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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.

Chaotic dynamics is interesting in the field of nonlinear systems. Real systems can present chaotic oscillations [

Dynamical properties of systems can be investigated using bifurcation diagrams. In a bifurcation diagram, various dynamics of the system can be seen as well as its bifurcation points [

Here, we study various dynamics of two chaotic maps. These maps show different bifurcations. The cobweb plot is used to study the dynamics of the chaotic map in which the transition of the time series is also shown in the map plot [

Two chaotic maps are studied in this paper. The first one is a one-dimensional chaotic map, which is called the sine map, as shown in the following equation:

To study various behavior of the system by varying initial values, cobweb diagrams of the sine map in parameter

Cobweb diagram of System (

To briefly explain the dynamic of System (

Bifurcation diagram by varying parameter

Cobweb diagram of System (

The sine function is symmetric concerning the origin. This property causes two different symmetric stabilities shown in the bifurcation diagram with two positive and negative initial conditions. Figure

Bifurcation diagram by changing parameter

The second studied system is the Gaussian map. This map is formulated as follows:

Bifurcation diagram of the Gaussian map by varying parameter

Here, the bifurcations of the symmetric sine map are studied. The studies of the previous section show that the system has various dynamics and many bifurcation points. Critical slowing down is observed before the tipping points. The slowness can be characterized using the critical slowing down predictors [

The modified autocorrelation (AC) method is applied to the dynamics of the map for varying parameter

The scaled bifurcation diagram of the sine map

(a) The bifurcation diagram of the sine map by changing

The modified variance method is applied to the dynamics of System (

Bifurcation diagram in black color and the scaled version of the logarithm of the modified variance

Another indicator of bifurcation points is the Lyapunov exponent [

Bifurcation diagram in black and Lyapunov exponent in blue color by changing (a)

The second studied system is the Gaussian map. It was shown that the system has various dynamics and bifurcation points. To predict the bifurcations, AC, variance, and Lyapunov exponent are used. Figure

The bifurcation diagram of the Gaussian map with respect to changing

Figure

Bifurcation diagram in black color and the scaled version of the logarithm of the modified variance

Another studied indicator of the paper is the Lyapunov exponent. The results of the Lyapunov exponent are shown in Figure

Bifurcation diagram in black and Lyapunov exponent in blue color by changing

This paper aimed to predict bifurcation points of chaotic maps. Two systems were studied: sine map and Gaussian map. Various dynamics of the sine map were studied, and its bifurcations were investigated. The results presented the multistability of the system because of its symmetric map. The system’s critical slowing down was indicated using the modified autocorrelation method, modified variance method, and Lyapunov exponent. These studies showed that the bifurcation points of the sine map could be predicted using the indicators. The other studied system was the Gaussian map. The system shows various dynamics in a period-doubling route to chaos. Bifurcation points of the system were predicted using modified autocorrelation, modified variance, and Lyapunov exponent. Some indicators, such as autocorrelation and Lyapunov exponent, have an exact value in the bifurcation points. So, approaching the exact values indicates approaching the bifurcations. However, the variance method does not have an exact value in the bifurcation points. In the variance method, increasing the variance shows approaching the bifurcation points; however, we cannot precisely determine when it happens. The chaotic attractors are very complex dynamics. In chaotic domain, we cannot determine the transient time and slowness of dynamics. So, we cannot trust the indicators in the chaotic dynamics.

This study shows that some indications can alarm approaching the various bifurcation points. Those indicators had various natures. For example, autocorrelation calculates the short-term memory of the time series. Variance is based on the variations which increase by approaching the bifurcation points. The third method was the Lyapunov exponent, which shows the speed of the system approaching its final dynamic. The results, tested on two discrete systems, showed that these indicators had a proper trend when approaching bifurcation points that alarms their occurrences. In the future works, prediction of bifurcation points of systems with other types of bifurcations than period-doubling route to chaos can be investigated. As a suggestion for future works, some can consider the application of deep learning and reinforcement learning in the prediction of tipping points.

All the numerical simulation parameters are mentioned in the respective text part, and there are no additional data requirements for the simulation results.

The authors declare that they have no conflicts of interest to report regarding the present study.

This work was supported by the High Level Innovation Team Program from Guangxi Higher Education Institutions of China (Document No. [2018] 35). This work was funded by the Center for Computational Biology, Chennai Institute of Technology, India (funding no. CIT/CCB/2021/RD/007).