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Switching-induced bifurcation analysis for piecewise nonlinear dynamical systems: A semi-analytical approach
Kai Jiang1, Jianzhe Huang1, Xilin Fu2
1School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China.
Chaos (Woodbury, N.Y.)
|July 23, 2025
Summary
This study introduces a novel semi-analytical framework to fully analyze piecewise nonlinear dynamical systems. It enables the discovery of hidden bifurcation routes and complex dynamics in systems with switching behaviors.
Area of Science:
- Nonlinear Dynamics and Control Systems
- Chaos Theory and Bifurcation Analysis
- Circuit Theory and Electronic Systems
Background:
- Piecewise nonlinear dynamical systems exhibit unique behaviors due to vector field changes at discontinuous boundaries.
- Transient dynamics persist in steady-state responses, complicating analysis and hindering the identification of hidden bifurcation routes.
- Existing analytical methods struggle to capture the complete dynamics, including transient components, in these systems.
Purpose of the Study:
- To develop a comprehensive framework for analyzing piecewise nonlinear dynamical systems.
- To systematically characterize flow-switching dynamics and uncover hidden bifurcation routes.
- To enable the complete analysis of unconventional bifurcations induced by flow switching.
Main Methods:
- A semi-analytical framework integrating generalized mapping structures and local singularity theory.
- Development of a generalized mapping formalism with closed-form constraint conditions at switching points.
- Parameterization of periodic motions with higher-order singularities.
Main Results:
- The proposed framework effectively characterizes flow-switching dynamics at discontinuous boundaries.
- Complete bifurcation trees for piecewise nonlinear dynamical systems can be obtained.
- Analysis of a piecewise nonlinear memristor circuit revealed complex bifurcation and chaotic behaviors.
Conclusions:
- The developed semi-analytical framework provides a systematic approach to analyze complex dynamics in piecewise nonlinear systems.
- This method facilitates the discovery of previously inaccessible unstable hidden bifurcation routes.
- The approach is versatile and applicable to a wide range of piecewise nonlinear systems, including memristor circuits.
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