# American Institute of Mathematical Sciences

December  2016, 36(12): 6715-6736. doi: 10.3934/dcds.2016092

## Normal forms of planar switching systems

 1 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China, China

Received  November 2014 Revised  August 2016 Published  October 2016

In this paper we study normal forms of planar differential systems with a non-degenerate equilibrium on a single switching line, i.e., the equilibrium is a non-degenerate equilibrium of both the upper system and the lower one. In the sense of $C^0$ conjugation we find all normal forms for linear switching systems and use them together with switching near-identity transformations to normalize second order terms, showing the reduction of normal forms. We prove that only one of those 19 types of linear normal form decides if the system is monodromic. With the monodromic linear normal form, we compute the second order monodromic normal form, which gives a condition under which exactly one limit cycle arises from a Hopf bifurcation.
Citation: Xingwu Chen, Weinian Zhang. Normal forms of planar switching systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6715-6736. doi: 10.3934/dcds.2016092
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