# American Institute of Mathematical Sciences

July  2016, 36(7): 3545-3601. doi: 10.3934/dcds.2016.36.3545

## Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems

 1 Dept. Matemàtiques, secció ETSEIB, UPC, Avda. Diagonal 647, Barcelona, 08028, Spain, Spain

Received  February 2015 Revised  January 2016 Published  March 2016

In this paper we study the Sotomayor-Teixeira regularization of a general visible fold singularity of a planar Filippov system. Extending Geometric Fenichel Theory beyond the fold with asymptotic methods, we determine the deviation of the orbits of the regularized system from the generalized solutions of the Filippov one. This result is applied to the regularization of global sliding bifurcations as the Grazing-Sliding of periodic orbits and the Sliding Homoclinic to a Saddle, as well as to some classical problems in dry friction.
Roughly speaking, we see that locally, and also globally, the regularization of the bifurcations preserve the topological features of the sliding ones.
Citation: Carles Bonet-Revés, Tere M-Seara. Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3545-3601. doi: 10.3934/dcds.2016.36.3545
##### References:
 [1] C. Bonet, Singular perturbation of relaxed periodic orbits, J. Differential Equations, 66 (1987), 301-339. doi: 10.1016/0022-0396(87)90024-6. [2] C. A. Buzzi, P. R. da Silva and M. A. Teixeira, A singular approach to discontinous vector fields on the plane, Journal of Differential Equations, 231 (2006), 633-655. The geometry of differential equations and dynamical systems. doi: 10.1016/j.jde.2006.08.017. [3] J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York, 1981. [4] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems, Applied Mathematical Sciences, Springer-Verlag London Ltd., London, 2008. Theory and applications. [5] F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996), x+100pp. With an appendix by Cheng Zhi Li. doi: 10.1090/memo/0577. [6] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9. [7] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9. [8] J. Giné and D. Peralta-Salas, Existence of inverse integrating factors and Lie symmetries for degenerate planar centers, J. Differential Equations, 252 (2012), 344-357. doi: 10.1016/j.jde.2011.08.044. [9] M. Guardia, T. M. Seara and M. A. Teixeira, Generic bifurcations of low codimension of planar Filippov systems, J. Differential Equations, 250 (2011), 1967-2023. doi: 10.1016/j.jde.2010.11.016. [10] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York-London, 1966. [11] C. K. R. T. Jones, Geometric singular perturbation theory, In Dynamical systems (Montecatini Terme, 1994), vol. 1609 of Lecture Notes in Math. Springer, Berlin, 1995, 44-118. doi: 10.1007/BFb0095239. [12] T. J. Kaper, An introduction to geometric methods and dynamical systems theory for singular perturbation problems, In Analyzing multiscale phenomena using singular perturbation methods (Baltimore, MD, 1998), vol. 56 of Proc. Sympos. Appl. Math. Amer. Math. Soc., Providence, RI, 1999, 85-131. doi: 10.1090/psapm/056/1718893. [13] K. U. Kristiansen and S. J. Hogan, On the use of blowup to study regularizations of singularities of piecewise smooth dynamical systems in $\mathbb{R}^3$, SIAM J. Appl. Dyn. Syst., 14 (2015), 382-422. doi: 10.1137/140980995. [14] M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), 286-314. doi: 10.1137/S0036141099360919. [15] M. Krupa and P. Szmolyan, Geometric analysis of the singularly perturbed planar fold, In Multiple-time-scale dynamical systems (Minneapolis, MN, 1997), vol. 122 of IMA Vol. Math. Appl., Springer, New York, 2001, 89-116. doi: 10.1007/978-1-4613-0117-2_4. [16] M. Kunze, Non-smooth Dynamical Systems, vol. 1744 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103843. [17] Y. 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. doi: 10.1142/S0218127403007874. [18] R. I. Leine, Bifurcations in Discontinuous Mechanical Systems of Filippov-type, ProQuest LLC, Ann Arbor, MI, 2000. Thesis (Dr.)-Technische Universiteit Eindhoven (The Netherlands). [19] R. I. Leine, D. H. Van Campen and B. L. Van de Vrande, Bifurcations in nonlinear discontinuous systems, Nonlinear Dynam., 23 (2000), 105-164. doi: 10.1023/A:1008384928636. [20] L. Mazzi and M. Sabatini, A characterization of centres via first integrals, J. Differential Equations, 76 (1988), 222-237. doi: 10.1016/0022-0396(88)90072-1. [21] E. F. Mishchenko and N. K. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations, vol. 13 of Mathematical Concepts and Methods in Science and Engineering, Plenum Press, New York, 1980. Translated from the Russian by F. M. C. Goodspeed. doi: 10.1007/978-1-4615-9047-7. [22] J. Sijbrand, Properties of center manifolds, Trans. Amer. Math. Soc., 289 (1985), 431-469. doi: 10.1090/S0002-9947-1985-0783998-8. [23] J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, In International Conference on Differential Equations (Lisboa, 1995). World Sci. Publ., River Edge, NJ, 1998, 207-223. [24] P. Szmolyan and M. Wechselberger, Canards in $\mathbb{R}^3$, J. Differential Equations, 177 (2001), 419-453. doi: 10.1006/jdeq.2001.4001. [25] M. A. Teixeira and P. R. da Silva, Regularization and singular perturbation techniques for non-smooth systems, Phys. D, 241 (2012), 1948-1955. doi: 10.1016/j.physd.2011.06.022. [26] V. I Utkin, Sliding Modes in Control and Optimization, Communications and Control Engineering Series. Springer-Verlag, Berlin, 1992. Translated and revised from the 1981 Russian original. doi: 10.1007/978-3-642-84379-2.

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##### References:
 [1] C. Bonet, Singular perturbation of relaxed periodic orbits, J. Differential Equations, 66 (1987), 301-339. doi: 10.1016/0022-0396(87)90024-6. [2] C. A. Buzzi, P. R. da Silva and M. A. Teixeira, A singular approach to discontinous vector fields on the plane, Journal of Differential Equations, 231 (2006), 633-655. The geometry of differential equations and dynamical systems. doi: 10.1016/j.jde.2006.08.017. [3] J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York, 1981. [4] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems, Applied Mathematical Sciences, Springer-Verlag London Ltd., London, 2008. Theory and applications. [5] F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996), x+100pp. With an appendix by Cheng Zhi Li. doi: 10.1090/memo/0577. [6] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9. [7] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, vol. 18 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9. [8] J. Giné and D. Peralta-Salas, Existence of inverse integrating factors and Lie symmetries for degenerate planar centers, J. Differential Equations, 252 (2012), 344-357. doi: 10.1016/j.jde.2011.08.044. [9] M. Guardia, T. M. Seara and M. A. Teixeira, Generic bifurcations of low codimension of planar Filippov systems, J. Differential Equations, 250 (2011), 1967-2023. doi: 10.1016/j.jde.2010.11.016. [10] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York-London, 1966. [11] C. K. R. T. Jones, Geometric singular perturbation theory, In Dynamical systems (Montecatini Terme, 1994), vol. 1609 of Lecture Notes in Math. Springer, Berlin, 1995, 44-118. doi: 10.1007/BFb0095239. [12] T. J. Kaper, An introduction to geometric methods and dynamical systems theory for singular perturbation problems, In Analyzing multiscale phenomena using singular perturbation methods (Baltimore, MD, 1998), vol. 56 of Proc. Sympos. Appl. Math. Amer. Math. Soc., Providence, RI, 1999, 85-131. doi: 10.1090/psapm/056/1718893. [13] K. U. Kristiansen and S. J. Hogan, On the use of blowup to study regularizations of singularities of piecewise smooth dynamical systems in $\mathbb{R}^3$, SIAM J. Appl. Dyn. Syst., 14 (2015), 382-422. doi: 10.1137/140980995. [14] M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), 286-314. doi: 10.1137/S0036141099360919. [15] M. Krupa and P. Szmolyan, Geometric analysis of the singularly perturbed planar fold, In Multiple-time-scale dynamical systems (Minneapolis, MN, 1997), vol. 122 of IMA Vol. Math. Appl., Springer, New York, 2001, 89-116. doi: 10.1007/978-1-4613-0117-2_4. [16] M. Kunze, Non-smooth Dynamical Systems, vol. 1744 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103843. [17] Y. 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. doi: 10.1142/S0218127403007874. [18] R. I. Leine, Bifurcations in Discontinuous Mechanical Systems of Filippov-type, ProQuest LLC, Ann Arbor, MI, 2000. Thesis (Dr.)-Technische Universiteit Eindhoven (The Netherlands). [19] R. I. Leine, D. H. Van Campen and B. L. Van de Vrande, Bifurcations in nonlinear discontinuous systems, Nonlinear Dynam., 23 (2000), 105-164. doi: 10.1023/A:1008384928636. [20] L. Mazzi and M. Sabatini, A characterization of centres via first integrals, J. Differential Equations, 76 (1988), 222-237. doi: 10.1016/0022-0396(88)90072-1. [21] E. F. Mishchenko and N. K. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations, vol. 13 of Mathematical Concepts and Methods in Science and Engineering, Plenum Press, New York, 1980. Translated from the Russian by F. M. C. Goodspeed. doi: 10.1007/978-1-4615-9047-7. [22] J. Sijbrand, Properties of center manifolds, Trans. Amer. Math. Soc., 289 (1985), 431-469. doi: 10.1090/S0002-9947-1985-0783998-8. [23] J. Sotomayor and M. A. Teixeira, Regularization of discontinuous vector fields, In International Conference on Differential Equations (Lisboa, 1995). World Sci. Publ., River Edge, NJ, 1998, 207-223. [24] P. Szmolyan and M. Wechselberger, Canards in $\mathbb{R}^3$, J. Differential Equations, 177 (2001), 419-453. doi: 10.1006/jdeq.2001.4001. [25] M. A. Teixeira and P. R. da Silva, Regularization and singular perturbation techniques for non-smooth systems, Phys. D, 241 (2012), 1948-1955. doi: 10.1016/j.physd.2011.06.022. [26] V. I Utkin, Sliding Modes in Control and Optimization, Communications and Control Engineering Series. Springer-Verlag, Berlin, 1992. Translated and revised from the 1981 Russian original. doi: 10.1007/978-3-642-84379-2.
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