Complete Dynamics and Synchronization

Abstract

This chapter presents a Ying–Yang theory for nonlinear discrete dynamical systems with consideration of positive and negative iterations of discrete iterative maps. In existing analysis, the solutions relative to “Yang” in nonlinear dynamical systems are extensively investigated. However, the solutions pertaining to “Ying” in nonlinear dynamical systems are presented. A set of concepts on “Ying” and “Yang” in discrete dynamical systems are introduced. Based on the Ying–Yang theory, the complete dynamics of discrete dynamical systems can be discussed. A discrete dynamical system with the Henon map is investigated as an example. The companion and synchronization of discrete dynamical systems will be introduced, and the corresponding conditions are developed. The synchronization dynamics of Duffing and Henon maps will be discussed.