Multistability and synchronization of chaos in maps with “Internal” coupling (original) (raw)

Synchronization in driven versus autonomous coupled chaotic maps

Physical Review E, 2005

The phenomenon of synchronization occurring in a locally coupled map lattice subject to an external drive is compared to the synchronization process in an autonomous coupled map system with similar local couplings plus a global interaction. It is shown that chaotic synchronized states in both systems are equivalent, but the collective states arising after the chaotic synchronized state becomes unstable can be different in these two systems. It is found that the external drive induces chaotic synchronization as well as synchronization of unstable periodic orbits of the local dynamics in the driven lattice. On the other hand, the addition of a global interaction in the autonomous system allows for chaotic synchronization which is not possible in a large coupled map system possessing only local couplings.

Generalized synchronization of chaos in noninvertible maps

Physical Review E, 2002

The properties of functional relation between a noninvertible chaotic drive and a response map in the regime of generalized synchronization of chaos are studied. It is shown that despite a very fuzzy image of the relation between the current states of the maps, the functional relation becomes apparent when a sufficient interval of driving trajectory is taken into account. This paper develops a theoretical framework of such functional relation and illustrates the main theoretical conclusions using numerical simulations.

Two-mode chaos and its synchronization properties

Physical Review E, 2005

Using a simple model with bimodal dynamics, we investigate the intra-and inter-system entrainment of the two different time scales involved in the chaotic oscillations. The transition between mode-locked and modeunlocked chaos is analyzed for a single system. For coupled oscillators, we demonstrate full and partial synchronization patterns depending on the adjustment between the fast and slow time scales and reveal the embedded structure of the corresponding synchronization regions.

Synchronization of chaos in coupled systems

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 2000

The stability of synchronous chaos of coupled oscillators with diffusive and gradient couplings is investigated. The stability boundaries of all transverse modes can be simultaneously drawn by justifying the boundary of a single mode, according to a scaling relation. Therefore, the distribution of stable and unstable regions can be explicitly shown in control parameter space. Bifurcations through different unstable modes, leading to different spatial orders, are analyzed.

Effect of parameter mismatch on the mechanism of chaos synchronization loss in coupled systems

Physical Review E, 1998

Using the example of two coupled logistic maps, we investigate the effect of nonidentical subsystems on the bifurcations of saddle periodic orbits embedded in a symmetric chaotic attractor. These bifurcations determine the process of loss of chaos synchronization. We show that if bifurcations conditioned by the symmetry of the system take part in the synchronization loss process, nonidentity changes the bifurcation scenario of the transition to a nonsynchronous regime. In this case, for example, the transition to the bubbling behavior is determined not by bifurcation of an orbit embedded in the chaotic attractor but by the smooth shift of it and the saddle-repeller bifurcation of the birth of new orbits in the vicinity of the quasisymmetric region.

In-phase and antiphase complete chaotic synchronization in symmetrically coupled discrete maps

Discrete Dynamics in Nature and Society, 2002

We consider in-phase and antiphase synchronization of chaos in a system of coupled cubic maps. Regions of stability and robustness of the regime of in-phase complete synchronization was found. It was demonstrated that the loss of the synchronization is accompanied by bubbling and riddling phenomena. The mechanisms of these phenomena are connected with bifurcations of the main family of periodic orbits and orbits appeared from them. We found that in spite of the in-phase synchronization, the antiphase self-synchronization of chaos is impossible for discrete maps with symmetric diffusive coupling. For achieving antiphase synchronization we used method of controlled synchronization by addition feedback. The region of the controlled antiphase synchronization and phenomena which accompany the loss of the synchronization are presented.

Mechanism for the partial synchronization in three coupled chaotic systems

Physical Review E, 2005

We investigate the dynamical mechanism for the partial synchronization in three coupled one-dimensional maps. A completely synchronized attractor on the diagonal becomes transversely unstable via a blowout bifurcation, and then a two-cluster state, exhibiting on-off intermittency, appears on an invariant plane. If the newly created two-cluster state is transversely stable, then partial synchronization occurs on the invariant plane; otherwise, complete desynchronization takes place. It is found that the transverse stability of the intermittent two-cluster state may be determined through the competition between its laminar and bursting components. When the laminar ͑bursting͒ component is dominant, partial synchronization ͑complete desynchroni-zation͒ occurs through the blowout bifurcation. This mechanism for the occurrence of partial synchronization is also confirmed in three coupled multidimensional invertible systems, such as coupled Hénon maps and coupled pendula.

Coupling effect on the occurrence of partial synchronization in four coupled chaotic systems

Physics Letters A, 2006

We study the coupling effect on the occurrence of partial synchronization in four coupled one-dimensional maps by varying a parameter w (0 w 1) which tunes the "weight" in the next-nearest-neighbor coupling from the local nearest-neighbor coupling (w = 0) to the global coupling (w = 1). As the coupling parameter ε decreases and passes a threshold value ε * , the fully synchronized attractor on the diagonal becomes transversely unstable via a blowout bifurcation, and then a partially synchronized or completely desynchronized attractor appears depending on the value of w. For the case of local coupling (w = 0), partial synchronization occurs on an invariant plane. However, as w increases and passes a threshold value w * , a transition from partial synchronization to complete desynchronization takes place. Thus, for w * < w 1, a fully desynchronized attractor, occupying a finite four-dimensional volume, appears. The dynamical mechanism for the occurrence of partial synchronization is investigated through competition between the laminar and bursting components of the intermittent two-cluster state born via the blowout bifurcation. Another type of partial synchronization, which occurs through a dynamical stabilization of an unstable orbit, is also discussed for both the local and global couplings. These results for the partial synchronization are also confirmed in a system of four coupled pendula.

Generalized synchronization of chaos in directionally coupled chaotic systems

Physical Review E - PHYS REV E, 1995

Synchronization of chaotic systems is frequently taken to mean actual equality of the variables of the coupled systems as they evolve in time. We explore a generalization of this condition, which equates dynamical variables from one subsystem with a function of the variables of another subsystem. This means that synchronization implies a collapse of the overall evolution onto a subspace of the system attractor in full space. We explore this idea in systems where a response system y(t) is driven with the output of a driving system x(t), but there is no feedback to the driver. We lose generality but gain tractability with this restriction. To investigate the existence of the synchronization condition y(t)=P(x(t)) we introduce the idea of mutual false nearest neighbors to determine when closeness in response space implies closeness in driving space. The synchronization condition also implies that the response dynamics is determined by the drive alone, and we provide tests for this as well. Examples are drawn from computer simulations on various known cases of synchronization and on data from nonlinear electrical circuits. Determining the presence of generalized synchronization will be quite important when one has only scalar observations from the drive and from the response systems since the use of time delay (or other) embedding methods will produce "imperfect" coordinates in which strict equality of the synchronized variables is unlikely to transpire. PACS number(s): 05.45. +b, 84.30. Ng, 07.05.t variables of the two systems are proportional to each other. There are two categories of systems where this 'Electronic address: rulkov Ihamilton.