ABSTRACT
This chapter describes procedures for quantifying attractors generated by nonlinear systems. Nonlinear dynamical systems are characterized by attractors. Gencay demonstrated that the nonlinear dynamics of the Henon attractor, including bifurcation behavior as one of the parameters changed, is approximated accurately by the neural network. Hyperchaos is represented by an attractor that expands in more than one orthogonal dimension. It is characterized by having more than one positive Lyapunov exponent. The chapter describes several useful quantitative indices for summarizing the dynamical characteristics of possibly nonlinear time series. Numerical examples have illustrated how these indices can be estimated from experimental data. Dimensionality indices provide evidence for low-dimensional, possibly fractal, systems but cannot by themselves determine whether a time series is chaotic. The frequent concern expressed about the possibly contaminating effects of noise on parameter estimates and the need to test hypotheses about various dynamical properties of systems.
