Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity

Luca Dieci; Cinzia Elia; Dingheng Pi

doi:10.3934/dcdsb.2017165

Article Contents

2017, Volume 22, Issue 8: 3091-3112. Doi: 10.3934/dcdsb.2017165

This issue Previous Article An instability theorem for nonlinear fractional differential systems Next Article On analyticity for Lyapunov exponents of generic bounded linear random dynamical systems

Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity

1.
School of Mathematics, Georgia Tech, Atlanta, GA 30332, USA
2.
Dipartimento di Matematica, University of Bari, I-70100, Bari, Italy
3.
School of Mathematical Sciences, Huaqiao University, Fujian 362021, China

This work was performed while the last two authors were visiting the School of Mathematics, Georgia Institute of Technology. The second author visit was made possible by the support of the GNCS (Italy), and the third author visit was made possible by the support of the CSC (China) 201508350024 and NNSF of China grant 11401228; the sponsorship of these agencies, and the hospitality of the School of Mathematics at Georgia Tech, are gratefully acknowledged. We also gratefully acknowledge an anonymous referee for pointing out to us the work [2]

Received: August 31, 2016

Revised: December 31, 2016

Published: October 2017

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Abstract

We consider an $n$ dimensional dynamical system with discontinuous right-hand side (DRHS), whereby the vector field changes discontinuously across a co-dimension 1 hyperplane $S$. We assume that this DRHS system has an asymptotically stable periodic orbit $γ$, not fully lying in $S$. In this paper, we prove that also a regularization of the given system has a unique, asymptotically stable, periodic orbit, converging to $γ$ as the regularization parameter goes to $0$.

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Mathematics Subject Classification: 34A36.

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Figure 1. Graph of transition function $\phi(x_1)$

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Figure 2. Periodic orbits of (1)

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Figure 3. P and $P_{\epsilon}$

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Figure 4. invariant region $V_{\epsilon}$

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Figure 5. Sliding periodic orbit

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