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  • Print publication year: 2013
  • Online publication date: June 2014

2 - Complexity and chaos in dynamical systems

Summary

Abstract

Dynamical systems are represented by mathematical models that describe different phenomena whose state (or instantaneous description) changes over time. Examples are mechanics in physics, population dynamics in biology and chemical kinetics in chemistry. One basic goal of the mathematical theory of dynamical systems is to determine or characterise the long-term behaviour of the system using methods of bifurcation theory for analysing differential equations and iterated mappings. Interestingly, some simple deterministic nonlinear dynamical systems and even piecewise linear systems can exhibit completely unpredictable behaviour, which might seem to be random. This behaviour of systems is known as deterministic chaos.

This chapter aims to introduce some of the techniques used in the modern theory of dynamical systems and the concepts of chaos and strange attractors, and to illustrate a range of applications to problems in the physical, biological and engineering sciences. The material covered includes differential (continuous-time) and difference (discrete-time) equations, first- and higher order linear and nonlinear systems, bifurcation analysis, nonlinear oscillations, perturbation methods, chaotic dynamics, fractal dimensions, and local and global bifurcation.

Readers are expected to know calculus and linear algebra and be familiar with the general concept of differential equations.

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Alligood, K.T., Sauer, T. D. and Yorke, J. A. (1996). Chaos:AnIntroduction to Dynamical Systems. Springer.
Boccara, N. (2004). Modeling Complex Systems. Springer.
Glendinning, P. (1994). Stability, Instability and Chaos. Cambridge University Press.
Strogatz, S.H. (2000). Nonlinear Dynamics and Chaos. Westview Press.