Predicting tipping points of dynamical systems during a
period-doubling route to chaos
Fahimeh Nazarimehr,1,a) Sajad Jafari,1,b)
Seyed Mohammad Reza Hashemi Golpayegani,1,c) Matjaž Perc,2,3,d)
and Julien Clinton Sprott4,e)
1Biomedical Engineering Department, Amirkabir
University of Technology, Tehran 15875-4413, Iran
2Faculty of Natural Sciences and Mathematics,
University of Maribor, Koroška cesta 160, Maribor SI-2000,
Slovenia
3School of Electronic and Information Engineering,
Beihang University, Beijing 100191, People’s Republic of China
4Department of Physics, University of Wisconsin,
Madison, Wisconsin 53706, USA
(Received 05 May 2018; accepted 26 June 2018; published online 18
July 2018)
ABSTRACT
Classical indicators of tipping points have limitations when they
are applied to an ecological and a biological model. For example,
they cannot correctly predict tipping points during a
period-doubling route to chaos. To counter this limitation, we here
try to modify four well-known indicators of tipping points, namely
the autocorrelation function, the variance, the kurtosis, and the
skewness. In particular, our proposed modification has two steps.
First, the dynamic of the considered system is estimated using its
time-series. Second, the original time-series is divided into some
sub-time-series. In other words, we separate the time-series into
different period-components. Then, the four different tipping point
indicators are applied to the extracted sub-time-series. We test our
approach on an ecological model that describes the logistic growth
of populations and on an attention-deficit-disorder model. Both
models show different tipping points in a period-doubling route to
chaos, and our approach yields excellent results in predicting these
tipping points.
Ref: F. Nazarimehr, S. Jafari, S. M. R. H. Golpayegani, M. Perc, and
J. C. Sprott, Chaos 28,
073102-1--10 (2018)
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