermittency and the bursts from lag synchronization were found to result from a small, but negative value of the second global Lyapunov exponent of the system, so that the trajec- tory visits attractor regions where the local Lyapunov exponent is still positive. In periodically driven systems, this condition should be modified to satisfy
∆≡ |x3(t)− hx3(t)i −η[x1(t−τ)− hx1(t)i]| ¿1, (9) whereη=¡
xmax3 −xmin3 ¢ /¡
xmax1 −xmin1 ¢
is a proportionality coefficient between the alternative amplitudes of the variables in the synchronous regime. The coefficient
Figure 5. Time series of ∆0(τ0 = 116) in period-1 intermittent lag syn- chronization regime. The windows with ∆0 ≈ 0 can be viewed as the low-dimensional ‘lag synchronous’ attractor. (b) Similarity function S2(τ) vs. lag timeτ. There exists a global minimum atτ0= 116 and local minima for smaller and larger timesτn(n= 1,2,3, ...).
η is introduced because the modulation is applied only to one of the oscillators (x1) and hence x3 < x1. Therefore, the averages of the two variables are also different and hence they should be normalized. Of course, the criterion (eq. (9)) can be used only for characterization of the simplest case of P1 ILS shown in figure 3. For higher periods (P2, P3,...) of ILS, the shapes of the oscillations in the periodic windows are different for two oscillators, and hence more complex relationship is required.
The temporal behavior of ∆0(τ0= 116) for the case of P1 ILS is shown in figure 5a.
If the criterion eq. (9) is correct, the function ∆0should be approximately equal to zero in the windows of the lag synchronous regime, as seen in figure 5a.
Similarly to Rosenblum et al. [6], we may characterize lag synchronization by similarity function S(τ), defined as time averaged difference ∆, conveniently nor- malized to the geometrical average of the two mean signals
S2(τ) =
∆2®
[hx21(t)ihx23(t)i]1/2, (10)
and search for its global minimum σ = minτS(τ), for τ0 6= 0. The similarity function vs. the lag time shown in figure 5b resembles the dependence similar to
those reported previously by Boccaletti and Valladares [21] for two coupled R¨ossler systems. Looking at figure 5b, one can see that, besides a global minimum at τ0 = 116, S(τ) displays many other local minima at smaller and larger lag times τn (n= 1,2,3, ...). The depth of the nth local minimum is closely related to the fraction of time when the corresponding lag configuration is closely visited by the system. The different lag timesτn can be expressed by the relationτn ≈τ0+nT, where T is the period of external modulation or the return time of the limit cycle onto the Poincar´e section. The anharmonicity in function S2(τ) results from the anharmonicity of the periodic oscillations due to high nonlinearity of the system.
In summary, we have studied synchronization properties of two mutually coupled Duffing oscillators with parametric modulation in one of them and have found syn- chronous states, which we identify with intermittent lag synchronization. In the intermittent states, the system during its temporal evolution occasionally changes its behavior from phase synchronization to lag synchronization. The regime of in- termittent lag synchronization appears in regions of the parameter space close to saddle-node bifurcations and is associated with on–off intermittency. Recently, methods of closed-loop control [28] and open-loop control [29] of this phenomenon have been suggested. We believe that the main features of the synchronization phe- nomena obtained in the coupled Duffing oscillators are common for a wide class of parametrically driven systems and may be observed in experiments.
Acknowledgements
ANP thanks S Boccaletti for valuable discussions. This work has been supported through a grant from the Institute Mexico-USA of the University of California (UC-MEXUS) and Consejo Nacional de Ciencia y Tecnologia (CONACyT).
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