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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 and 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.

[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.

[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.

[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].

[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.

[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.

[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.

[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.

[9]

A. F. Filippov, Differential Equations with Discontinuous Righthand Side Kluwer Academic, Netherlands, 1988. doi: 10.1007/978-94-015-7793-9.

[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. 

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[19]

J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, International Conference on Differential Equations, Lisboa, (1995), 207-223. 

[20]

W. Wasow, Asymptotic Expansions for Ordinary Differential Equations Dover Publications, Inc. , New York, 1987.

[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.

show all references

References:
[1]

A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations Pergamon Press, Oxford, UK, 1996.

[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.

[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.

[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].

[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.

[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.

[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.

[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.

[9]

A. F. Filippov, Differential Equations with Discontinuous Righthand Side Kluwer Academic, Netherlands, 1988. doi: 10.1007/978-94-015-7793-9.

[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. 

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[19]

J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, International Conference on Differential Equations, Lisboa, (1995), 207-223. 

[20]

W. Wasow, Asymptotic Expansions for Ordinary Differential Equations Dover Publications, Inc. , New York, 1987.

[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.

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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