Determinism: uniqueness in phase space

The nonlinear time series methods discussed in this book are motivated and based on the theory of dynamical systems; that is, the time evolution is defined in some phase space. Since such nonlinear systems can exhibit deterministic chaos, this is a natural starting point when irregularity is present in a signal. Eventually, one might think of incorporating a stochastic component into the description as well. So far, however, we have to assume that this stochastic component is small and essentially does not change the nonlinear properties. Thus all the successful approaches we are aware of either assume the nonlinearity to be a small perturbation of an essentially linear stochastic process, or they regard the stochastic element as a small contamination of an essentially deterministic, nonlinear process. If a given data set is supposed to stem from a genuinely non-linear stochastic processes, time series analysis tools are still very limited and their discussion will be postponed to Section 12.1.

Consider for a moment a purely deterministic system. Once its present state is fixed, the states at all future times are determined as well. Thus it will be important to establish a vector space (called a state space or phase space) for the system such that specifying a point in this space specifies the state of the system, and vice versa. Then we can study the dynamics of the system by studying the dynamics of the corresponding phase space points. In theory, dynamical systems are usually defined by a set of first-order ordinary differential equations (see below) acting on a phase space.