On the Lyapunov Spectra and Synchronization Phenomena of Time-Delayed Chaos System

Abstract

In the present work, we shall investigate the Lyapunov spectrum as well as synchronization phenomena of the time-delayed chaos system with nonlinear Tent mapping to involve a chaotic dynamics. In practice we shall examine chaotic property in such a time-delayed chaos model in terms of the Lyapunov spectrum analyses through the Gram-Schmidt scheme. Then one shall examine the relation between the chaotic behavior and the characteristic solutions in terms of the Lambert function. In addition it will be confirmed that there exists a synchronization regime with respect to a coupling parameter for such a coupled time-delayed chaos system on the basis of the Lyapunov spectrum and the characteristic exponent solutions. According to our computer simulations, it will be also found that the synchronization phenomena in the coupled chaos system may be involved not only in chaotic states but also in non-chaotic states depending on both the delay-time and the coupling parameters. As future works, such a time-delayed chaos system with the multi-delay time parameters and/or with other nonlinear mapping functions such as Chebysheff function recently investigated are considered to be worthwhile from the view point of attractive applications to the secure-communications.