Normal forms of piecewise-smooth systems with a monodromic singular point
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918060115361792 |
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| author | Li, Jiahao Chen, Xingwu Zhang, Weinian |
| author_facet | Li, Jiahao Chen, Xingwu Zhang, Weinian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_13200 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normal forms of piecewise-smooth systems with a monodromic singular point Li, Jiahao Chen, Xingwu Zhang, Weinian Dynamical Systems 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. |
| title | Normal forms of piecewise-smooth systems with a monodromic singular point |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2506.13200 |