Characterization of Regular and Chaotic Motions

Abstract

We have pointed out in the previous chapters that many important nonlinear dynamical systems in physics, chemistry and biology clearly display different types of regular and chaotic behaviours, depending upon factors such as the strength of control parameters, initial conditions, nature of external forcings, and so on. Thus in order to identify definitively whether a given motion of a typical dynamical system is, for example, periodic or quasiperiodic or chaotic, one would like to have specific quantitative measures in addition to the various qualitative features such as the geometric structures of attractors, stability features and so on. Moreover we have seen earlier that the motion of typical nonlinear systems undergo characteristic qualitative changes as certain control parameters smoothly change, namely bifurcations. These are again identified by means of the changes of the system attractors or phase space structures and stability properties. However, since deterministic chaos is associated with random behaviour arising from sensitive dependence on initial conditions, it is quite natural to expect quantitative criteria to distinguish between chaotic and regular motions should be based on statistical measures. Indeed there are many such measures available in the literature for this purpose and some of the most prominent of them are

• (1) Lyapunov exponents

• (2) power spectrum

• (3) correlation function and

• (4) dimension.