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

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 2N 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).