Potential risk of variable parameter cascade chaos system

Abstract

Lyapunov index is one of criteria for testing whether the system is in a chaotic state, and its value represents the developed level of system chaotic state. To study the Lyapunov index characteristic of cascade chaotic system and reveal disturbance mechanism among subsystems in cascade chaotic system, the following researches are carried out. First, the disturbance model among subsystems is constructed from the viewpoint of pseudo noise disturbance, Lyapunov index difference between without and with external noise influence is investigated. Then the conclusion that disturbances among subsystems can be considered as pseudo noise influence is drawn. Second, the conclusion is proved that cascade system Lyapunov index is not the algebraic sum of each independent subsystems, but the one of each subsystems which consist of pre disturbances. Then taking the logistic representation for example, nine cascade systems are designed to prove this conclusion. And some novel characteristics and phenomena are found from the above investigations. They are (a) the phenomenon of “more is less”, that is, Lyapunov index will decrease with the increase of cascade levels, and the phenomenon of “A miss is as good as mile”; (b) even each independent subsystems is chaotic, the cascade system needs not to be chaotic; conversely, even each independent subsystems is not chaotic, the cascade system may be chaotic; (c) whether the cascade system is chaotic is associated with the order of subsystem. Finally, it is pointed out that cascade level has the influences of pros and cons on cascade system, thus revealing the latent hazard of parametric cascade chaotic system. The research result can provide important theoretic foundation for system security and the scientific evaluation of encryption keys.