# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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. 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. doi: 10.1016/j.jde.2006.08.017. [3] J. Carr, Applications of Centre Manifold Theory,, Springer-Verlag, (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., (2008). [5] F. Dumortier and R. Roussarie, Canard cycles and center manifolds,, Mem. Amer. Math. Soc., 121 (1996). doi: 10.1090/memo/0577. [6] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations,, J. Differential Equations, 31 (1979), 53. 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, (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. 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. doi: 10.1016/j.jde.2010.11.016. [10] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags,, Academic Press, (1966). [11] C. K. R. T. Jones, Geometric singular perturbation theory,, In Dynamical systems (Montecatini Terme, (1994), 44. 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, (1998), 85. 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 $\mathbbR^3$,, SIAM J. Appl. Dyn. Syst., 14 (2015), 382. 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. 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, (1997), 89. 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, (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. doi: 10.1142/S0218127403007874. [18] R. I. Leine, Bifurcations in Discontinuous Mechanical Systems of Filippov-type,, ProQuest LLC, (2000). [19] R. I. Leine, D. H. Van Campen and B. L. Van de Vrande, Bifurcations in nonlinear discontinuous systems,, Nonlinear Dynam., 23 (2000), 105. doi: 10.1023/A:1008384928636. [20] L. Mazzi and M. Sabatini, A characterization of centres via first integrals,, J. Differential Equations, 76 (1988), 222. 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, (1980). doi: 10.1007/978-1-4615-9047-7. [22] J. Sijbrand, Properties of center manifolds,, Trans. Amer. Math. Soc., 289 (1985), 431. 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), 207. [24] P. Szmolyan and M. Wechselberger, Canards in $\mathbbR^3$,, J. Differential Equations, 177 (2001), 419. 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. 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, (1992). doi: 10.1007/978-3-642-84379-2.

show all references

##### References:
 [1] C. Bonet, Singular perturbation of relaxed periodic orbits,, J. Differential Equations, 66 (1987), 301. 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. doi: 10.1016/j.jde.2006.08.017. [3] J. Carr, Applications of Centre Manifold Theory,, Springer-Verlag, (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., (2008). [5] F. Dumortier and R. Roussarie, Canard cycles and center manifolds,, Mem. Amer. Math. Soc., 121 (1996). doi: 10.1090/memo/0577. [6] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations,, J. Differential Equations, 31 (1979), 53. 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, (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. 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. doi: 10.1016/j.jde.2010.11.016. [10] A. Halanay, Differential Equations: Stability, Oscillations, Time Lags,, Academic Press, (1966). [11] C. K. R. T. Jones, Geometric singular perturbation theory,, In Dynamical systems (Montecatini Terme, (1994), 44. 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, (1998), 85. 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 $\mathbbR^3$,, SIAM J. Appl. Dyn. Syst., 14 (2015), 382. 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. 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, (1997), 89. 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, (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. doi: 10.1142/S0218127403007874. [18] R. I. Leine, Bifurcations in Discontinuous Mechanical Systems of Filippov-type,, ProQuest LLC, (2000). [19] R. I. Leine, D. H. Van Campen and B. L. Van de Vrande, Bifurcations in nonlinear discontinuous systems,, Nonlinear Dynam., 23 (2000), 105. doi: 10.1023/A:1008384928636. [20] L. Mazzi and M. Sabatini, A characterization of centres via first integrals,, J. Differential Equations, 76 (1988), 222. 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, (1980). doi: 10.1007/978-1-4615-9047-7. [22] J. Sijbrand, Properties of center manifolds,, Trans. Amer. Math. Soc., 289 (1985), 431. 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), 207. [24] P. Szmolyan and M. Wechselberger, Canards in $\mathbbR^3$,, J. Differential Equations, 177 (2001), 419. 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. 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, (1992). doi: 10.1007/978-3-642-84379-2.
 [1] Fabio Camilli, Annalisa Cesaroni. A note on singular perturbation problems via Aubry-Mather theory. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 807-819. doi: 10.3934/dcds.2007.17.807 [2] Nathan Glatt-Holtz, Mohammed Ziane. Singular perturbation systems with stochastic forcing and the renormalization group method. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1241-1268. doi: 10.3934/dcds.2010.26.1241 [3] Ruyuan Zhang. Hopf bifurcations of ODE systems along the singular direction in the parameter plane. Communications on Pure & Applied Analysis, 2005, 4 (2) : 445-461. doi: 10.3934/cpaa.2005.4.445 [4] Stéphane Chrétien, Sébastien Darses, Christophe Guyeux, Paul Clarkson. On the pinning controllability of complex networks using perturbation theory of extreme singular values. application to synchronisation in power grids. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 289-299. doi: 10.3934/naco.2017019 [5] Zongming Guo, Juncheng Wei. Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity. Communications on Pure & Applied Analysis, 2008, 7 (4) : 765-786. doi: 10.3934/cpaa.2008.7.765 [6] Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —. Communications on Pure & Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211 [7] Xiangtuan Xiong, Jinmei Li, Jin Wen. Some novel linear regularization methods for a deblurring problem. Inverse Problems & Imaging, 2017, 11 (2) : 403-426. doi: 10.3934/ipi.2017019 [8] Stefan Kindermann, Antonio Leitão. Convergence rates for Kaczmarz-type regularization methods. Inverse Problems & Imaging, 2014, 8 (1) : 149-172. doi: 10.3934/ipi.2014.8.149 [9] Timothy Blass, Rafael de la Llave. Perturbation and numerical methods for computing the minimal average energy. Networks & Heterogeneous Media, 2011, 6 (2) : 241-255. doi: 10.3934/nhm.2011.6.241 [10] Ilona Gucwa, Peter Szmolyan. Geometric singular perturbation analysis of an autocatalator model. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 783-806. doi: 10.3934/dcdss.2009.2.783 [11] Andrea Tellini. Imperfect bifurcations via topological methods in superlinear indefinite problems. Conference Publications, 2015, 2015 (special) : 1050-1059. doi: 10.3934/proc.2015.1050 [12] Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II. Networks & Heterogeneous Media, 2015, 10 (4) : 897-948. doi: 10.3934/nhm.2015.10.897 [13] Wei Wang, Yan Lv. Limit behavior of nonlinear stochastic wave equations with singular perturbation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 175-193. doi: 10.3934/dcdsb.2010.13.175 [14] Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I. Networks & Heterogeneous Media, 2013, 8 (4) : 1009-1034. doi: 10.3934/nhm.2013.8.1009 [15] Eugene Kashdan, Dominique Duncan, Andrew Parnell, Heinz Schättler. Mathematical methods in systems biology. Mathematical Biosciences & Engineering, 2016, 13 (6) : i-ii. doi: 10.3934/mbe.201606i [16] Zongming Guo, Zhongyuan Liu, Juncheng Wei, Feng Zhou. Bifurcations of some elliptic problems with a singular nonlinearity via Morse index. Communications on Pure & Applied Analysis, 2011, 10 (2) : 507-525. doi: 10.3934/cpaa.2011.10.507 [17] Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623 [18] Ryo Ikehata, Marina Soga. Asymptotic profiles for a strongly damped plate equation with lower order perturbation. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1759-1780. doi: 10.3934/cpaa.2015.14.1759 [19] Krešimir Burazin, Marko Vrdoljak. Homogenisation theory for Friedrichs systems. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1017-1044. doi: 10.3934/cpaa.2014.13.1017 [20] Per Christian Moan, Jitse Niesen. On an asymptotic method for computing the modified energy for symplectic methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1105-1120. doi: 10.3934/dcds.2014.34.1105

2017 Impact Factor: 1.179