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Detection of a nonlinear oscillator underlying experimental time series:
the sunspot cycle

Milan Palus
Institute of Computer Science, Academy of Sciences of the Czech Republic
Pod vodárenskou vezí 2, 182 07 Prague 8, Czech Republic


After a brief review of a nonlinearity test based on information-theoretic functionals (redundancies) and surrogate data technique, we discuss problems of this and similar tests for nonlinearity. In particular, we stress that a formal rejection of a linear stochastic null hypothesis does not automatically mean an evidence for nonlinear dynamical origin of studied data. In an example we show how a variable variance could be mistaken for nonlinearity in a series of surface air pressure. Therefore we find a detection of nonlinearity in a series of sunspot numbers insufficient for an identification of a mechanism underlying the sunspot cycle. As a solution in this case we propose to test for a property of nonlinear oscillators -- mutual dependence between their instantaneous amplitude and frequency. This behavior is detected in yearly and monthly records of the sunspot numbers using histogram-adjusted isospectral surrogate data and Barnes model as ARMA surrogates. The instantaneous amplitudes and frequencies are obtained by means of the analytic signal approach using the discrete Hilbert transform. In several tests the amplitude-frequency correlation has been found significant on levels ranging from p<0.03 to p<0.07, which supports the hypothesis of a driven nonlinear oscillator as a mechanism underlying the sunspot cycle.

Nonlinear Dynamics and Statistics ed. A.I. Mees, Birkhauser, Boston, 2001, pp. 453-473.

Milan Palus 2000