Possibilities of the Discrete Fourier Transform for Determining the Order-Chaos Transition in a Dynamical System

Abstract

This paper is devoted to a discussion of the Discrete Fourier Transform (DFT) representation of a chaotic finite-duration sequence. This representation has the advantage that is itself a finite-duration sequence corresponding to samples equally spaced in the frequency domain. The Fast Fourier Transform (FFT) algoritm allows us an effective computation, and it can be applied to a relatively short time series. DFT representation requirements were analized and applied for determining the order-chaos transition in a nonlinear system described by the equation $x[n+1]=rx[n](1-x[n])$. Its effectiveness was demonstrated by comparing the results with those obtained by calculating the largest Lyapounov exponent for the time series set, obtained from the logistic equation.