Normal forms of piecewise-smooth systems with a monodromic singular point
Abstract
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.
Cite
@article{arxiv.2506.13200,
title = {Normal forms of piecewise-smooth systems with a monodromic singular point},
author = {Jiahao Li and Xingwu Chen and Weinian Zhang},
journal= {arXiv preprint arXiv:2506.13200},
year = {2025}
}