Stability Analysis of Equilibrium Point and Limit Cycle of Two-Dimensional Nonlinear Dynamical Systems—A Tutorial

• Published in: Applied sciences Vol. 13; no. 2; p. 1136
• Main Author: Wei, Bin
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01-01-2023
• Subjects: Dynamic stability; Dynamical systems; Energy; Equilibrium; limit cycle; linearization; Lyapunov theorem; phrase plane; Stability analysis; Two dimensional analysis
• ISSN: 2076-3417; 2076-3417
• DOI: 10.3390/app13021136

The equilibrium state of a dynamical system can be divided into the equilibrium point and limit cycle. In this paper, the stability analysis of the equilibrium point and limit cycle of dynamical systems are presented through different and all possible approaches, and those approaches are compared as well. In particular, the author presented the stability analysis of the equilibrium point through phase plane approach, Lyapunov–LaSalle energy-based approach, and linearization approach, respectively, for two-dimensional nonlinear system, while the stability analysis of the limit cycle is analyzed by using the LaSalle local invariant set theorem and Poincaré–Bendixson theorem, which is only valid in two-dimensional systems. Different case studies are used to demonstrate the stability analysis of equilibrium point and limit cycle.