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

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October 2017, 22(8): 3091-3112. doi: 10.3934/dcdsb.2017165

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

Luca Dieci ^1, , Cinzia Elia ^2, and Dingheng Pi ^3,

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 September 2016 Revised January 2017 Published June 2017

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 $0$.

Keywords: Piecewise smooth vector fields, regularization, limit cycle, monodromy matrix, asymptotic stability.

Mathematics Subject Classification: 34A36.

Citation: Luca Dieci, Cinzia Elia, Dingheng Pi. Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3091-3112. doi: 10.3934/dcdsb.2017165

References:

[1]	A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations Pergamon Press, Oxford, UK, 1996. Google Scholar
[2]	J. Awrejcewicz, M. Feckan and P. Olejnik, On continuous approximation of discontinuous systems, Nonlinear Analysis, 62 (2005), 1317-1331. doi: 10.1016/j.na.2005.04.033. Google Scholar
[3]	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. Google Scholar
[4]	C. A. Buzzi, T. De Carvalho and R. D. Euzébio, On Poincaré-Bendixson Theorem and nontrivial minimal sets in planar nonsmooth vector fields, arxiv: 1307.6825v1 [math. DS].Google Scholar
[5]	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. Google Scholar
[6]	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. Google Scholar
[7]	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. Google Scholar
[8]	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. Google Scholar
[9]	A. F. Filippov, Differential Equations with Discontinuous Righthand Side Kluwer Academic, Netherlands, 1988. doi: 10.1007/978-94-015-7793-9. Google Scholar
[10]	Yu. A. Kuznetsov, S. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2157-2188. Google Scholar
[11]	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. Google Scholar
[12]	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. Google Scholar
[13]	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. Google Scholar
[14]	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. Google Scholar
[15]	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. Google Scholar
[16]	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. Google Scholar
[17]	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. Google Scholar
[18]	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. Google Scholar
[19]	J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, International Conference on Differential Equations, Lisboa, (1995), 207-223. Google Scholar
[20]	W. Wasow, Asymptotic Expansions for Ordinary Differential Equations Dover Publications, Inc. , New York, 1987. Google Scholar
[21]	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. Google Scholar

show all references

References:

[1]	A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations Pergamon Press, Oxford, UK, 1996. Google Scholar
[2]	J. Awrejcewicz, M. Feckan and P. Olejnik, On continuous approximation of discontinuous systems, Nonlinear Analysis, 62 (2005), 1317-1331. doi: 10.1016/j.na.2005.04.033. Google Scholar
[3]	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. Google Scholar
[4]	C. A. Buzzi, T. De Carvalho and R. D. Euzébio, On Poincaré-Bendixson Theorem and nontrivial minimal sets in planar nonsmooth vector fields, arxiv: 1307.6825v1 [math. DS].Google Scholar
[5]	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. Google Scholar
[6]	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. Google Scholar
[7]	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. Google Scholar
[8]	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. Google Scholar
[9]	A. F. Filippov, Differential Equations with Discontinuous Righthand Side Kluwer Academic, Netherlands, 1988. doi: 10.1007/978-94-015-7793-9. Google Scholar
[10]	Yu. A. Kuznetsov, S. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2157-2188. Google Scholar
[11]	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. Google Scholar
[12]	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. Google Scholar
[13]	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. Google Scholar
[14]	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. Google Scholar
[15]	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. Google Scholar
[16]	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. Google Scholar
[17]	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. Google Scholar
[18]	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. Google Scholar
[19]	J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, International Conference on Differential Equations, Lisboa, (1995), 207-223. Google Scholar
[20]	W. Wasow, Asymptotic Expansions for Ordinary Differential Equations Dover Publications, Inc. , New York, 1987. Google Scholar
[21]	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. Google Scholar

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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