Possibilities of the Discrete Fourier Transform for Determining the Order-Chaos Transition in a Dynamical System
Carlos R. Fadragas, Juan V. Lorenzo-Ginori, Ruben Orozco-Morales
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.
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