Optimal delay time and embedding dimension for delay-time coordinates by analysis of the global static and local dynamical behavior of strange attractors

Abstract

Fractal dimensions and Lyapunov exponents can be estimated faster and more accurately and efficiently with the knowledge of the optimal embedding dimension and delay time. Two methods are presented to calculate the optimal embedding parameters for Takens's delay-time coordinates. The first, called the fill factor, is a procedure that yields a global measure of phase-space utilization for (quasi) periodic and strange attractors and leads to a maximum separation of trajectories within the phase space. The second, which we call local deformation, is complementary to the fill factor. It describes the local dynamical behavior of points on the attractor and gives a measure of homogeneity of the local flow. Both methods were tested on numerical and experimental (Taylor-Couette flow) time series.