Basis of the time series analysis on a nonequilibrium process

To extend and deepen knowledge of time series analysis, this paper discusses the question of fractional frequencies with chaos signals. It gives the multiple-periodic time function x_(N)(t) with the N-stage bifurcation, and puts forth the ultra-Fourier series to express the frequency spectrum of new time function x_(N)(t). It extends the Shannon sampling theorem by the fractional frequency f/z(n), to put forward the theorems of the time-domain sampling and frequency-domain sampling used in the processing of nonequilibrium signal