LOCAL OR DYNAMICAL DIMENSIONS OF NONLINEAR SYSTEMS INFERRED FROM OBSERVATIONS

Abstract

In time delay reconstruction of the phase space of a system from observed scalar data, one requires a time lag and an integer embedding dimension. The minimum embedding dimension, dE, may be larger than the actual local dimension of the underlying dynamics, dL. The embedding theorem only guarantees that the attractor of the system is unfolded in the integer dE greater than 2dA with dA being the attractor dimension. We present two methods for determining the dimension, dL≤dE, of the underlying dynamics. The first relies on the local Lyapunov exponents of the dynamics, and the second seeks an optimum dimension for prediction of the time series for steps forward and then backward in time. We demonstrate these methods on several examples. Model building of the dynamics should take place in the dL-dimensional space.