Spectral analysis of time series generated by nonlinear processes

The theory of nonlinear dynamics can assist in the selection of appropriate models to guide the analysis of observational time series. Weak and moderate nonlinearities lead to periodic or quasi‐periodic functions having discrete spectral lines. These functions are best analyzed by using the classical periodogram for equally spaced data and Lomb's extension of the periodogram for unequally spaced data. Strong nonlinearities can lead to aperiodic functions that have the continuous spectra characteristic of chaos. Power spectral analysis alone cannot be used to distinguish stochastic input to the time series from contributions to the continuum caused by deterministic nonlinearities. The evaluation of higher‐order spectra (bispectra, etc.) provides one means of distinguishing between those parts of the continuum due to hidden determinism and those due to stochastic input. This paper outlines a procedure for the analysis of mixed spectra modeled by a nonlinear system. First, an acceptable false alarm rate for misidentifying noise as a spectral peak is set. A normalized periodogram is then evaluated, and the highest peak is tested against the limit set by the false alarm criteria. The phase and amplitude of the maximum peak are estimated by least squares, provided the peak is judged to be significant. These parameters, together with the periodogram‐determined frequency, are used for a point‐by‐point subtraction of the sinusoid from the record. The prewhitening procedure is continued until all significant peaks are removed, leaving the estimated continuum part of the spectrum. In order to illustrate the procedure and general concepts, three examples are analyzed. A calculated time series showing the long‐term variation of Earth's obliquity demonstrates the difficulty of analyzing a quasi‐periodic function when the length of the record is insufficient to separate nearby lines in the spectrum. Quasi‐periodic behavior is shown by the sunspot record, which has 15 significant peaks at the 0.05 level; 12 of the peaks are simple combination tones of the other three. Analysis of unequally spaced observations is illustrated by the Vostok ice record of CO2 variation. The detected peaks of CO2 variation include the 100,000‐, 40,000‐, and 20,000‐year peaks detected in oxygen isotope records.