Spectral analysis and generation of certain highly oscillatory curves related to chaos, part 2: Calculation aspects

The Fourier spectrum of a curve on a given interval, as represented by the Fourier series, is very reflective of the oscillatory behavior of the curve as the “weights” of the Fourier coefficients can indicate the individual prominence of frequencies of oscillations. But, unfortunately, the calculations of the Fourier coefficients can often get quite cumbersome. s paper, we first review the oscillatory behavior of chaotic maps as time series, and discuss certain properties between the exponential growth of total variations and the Fourier coefficients. Then we will discuss some simple techniques that can simplify the calculations of Fourier coefficients. In the process, one can also obtain valuable information about the asymptotic properties of the Fourier coefficients.