Determining higher order periods of the Logistic equation using a Chebyshev Transform

Abstract

The Logistic Equation is an iterative equation which exhibits complicated dynamics when the control parameter r is varied. When r is in between 3 and 3.57, a period doubling phenomena occurs in which periods of 2^N is observed, where N∈(1,∞). The Time Series of the Logistic Equation can be interpreted using Power Spectrum analysis method in which one can determine the periods present with a given r value. A popular method in determining the Power Spectrum is the Fast Fourier Transform (FFT). A difficulty encountered by the FFT however is presenting the Power Spectrum of higher order periods since the lower order periods have larger magnitudes, making the higher periods barely visible in the plot. In this paper, we explore the possibility of determining the Power Spectrum of the Logistic Equation using a Chebyshev Transform. The transform was constructed with the help of a package in Matlab called "Chebfun" in extracting the Chebyshev coefficients. The coefficients together with the scaled frequency bin was used in comparison to the power spectrum of the FFT. It was seen that the Chebyshev Transform had larger peaks compared to that of the FFT and had better numerical accuracy when it comes to determining the periods of the time series.