Detecting chaotic dynamics in observed time series

Deterministic chaos is about the erratic or stochastic behavior of solutions to deterministic equations of motion, i. e. deterministic dynamical systems, and has received much attention over recent years in many diverse fields including economics and finance.

Broadly speaking, a dynamical system is said to be deterministic when it comprises no random variables. In contrast, the observable behavior, i. e. the observed time series, of a dynamical system is said to be stochastic when the transitionof the system from one state to another can only be given a probabilistic description. One important implication of chaos is the limited predictability of the dynamics.

This is because a chaotic dynamical system has a property of "sensitive dependenceon initial conditions" : any two solution paths with arbitrarily close but not equal initial conditions will diverge at exponential rates. Globally, however, the solutionpaths remain within a bounded set if the system is dissipative, i. e. if the system has internal "friction".

Thus, the future development of the dynamics is essentially unpredictable even though the underlying equations of motion are deterministic.Because predictability is important, especially in economics and finance, the question of whether observed time series, e.g. exchange rate series, are generated by a chaotic dynamical system or not is of special importance