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A novel method to identify boundaries of basins of attraction in a dynamical system using Lyapunov exponents and Monte Carlo techniques

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Abstract

Stability analysis is a critical part of the study of a dynamical system that includes: finding the attractors of the system, determining their stability or instability, and identifying basins of attraction (BoAs) of the attractors. There are some classical methods for finding parts of BoAs, like Lyapunov-based and Non-Lyapunov-based methods; however, due to the limitations of the current methods, the identification of a complex boundary of a BoA is not feasible in addition to the high computational load of these methods. The concept of Lyapunov exponents is a powerful tool for performing stability analysis of highly nonlinear systems. In this paper, a new and intelligent approach for identifying the boundary of the entire BoA using Lyapunov exponents, Monte Carlo techniques, and support vector machine learning algorithm is developed. The proposed approach can identify the boundary of BoAs due to the intelligent sampling brought by Monte Carlo technique, which also dramatically reduces the computational load. Three illustrative examples are presented to demonstrate the effectiveness of the proposed approach.

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  1. Nonlinear Systems Research Laboratory, University of Manitoba, Winnipe, MB, Canada

    Ali Reza Armiyoon & Christine Q. Wu

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  1. Ali Reza Armiyoon
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  2. Christine Q. Wu
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Correspondence to Christine Q. Wu.

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Armiyoon, A.R., Wu, C.Q. A novel method to identify boundaries of basins of attraction in a dynamical system using Lyapunov exponents and Monte Carlo techniques. Nonlinear Dyn 79, 275–293 (2015). https://doi.org/10.1007/s11071-014-1663-z

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