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An instability theorem for nonlinear fractional differential systems
Limit cycles for regularized discontinuous dynamical systems with a hyperplane of discontinuity
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 |
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 $0$.
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. |





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