An analytic criterion for generalized synchronization in unidirectionally coupled systems based on the auxiliary system approach

Synchronization in area-preserving maps: Effects of mixed phase space and coherent structures

The problem of synchronization of coupled Hamiltonian systems presents interesting features due to the mixed nature (regular and chaotic) of the phase space. We study these features by examining the synchronization of unidirectionally coupled area-preserving maps coupled by the Pecora-Caroll method. The master stability function approach is used to study the stability of the synchronous state and to identify the percentage of synchronizing initial conditions. The transient to synchronization shows intermittency with an associated power law. The mixed nature of the phase space of the studied map has notable effects on the synchronization times as is seen in the case of the standard map. Using finite-time Lyapunov exponent analysis, we show that the synchronization of the maps occurs in the neighborhood of invariant curves in the phase space. The phase differences of the coevolving trajectories show intermittency effects, due to the existence of stable periodic orbits contributing locally stable directions in the synchronizing neighborhoods. Furthermore, the value of the nonlinearity parameter, as well as the location of the initial conditions play an important role in the distribution of synchronization times. We examine drive response combinations which are chaotic-chaotic, chaotic-regular, regular-chaotic, and regular-regular. A range of scaling behavior is seen for these cases, including situations where the distributions show a power-law tail, indicating long synchronization times for at least some of the synchronizing trajectories. The introduction of coherent structures in the system changes the situation drastically. The distribution of synchronization times crosses over to exponential behavior, indicating shorter synchronization times, and the number of initial conditions which synchronize increases significantly, indicating an enhancement in the basin of synchronization. We discuss the implications of our results.

Network-scale effect on synchronizability of fully coupled network with connection delay

Network-scale effect on synchronizability of fully coupled network with connection delay is investigated in this paper. The master stability function, which governs the stability of synchronization manifold, is first obtained by separating the synchronization manifold direction from other transverse directions. Then, by introducing a new time variable in the master stability function, it is shown the effect of connection delay can be weakened with the increase of network scale, and thus, in contrast to the situation without connection delay, large network scale is more positive to the synchronizability of fully coupled network with connection delay. Those findings are confirmed by the studies on two specific networks with nodes of typical nonlinear dynamical systems.

A simple approach to achieve modified projective synchronization between two different chaotic systems

A new approach, the projective system approach, is proposed to realize modified projective synchronization between two different chaotic systems. By simple analysis of trajectories in the phase space, a projective system of the original chaotic systems is obtained to replace the errors system to judge the occurrence of modified projective synchronization. Theoretical analysis and numerical simulations show that, although the projective system may not be unique, modified projective synchronization can be achieved provided that the origin of any of projective systems is asymptotically stable. Furthermore, an example is presented to illustrate that even a necessary and sufficient condition for modified projective synchronization can be derived by using the projective system approach.

Inapplicability of an auxiliary-system approach to chaotic oscillators with mutual-type coupling and complex networks

The auxiliary system approach being de facto the standard for the study of generalized synchronization in the unidirectionally coupled chaotic oscillators is also widely used to examine the mutually coupled systems and networks of nonlinear elements with the complex topology of links between nodes. In this Brief Report we illustrate by two simple counterexamples that the auxiliary-system approach gives incorrect results for the mutually coupled oscillators and therefore to study the generalized synchronization this approach may be used only for the drive-response configuration of nonlinear oscillators and networks.