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Chaos; time series analysis; Power spectrum; Aliasing

We investigate the nature of the numerically computed power spectral densityP(f, N, τ) of a discrete (sampling time interval,τ) and finite (length,N) scalar time series extracted from a continuous time chaotic dynamical system. We highlight howP(f, N, τ) differs from the true power spectrum and from the power spectrum of a general stochastic process. Non-zeroτ leads to aliasing;P(f, N, τ) decays at high frequencies as [πτ/sinπτf]², which is an aliased form of the 1/f² decay. This power law tail seems to be a characteristic feature of all continuous time dynamical systems, chaotic or otherwise. Also the tail vanishes in the limit ofN → ∞, implying that the true power spectral density must be band width limited. In striking contrast the power spectrum of a stochastic process is dominated by a term independent of the length of the time series at all frequencies.

M C Valsakumar¹ S V M Satyanarayana¹ V Sridhar¹

1. Theoretical Studies Section, Materials Science Division, Indira Gandhi Centre for Atomic Research, Kalpakkam - 603 102, India

Published

January 1997