Characterising chaos-hyperchaos transition using correlation dimension

Abstract

Transition to hyperchaos is uaually studied by computing the spectrum of Lyapunov Exponents (LE). But such a procedure can be employed mainly when the equations governing the dynamical system are known. However, if the information available on the system is only through time series, the method becomes difficult to implement. We show that the transition to hyperchaos is followed by a sudden change in the topological structure of the underlying attractor. Our numerical results indicate that the transition to hyperchaos can be characterized accurately through the computation of correlation dimension (D 2) from time series. We use two standard time delayed hyperchaotic systems as examples since, for such systems, D 2 varies smoothly as a function of the time delay τ which can be used as the control parameter.