2025/06/16 by Jiahao Li, Xingwu Chen, Li, Jiahao +3
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Quantum chaos and dynamical systems #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2506.13200
openalex publication_date 2025/06/16 · openalex created_date 2025/10/13 · openalex updated_date 2026/07/28
Normal form theory is developed deeply for planar smooth systems but has few results for piecewise-smooth systems because difficulties arise from continuity of the near-identity transformation, which is constructed piecewise. In this paper, we overcome the difficulties to study normal forms for piecewise-smooth systems with FF, FP, or PP equilibrium and obtain explicit any-order normal forms by finding piecewise-analytic homeomorphisms and deriving a new normal form for analytic systems. Our theorems of normal forms not only generalize previous results from second-order to any-order, from FF type to all FF, FP, PP types, but also provide a new method to compute Lyapunov constants, which are applied to solve the center problem and any-order Hopf bifurcations of piecewise-smooth systems.