Synchronization with positive conditional Lyapunov exponents

Abstract

Synchronization of chaotic system may occur only when the largest conditional Lyapunov exponent of the driven system is negative. The synchronization with positive conditional Lyapunov reported in a recent paper (Phys. Rev. E, 56, 2272Rev. E, 56, (1997) is a combined result of the contracting region of the system and the finite precision in computer simulations. PACS number(s): 05.45.+b; 1 Sensitivity to initial conditions is a generic feature of chaotic dynamical systems. Two chaotic orbits, starting from slightly different initial points in the state space, separate exponentially with time, and become totally uncorrelated. As a result, independent identical chaotic systems cannot synchronize with each other. The sensitivity is quantitatively described by positive Lyapunov exponent(s) in the Lyapunov exponent spectrum of the chaotic system.However, chaotic systems linked by common signal can synchronize with each other. Several cases could be distinguished. In the first case, a replica subsystem driven by chaotic signals of the chaotic system can synchronize identically with the drive system [1][2][3][4][5], if the largest conditional Lyapunov is negative. This is referred to as identical synchronization.Secondly, a driven system, which is not a replica of the drive system, however, may not achieve identical synchronization, but generalized synchronization [6][7][8], if the largest conditional Lyapunov exponent is negative. Two identical systems, driven by the same signal, thus may come to the same final state due to the negative largest conditional Lyapunov exponent.Lyapunov exponents are also employed to characterize behavior of random dynamical systems [9]: the system is chaotic (non-chaotic) when the largest Lyapunov exponent is positive (negative). The sensitivity of a chaotic system may also be suppressed by noise, and identical chaotic systems subjected to common noise can synchronize with each other. Maritan and Banavar[10] studied the behavior of the noise-driven logistic maps and reported synchronization phenomenon. It turned out that the observed synchronization was an outcome of finite precision in numerical simulations [11,12], while the Lyapunov exponent of the noisy logistic map is positive [11].Very recently, Shuai et al [13] claimed that synchronization can be achieved with positive conditional Lyapunov exponents. In a one-way coupled map lattice, they observed, through computer simulations, synchronization of spatiotemporal chaos with many positive components in the conditional Lyapunov exponent spectrum. Based on these results, they drew the conclusion that the conditional Lyapunov exponents cannot be used as a criterion for synchronous chaotic systems.Whether such a claim is true is of great importance for our understanding of synchronization. In this paper, we reexamined such synchronization phenomenon, revealing that it is yet another example of round-off induced phenomenon.In [13], Shuai et al studied a driven one-way coupled map latticewhere x 0 (t) is a hyperchaotic signal f...

Analytical and numerical studies of noise-induced synchronization of chaotic systems

“…The analysis of Pikovsky was confirmed by Longa et al [42] who studied the logistic map with arbitrary numerical precision. The criterion of negative Lyapunov exponent has also been shown to hold for other types of synchronization of chaotic systems and Zhou and Lai [43] noticed that previous results by Shuai, Wong and Cheng [44] showing synchronization with a positive Lyapunov exponent were again an artifact of the limited precision of the calculation. In addition to the above criticisms, Herzel and Freund [45] and Malescio [46] pointed out that the noise used to simulate Eq.…”

Frequency and phase synchronization of two coupled neurons with channel noise

“…For example, the identical neurons which are not coupled or weakly coupled but subjected to a common noise may achieve complete synchronization. Actually, this is a general results for all the dynamical system [8,9,10,11,12]. Both independent and correlated noises are found to enhance phase synchronization of two coupled chaotic oscillators below the synchronization threshold [13,14].…”

Phase synchronization in fractional differential chaotic systems

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