Predicting Tipping Points in Chaotic Maps with Period-Doubling Bifurcations

Changzhi Li,Sajad Jafari,Yongjian Liu,Dhanagopal Ramachandran,Karthikeyan Rajagopal,Guillermo Huerta Cuellar

doi:10.1155/2021/9927607

Abstract

In this paper, bifurcation points of two chaotic maps are studied: symmetric sine map and Gaussian map. Investigating the properties of these maps shows that they have a variety of dynamical solutions by changing the bifurcation parameter. Sine map has symmetry with respect to the origin, which causes multistability in its dynamics. The systems’ bifurcation diagrams show various dynamics and bifurcation points. Predicting bifurcation points of dynamical systems is vital. Any bifurcation can cause a huge wanted/unwanted change in the states of a system. Thus, their predictions are essential in order to be prepared for the changes. Here, the systems’ bifurcations are studied using three indicators of critical slowing down: modified autocorrelation method, modified variance method, and Lyapunov exponent. The results present the efficiency of these indicators in predicting bifurcation points.

Highlights

Chaotic dynamics is interesting in the field of nonlinear systems
We study various dynamics of two chaotic maps. ese maps show different bifurcations. e cobweb plot is used to study the dynamics of the chaotic map in which the transition of the time series is shown in the map plot [60]. en, using critical slowing down indicators, various tipping points of the systems are investigated
The bifurcations of the symmetric sine map are studied. e studies of the previous section show that the system has various dynamics and many bifurcation points

Summary

Introduction

Chaotic dynamics is interesting in the field of nonlinear systems. Real systems can present chaotic oscillations [1]. Critical points of the bifurcation diagram in a chaotic map were investigated in [19]. Dynamical properties of systems can be investigated using bifurcation diagrams. The study of bifurcation points and their predictions is interesting [45, 46]. Prediction of bifurcation points of biological systems has been studied in [50]. In [56], some issues in those indicators in predicting bifurcation points during a period-doubling route to chaos were studied. In [57, 58], the Lyapunov exponent was studied as an indicator of bifurcation points. We study various dynamics of two chaotic maps. En, using critical slowing down indicators, various tipping points of the systems are investigated We study various dynamics of two chaotic maps. ese maps show different bifurcations. e cobweb plot is used to study the dynamics of the chaotic map in which the transition of the time series is shown in the map plot [60]. en, using critical slowing down indicators, various tipping points of the systems are investigated

The Chaotic Maps

Critical Slowing Down Indicators of the Chaotic Maps

B Figure 12