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arXiv · 2506.13200

Normal forms of piecewise-smooth systems with a monodromic singular point

Jiahao Li·Xingwu Chen·Weinian Zhang
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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.

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Jiahao Li, Xingwu Chen, Weinian Zhang. 2025-06-16. Normal forms of piecewise-smooth systems with a monodromic singular point. https://arxiv.org/abs/2506.13200

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