#
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*

E-mail: mp@uivt.cas.cz

### Abstract:

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*