Dynamical Systems, Stability, and Chaos

In this expository and resources chapter we review selected aspects of the mathematics of dynamical systems, stability, and chaos, within a historical framework that draws together two threads of its early development: celestial mechanics and control theory, and focussing on qualitative theory. From this perspective we show how concepts of stability enable us to classify dynamical equations and their solutions and connect the key issues of nonlinearity, bifurcation, control, and uncertainty that are common to time-dependent problems in natural and engineered systems. We discuss stability and bifurcations in three simple model problems, and conclude with a survey of recent extensions of stability theory to complex networks.Comment: 28 pages, 10 figures. 26/04/2007: The book title was changed at the last minute. No other changes have been made. Chapter 1 in: J.P. Denier and J.S. Frederiksen (editors), Frontiers in Turbulence and Coherent Structures. World Scientific Singapore 2007 (in press

Adaptive Switching Control and Synchronization of Chaotic Systems with Uncertainties

In this paper, a new adaptive switching control scheme is presented to solve control and synchronization problems. Based on Lyapunov stability theory, an adaptive control law is applied to globally stabilize chaotic systems and achieve states synchronization of two chaotic systems whose dynamics are subjected to the system disturbances and/or some unknown parameters. Simulation examples, the chaotic Chen's system and Chua's circuit, are given to show the feasibility and effectiveness of the proposed theory and method