THE BREAKDOWN OF SYNCHRONIZATION IN SYSTEMS OF NONIDENTICAL CHAOTIC - PDF document

International Journal of Bifurcation and Chaos, Vol. 11, No. 10 (2001) 2705–2713 c � World Scientific Publishing Company THE BREAKDOWN OF SYNCHRONIZATION IN SYSTEMS OF NONIDENTICAL CHAOTIC OSCILLATORS: THEORY AND EXPERIMENT JENNIFER CHUBB ∗ , ERNEST BARRETO ∗ , † , PAUL SO and BRUCE J. GLUCKMAN Departments of Mathematics, Physics & Astronomy, and The Krasnow Institute for Advanced Study, Mail Stop 2A1, George Mason University, Fairfax, VA 22030, USA Received October 17, 2000; Revised January 24, 2001 The synchronization of chaotic systems has received a great deal of attention. However, most of the literature has focused on systems that possess invariant manifolds that persist as the coupling is varied. In this paper, we describe the process whereby synchronization is lost in systems of nonidentical coupled chaotic oscillators without special symmetries. We qualitatively and quantitatively analyze such systems in terms of the evolution of the unstable periodic orbit structure. Our results are illustrated with data from physical experiments. 1. Introduction two individual neurons are identical. Our methods, which are applicable to experimental data, form the Systems of several interacting nonlinear elements foundation for discussing synchronization in non- present a very rich variety of behavior. Of par- identical coupled chaotic systems in a more general ticular interest has been the phenomenon of chaos context without making reference to special sym- synchronization. Most of the relevant literature has metries or invariant manifolds. We illustrate our considered coupled systems of identical elements for results with both numerical calculations and exper- which the dynamics can be understood in terms imental data from electronic circuits. of an invariant synchronization manifold. In this The synchronization of coupled chaotic oscilla- paper, we discuss a more general method of analysis tors, a phenomenon first noticed many years ago of coupled systems and apply it to an experimen- [Fujisaka & Yamada, 1983] is most conveniently tal system. In particular, we focus on the process described in terms of a synchronization manifold: of desynchronization, with special emphasis on sys- when synchronized, the time evolution occurs on a tems of nonidentical coupled oscillators. We draw restricted set embedded in the full state space. For particular attention to this case, since it represents systems of coupled identical elements, this synchro- almost every experimental situation of interest: in nization manifold is contained within a plane (or hy- practice, it is very difficult to prepare sets of truly perplane) of symmetry and exists for a wide range identical oscillators in physical systems. Further- of coupling. However, for systems that do not pos- more, in biological systems, natural oscillators oc- sess special symmetries, such as systems of coupled cur with considerable variability. For example, even nonidentical elements, this invariant synchroniza- within each of several different classes of neurons, no tion manifold may become extremely complicated ∗ JC (experiment) and EB (theory) contributed equally to the authorship of this work. † Author for correspondence. E-mail: ebarreto@gmu.edu 2705

2706 J. Chubb et al. or even be destroyed as the degree of coupling is 2. Systems decreased. We will describe the phenomenology of desyn- Previous work has focused on systems of identi- chronization in a general unidirectionally coupled cal coupled elements for which the synchronization system of nonidentical chaotic maps. 1 Consider a manifold M persists for a large range of coupling system of the form: and can be easily identified. On M , the individ- ual components evolve identically in time, and are � x → f ( x ) said to exhibit identical synchrony [Pecora & Car- (1a) y → G ( x, y ; c ) . roll, 1990]. As the coupling decreases from a fully synchronized state, a bubbling bifurcation [Ashwin Systems such as Eq. (1) are known in the mathe- et al. , 1994, 1996; Venkataramani et al. , 1996a, matical literature as skew products or extensions. 1996b] occurs when an orbit within M (usually of Here we assume that the coupling is such that at low period [Hunt & Ott, 1996]) loses transverse sta- c = 1, the x and y dynamics are in a state of gener- bility. In the presence of noise or small asymme- alized synchrony (i.e. y = φ ( x )) [Afraimovich et al. , tries, a typical trajectory quickly approaches and 1986; Rulkov et al. , 1995; Kocarev & Parlitz, 1996] spends a long time in the vicinity of M , but makes and that at c = 0, the x and y dynamics are com- occasional excursions. As the coupling is further de- pletely independent of one another. f and G may creased, the blowout bifurcation [Ott & Sommerer, be of any dimension. For illustration of our theo- 1994] is observed when M itself becomes trans- retical results, we use in our discussion below the versely unstable (on average). simplest case The concept of (differentiable) generalized syn- chrony (GS) [Afraimovich et al. , 1986; Rulkov et al. , G ( x, y ; c ) = cf ( x ) + (1 − c ) g ( y ) (1b) 1995; Kocarev & Parlitz, 1996; Hunt et al. , 1997] ex- tends these ideas. GS relaxes the condition that the and take f and g to be quadratic maps with differ- state variables evolve identically, and only requires ent parameters. (Another simple option is to use that they be functionally related. However, as the dissimilar H´ enon maps.) Our arguments are not coupling is reduced, this function may become ex- specific to these choices, and our experimental sys- tremely complicated. In particular, if the system tem is in fact more complicated. lacks special symmetries (as in the case when the For our experimental system, we constructed coupled elements are not identical), M may not ex- two nearly identical circuits D and R based on the ist, or its structure may be so complicated that the generalized Duffing equation practical identification of bubbling-type or blowout- type bifurcations is impossible. In this situation, d 2 x dt 2 + ν dx the work described above does not carry over, and dt + N L ( x ) = A sin( t ) (2) a more general description of the desynchronization process beyond the state of generalized synchrony where N L ( x ) is a nonlinear term, typically N L ( x ) = is needed. ( x 3 − x ). In our circuits the nonlinear element We find that the entire desynchronization was constructed with standard resistors and diodes. process can be fruitfully studied by considering the The response of this nonlinear element is shown in evolution of the system’s unstable periodic orbit Fig. 1. A nondimensional parameterization of the (UPO) structure as the coupling is varied over a equations of our circuits is large range [Barreto et al. , 2000; So et al. , 2000]. Our analysis, discussed in Sec. 3, provides both a d 2 x dt 2 + αdx dt − ( βx 3 + γx ) = δ sin( t ) (3) qualitative and a quantitative understanding of the desynchronization process, with the advantage of where α = 0 . 124, β = 0 . 238 and γ = 1 . 00. Each cir- not making reference to invariant manifolds. We cuit received a common sinusoidal input, the zero- introduce in Sec. 2 the numerical and experimental phase of which was also used to trigger stroboscopic models that have been used in this work, and in measurements. In order to break the symmetry, the Sec. 4 we report the first experimental verification of these theoretical results. amplitude δ of the sinusoidal input to the response 1 It has been shown that unidirectional and bidirectionally coupled systems are locally equivalent. See [Josi´ c, 1998].