Normal forms of planar switching systems

Xingwu  Chen; Weinian  Zhang

doi:10.3934/dcds.2016092

Article Contents

2016, Volume 36, Issue 12: 6715-6736. Doi: 10.3934/dcds.2016092

This issue Previous Article Calderón-Zygmund estimate for homogenization of parabolic systems Next Article Eigenvalues for a nonlocal pseudo $p-$Laplacian

Normal forms of planar switching systems

Xingwu Chen^1, and
Weinian Zhang^1,

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

Received: November 2014

Revised: August 2016

Published: May 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 37G05, 37G10; Secondary: 34C07.

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