Periodic orbits for double regularization of piecewise smooth systems with a switching manifold of codimension two

Dingheng Pi

doi:10.3934/dcdsb.2021080

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2022, Volume 27, Issue 2: 1055-1073. Doi: 10.3934/dcdsb.2021080

This issue Previous Article Normal deviation of synchronization of stochastic coupled systems Next Article Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control

Periodic orbits for double regularization of piecewise smooth systems with a switching manifold of codimension two

Dingheng Pi^,

Fujian Province University Key Laboratory of Computational Science, School of Mathematical Sciences, Huaqiao University, Fujian 362021, China

^* Corresponding author: Dingheng Pi
^* Corresponding author: Dingheng Pi

Received: November 2019

Revised: January 2021

Early access: March 2021

Published: February 2022

This work was partially supported by NNSF of China grant 11671040, Cultivation Program for Outstanding Young Scientific talents of Fujian Province in 2017, Program for Innovative Research Team in Science and Technology in Fujian Province University, Quanzhou High-Level Talents Support Plan under Grant 2017ZT012 and Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (ZQN-YX401)

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Abstract

In this paper we consider an $ n $ dimensional piecewise smooth dynamical system. This system has a co-dimension 2 switching manifold $ \Sigma $ which is an intersection of two hyperplanes $ \Sigma_1 $ and $ \Sigma_2 $. We investigate the relation between periodic orbit of PWS system and periodic orbit of its double regularized system. If this PWS system has an asymptotically stable sliding periodic orbit(including type Ⅰ and type Ⅱ), we establish conditions to ensure that also a double regularization of the given system has a unique, asymptotically stable, periodic orbit in a neighbourhood of $ \gamma $, converging to $ \gamma $ as both of the two regularization parameters go to $ 0 $ by applying implicit function theorem and geometric singular perturbation theory.

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Mathematics Subject Classification: Primary: 34A36.

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Figure 1. Sliding periodic orbit of type Ⅰ

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Figure 2. Sliding periodic orbit of type Ⅱ

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