Limit cycles for regularized piecewise smooth systems with a switching manifold of codimension two

Dingheng Pi

doi:10.3934/dcdsb.2018211

Article Contents

2019, Volume 24, Issue 2: 881-905. Doi: 10.3934/dcdsb.2018211

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Limit cycles for regularized piecewise smooth systems with a switching manifold of codimension two

Dingheng Pi^,

School of Mathematical Sciences, Huaqiao University, Fujian 362021, China

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

Received: September 30, 2017

Revised: December 31, 2017

Early access: June 2018

Published: January 2019

This work was partially supported NNSF of China grants 11401228 and 11671040, Cultivation Program for Outstanding Young Scientific talents of Fujian Province in 2017 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 Σ which is an intersection of two co-dimension one switching manifolds Σ₁ and Σ₂. We investigate the relation of periodic orbit of PWS between periodic orbit of its regularized system. If this PWS system has an asymptotically stable crossing periodic orbit γ or sliding periodic orbit, we establish conditions to ensure that also a regularization of the given system has a unique, asymptotically stable, limit cycle in a neighbourhood of γ, converging to γ as the regularization parameter goes to 0.

Keywords:

Piecewise smooth systems,
regularization,
limit cycle,
co-dimension two discontinuity boundary,
asymptotic stability.

Mathematics Subject Classification: Primary: 34A36.

Citation:

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Figure 2. Sliding periodic orbit

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Figure 1. Crossing periodic orbit

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References

	J. Alexander and T. Seidman , Sliding modes in intersecting switching surfaces. I. Blending, Houston J. Math., 24 (1998) , 545-569.
	A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, UK, 1996.
	M. Di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, Appl. Math. Sci. 163, Springer-Verlag, London, 2008.
	C. A. Buzzi, T. De Carvalho and R. D. Euzébio, On Poincaré-Bendixson Theorem and nontrivial minimal sets in planar nonsmooth vector fields, Publ. Mat., 62 (2018), 113–131, arXiv: 1307.6825v1 [math. DS]. doi: 10.5565/PUBLMAT6211806.
	C. A. Buzzi , T. De Carvalho and P. R. Da Silva , Closed Poly-trajectories and Poincaré index of non-smooth vector fields on the plane, J. Dyn. Control. Sys., 19 (2013) , 173-193. doi: 10.1007/s10883-013-9169-4.
	C. A. Buzzi , T. De Carvalho and M. A. Teixeira , Birth of limit cycles bifurcating from a nonsmooth center, J. Math. Pure. Appl., 102 (2014) , 36-47. doi: 10.1016/j.matpur.2013.10.013.
	L. Dieci and F. Difonzo , A comparison of Filippov sliding vector fields in codimension 2, J. Comput. Appl. Math., 262 (2014) , 161-179. doi: 10.1016/j.cam.2013.10.055.
	L. Dieci and C. Elia , Piecewise smooth systems near a co-dimension 2 discontinuity manifold: can we say what should happen?, DCDS -S, 9 (2016) , 1039-1068. doi: 10.3934/dcdss.2016041.
	L. Dieci , C. Elia and L. Lopez , A Filippov sliding vector field on an attracting co-dimension 2 discontinuity surface, and a limited loss-of-attractivity analysis, J. Differential Equations, 254 (2013) , 1800-1832. doi: 10.1016/j.jde.2012.11.007.
	L. Dieci , C. Elia and L. Lopez , Uniqueness of Filippov sliding vector field on the intersection of two surfaces in $\mathbb{R}^3$ and implications for stability of periodic orbits, J. Nonlinear Sci., 25 (2015) , 1453-1471. doi: 10.1007/s00332-015-9265-6.
	L. Dieci , C. Elia and D. Pi , Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity, DCDS-B, 22 (2017) , 3091-3112. doi: 10.3934/dcdsb.2017165.
	L. Dieci and L. Lopez , Fundamental matrix solutions of piecewise smooth differential systems, Math. Comput. Simulation, 81 (2011) , 932-953. doi: 10.1016/j.matcom.2010.10.012.
	L. Dieci and N. Guglielmi , Regularizing piecewise smooth differential systems: Co-dimension 2 discontinuity surface, J. Dynam. Differential Equations, 25 (2013) , 71-94. doi: 10.1007/s10884-013-9287-4.
	Z. Du and Y. Li , Bifurcation of periodic orbits with multiple crossings in a class of planar Filippov systems, Math. Comput. Modelling, 55 (2012) , 1072-1082. doi: 10.1016/j.mcm.2011.09.032.
	A. F. Filippov, Differential Equations with Discontinuous Righthand Side, Kluwer Academic, Netherlands, 1988. doi: 10.1007/978-94-015-7793-9.
	R. I. Leine and H. Nijmeijer, Dynamics and Bifurcations of Non-Smooth Mechanical Systems, Lecture Notes in Applied and Computational Mechanics, 18, Springer-verlag, Berlin, 2004. doi: 10.1007/978-3-540-44398-8.
	J. Llibre , P. da Silva and M. A. Teixeira , Regularization of discontinuous vector fields on $\mathbb{R}^3$ via singular perturbation, J. Dynam. Differential Equations, 19 (2007) , 309-331. doi: 10.1007/s10884-006-9057-7.
	J. Llibre , P. R. da Silva and M. A. Teixeira , Sliding vector fields via slow-fast systems, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008) , 851-869.
	J. Llibre , P. R. da Silva and M. A. Teixeira , Study of singularities in nonsmooth dynamical systems via singular perturbation, SIAM. J. Applied Dynam. Sys., 8 (2009) , 508-526. doi: 10.1137/080722886.
	P. C. Müller , Calculation of lyapunov exponents for dynamic systems with discontinuities, Chaos, Solitons and Fractals, 5 (1995) , 1671-1681. doi: 10.1016/0960-0779(94)00170-U.
	D. Pi and X. Zhang , The sliding bifurcations in planar piecewise smooth differential systems, J. Dynam. Differential Equations, 25 (2013) , 1001-1026. doi: 10.1007/s10884-013-9327-0.
	L. A. Sanchez , Cones of rank 2 and the Poincaré-Bendixson property for a new class of monotone systems, J. Differential Equations, 246 (2009) , 1978-1990. doi: 10.1016/j.jde.2008.10.015.
	J. Sotomayor and A. L. Machado , Sructurally stable discontinuous vector fields on the plane, Qual. Theory of Dynamical Systems, 3 (2002) , 227-250. doi: 10.1007/BF02969339.
	J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, International Conference on Differential Equations, Lisboa, (1995), 207–223.
	S. Tang , J. Liang , Y. Xiao and A. Robert , Sliding bifurcations of Filippov two stage pest control models with economic thresholds, SIAM J.Appl.Math., 72 (2012) , 1061-1080. doi: 10.1137/110847020.
	Y. Wang , M. Han and D. Constantinescu , On the limit cycles of perturbed discontinuous planar systems with 4 switching lines, Chaos Solitons Fractals, 83 (2016) , 158-177.
	H. R. Zhu and H. L. Smith , Stable periodic orbits for a class of three-dimensional competitive systems, J. Differential Equations, 110 (1994) , 143-156. doi: 10.1006/jdeq.1994.1063.