American Institute of Mathematical Sciences

Advanced
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 $\phi(x_1)$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Periodic orbits of (1)
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

Figure 5.  Sliding periodic orbit
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123
[2]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020257
[3]

Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803
[4]

Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114
[5]

David J. W. Simpson. On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3803-3830. doi: 10.3934/dcds.2014.34.3803
[6]

Tiago de Carvalho, Bruno Freitas. Birth of an arbitrary number of T-singularities in 3D piecewise smooth vector fields. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4851-4861. doi: 10.3934/dcdsb.2019034
[7]

Jaume Llibre, Marco Antonio Teixeira. Regularization of discontinuous vector fields in dimension three. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 235-241. doi: 10.3934/dcds.1997.3.235
[8]

José Luis Bravo, Manuel Fernández, Ignacio Ojeda, Fernando Sánchez. Uniqueness of limit cycles for quadratic vector fields. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 483-502. doi: 10.3934/dcds.2019020
[9]

Dingheng Pi. Limit cycles for regularized piecewise smooth systems with a switching manifold of codimension two. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 881-905. doi: 10.3934/dcdsb.2018211
[10]

Shanshan Liu, Maoan Han. Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3115-3124. doi: 10.3934/dcdss.2020133
[11]

Vincent Naudot, Jiazhong Yang. Finite smooth normal forms and integrability of local families of vector fields. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 667-682. doi: 10.3934/dcdss.2010.3.667
[12]

J. C. Artés, Jaume Llibre, J. C. Medrado. Nonexistence of limit cycles for a class of structurally stable quadratic vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 259-270. doi: 10.3934/dcds.2007.17.259
[13]

Isaac A. García, Jaume Giné, Susanna Maza. Linearization of smooth planar vector fields around singular points via commuting flows. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1415-1428. doi: 10.3934/cpaa.2008.7.1415
[14]

Michael Schönlein. Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4053-4069. doi: 10.3934/dcds.2017172
[15]

Jianfeng Huang, Haihua Liang. Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020145
[16]

José Ginés Espín Buendía, Víctor Jiménez Lopéz. A topological characterization of the $\omega$-limit sets of analytic vector fields on open subsets of the sphere. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1143-1173. doi: 10.3934/dcdsb.2019010
[17]

Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481
[18]

Jifeng Chu, Zhaosheng Feng, Ming Li. Periodic shadowing of vector fields. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3623-3638. doi: 10.3934/dcds.2016.36.3623
[19]

Jaume Llibre, Jesús S. Pérez del Río, J. Angel Rodríguez. Structural stability of planar semi-homogeneous polynomial vector fields applications to critical points and to infinity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 809-828. doi: 10.3934/dcds.2000.6.809
[20]

Lev M. Lerman, Elena V. Gubina. Nonautonomous gradient-like vector fields on the circle: Classification, structural stability and autonomization. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1341-1367. doi: 10.3934/dcdss.2020076

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (63)
  • HTML views (61)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences