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)


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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