On the number of limit cycles in general planar piecewise linear systems

Song-Mei Huan; Xiao-Song Yang

doi:10.3934/dcds.2012.32.2147

Article Contents

2012, Volume 32, Issue 6: 2147-2164. Doi: 10.3934/dcds.2012.32.2147

This issue Previous Article Generalized Stokes system in Orlicz spaces Next Article Collasping behaviour of a singular diffusion equation

On the number of limit cycles in general planar piecewise linear systems

Song-Mei Huan^1, and
Xiao-Song Yang^2,

1.
Department of Mathematics, Department of Electronics and Information Engineering, Huazhong University of Science and Technology, Wuhan, 430074
2.
Department of Mathematics, Huazhong University of Science and Technology, Wuhan, 430074

Received: December 31, 2010

Revised: May 31, 2011

Published: May 2012

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Abstract

Much progress has been made in planar piecewise smooth dynamical systems. However there remain many important problems to be solved even in planar piecewise linear systems. In this paper, we investigate the number of limit cycles of planar piecewise linear systems with two linear regions sharing the same equilibrium. By studying the implicit Poincaré map induced by the discontinuity boundary, some cases when there exist at most 2 limit cycles is completely investigated. Especially, based on these results we provide an example along with numerical simulations to illustrate the existence of 3 limit cycles thus have a negative answer to the conjecture by M. Han and W. Zhang [11](J. Differ.Equations 248 (2010) 2399-2416) that piecewise linear systems with only two regions have at most 2 limit cycles.

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Mathematics Subject Classification: Primary: 34C99, 34C07; Secondary: 37E99.

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