SIGNATURE OF CHAOS IN POWER SPECTRUM

We investigate the nature of the numerically computed power spectral d ensity P(f, N, tau) Of a discrete (sampling time interval, tau) and fi nite (length, N) scalar time series extracted from a continuous time c haotic dynamical system. We highlight how P(f, N, tau) differs from th e true power spectrum and from the power spectrum of a general stochas tic process. Non-zero tau leads to aliasing; P(f, N, tau) decays at hi gh frequencies as [pi tau/sin pi tau f](2), which is an aliased form o f the 1/f(2) decay. This power law tail seems to be a characteristic f eature of all continuous time dynamical systems, chaotic or otherwise. Also the tail vanishes in the limit of N --> infinity, implying that the true power spectral density must be band width limited. In strikin g contrast the power spectrum of a stochastic process is dominated by a term independent of the length of the time series at all frequencies .