Using the Magnitude-Squared Coherence for Determining Order-Chaos Transition in a System Governed by Logistic Equation Dynamics

Abstract

This paper is devoted to show the results obtained by using the magnitude-squared coherence for determining order-chaos transition in a system described by the logistic equation dynamics. For determining the power spectral density of a chaotic finite-duration discrete-time sequence the Welch average periodogram method was used. This method has the advantage that can be applied to any stationary signal by using the discrete Fourier transform (DFT) representation of a discrete-time series which allows an effective computation via fast Fourier transform (FFT) algorithm, and that can be applied to a discrete-time series shorter than that required by nonlinear dynamical analysis methods. The estimate of the Inverse Average Magnitude-Squared Coherence Index (IAMSCI) for each discrete-time series in the set obtained from the logistic mapping was calculated. The control parameter, {\bf r}, ranges in the interval [2.8,4] for producing the discrete-time series set. When the condition $r \geq 3.57$ is satisfied, each discrete-time series exhibited a positive value for IAMSCI estimate, indicating a high level of coherence loss of the signal and corresponding to a chaotic behavior of the dynamical system. Its effectiveness was demonstrated by comparing the results with those obtained by calculating the largest Lyapounov exponent of the time series set obtained from the logistic equation.