The application of the cell mapping method in the characteristic diagnosis of nonlinear dynamical systems

• Published in: Nonlinear dynamics 2023-10, Vol.111 (19), p.18095-18112
• Main Authors: Zhang, Zhengyuan, Dai, Liming
• Format: Article
• Language: English
• Subjects: Attractors (mathematics); Automotive Engineering; Cell mapping method; Chaos theory; Classical Mechanics; Control; Convergence; Density; Dynamical Systems; Eigenvalues; Engineering; Invariants; Liapunov exponents; Mechanical Engineering; Nonlinear systems; Numerical methods; Original Paper; Vibration
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-023-08777-0

The cell mapping method is one of the most powerful tools in the global analysis of nonlinear dynamical systems. It accurately depicts the boundaries of the domains of attractors but is not the ideal method for diagnosing the characteristics of attractors. To improve the performance of the conventional cell mapping method, an approach telling quasiperiodicity from chaos is developed in the current research. The proposed method uses the convergence speed of a probability distribution on the attractor as the criterion. For a chaotic system, the distribution converges quickly to the invariant density in several iterations. As for a quasiperiodic system, the distribution converges extremely slowly to the invariant density. The difference can be explained by the eigenvalues of the transition matrices of different systems. The transition matrix of a chaotic system has irregular eigenvalues, and the second-largest eigenvalue is small enough, while the matrix of a quasiperiodic system has regular eigenvalues, many of which are close to the unit circle. A theoretical analysis of some simple mappings is addressed to support this explanation. Furthermore, as a supplement to the analysis, a numerical method that calculates the Lyapunov exponent from the transition matrix is also discussed.