Dynamical Systems, Stability, and Chaos

Details Dynamical Systems, Stability, and Chaos Dynamical Systems, Stability, and Chaos

2007-02-23

arxiv.org/abs/nlin/0702044v2

Nonlinear Sciences

Chaotic Dynamics

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

Scientific paper

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.

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