Lyapunov Functions in Piecewise Linear Systems: From Fixed Point to Limit Cycle

2013/06/28 by Yi-An Ma, Ma, Yian, Ruoshi Yuan +7
Engineering · Mathematics · #Advanced Differential Equations and Dynamical Systems #Chaotic Dynamics (nlin.CD) #Control and Stability of Dynamical Systems #FOS: Physical sciences #Stability and Control of Uncertain Systems

paper · pdf · doi:10.48550/arxiv.1306.6880

openalex publication_date 2013/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper provides a first example of constructing Lyapunov functions in a class of piecewise linear systems with limit cycles. The method of construction helps analyze and control complex oscillating systems through novel geometric means. Special attention is stressed upon a problem not formerly solved: to impose consistent boundary conditions on the Lyapunov function in each linear region. By successfully solving the problem, the authors construct continuous Lyapunov functions in the whole state space. It is further demonstrated that the Lyapunov functions constructed explain for the different bifurcations leading to the emergence of limit cycle oscillation.

Citations

• Sustained oscillations generated by mutually inhibiting neurons with adaptation
• Multiple Lyapunov functions and other analysis tools for switched and hybrid systems

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