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 & 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. Google Scholar

[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. Google Scholar

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J. Carr, Applications of Centre Manifold Theory,, Springer-Verlag, (1981). Google Scholar

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M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems, Applied Mathematical Sciences,, Springer-Verlag London Ltd., (2008). Google Scholar

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F. Dumortier and R. Roussarie, Canard cycles and center manifolds,, Mem. Amer. Math. Soc., 121 (1996). doi: 10.1090/memo/0577. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

A. Halanay, Differential Equations: Stability, Oscillations, Time Lags,, Academic Press, (1966). Google Scholar

[11]

C. K. R. T. Jones, Geometric singular perturbation theory,, In Dynamical systems (Montecatini Terme, (1994), 44. doi: 10.1007/BFb0095239. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

M. Kunze, Non-smooth Dynamical Systems, vol. 1744 of Lecture Notes in Mathematics,, Springer-Verlag, (2000). doi: 10.1007/BFb0103843. Google Scholar

[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. Google Scholar

[18]

R. I. Leine, Bifurcations in Discontinuous Mechanical Systems of Filippov-type,, ProQuest LLC, (2000). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

J. Sijbrand, Properties of center manifolds,, Trans. Amer. Math. Soc., 289 (1985), 431. doi: 10.1090/S0002-9947-1985-0783998-8. Google Scholar

[23]

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

[24]

P. Szmolyan and M. Wechselberger, Canards in $\mathbbR^3$,, J. Differential Equations, 177 (2001), 419. doi: 10.1006/jdeq.2001.4001. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

J. Carr, Applications of Centre Manifold Theory,, Springer-Verlag, (1981). Google Scholar

[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). Google Scholar

[5]

F. Dumortier and R. Roussarie, Canard cycles and center manifolds,, Mem. Amer. Math. Soc., 121 (1996). doi: 10.1090/memo/0577. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

A. Halanay, Differential Equations: Stability, Oscillations, Time Lags,, Academic Press, (1966). Google Scholar

[11]

C. K. R. T. Jones, Geometric singular perturbation theory,, In Dynamical systems (Montecatini Terme, (1994), 44. doi: 10.1007/BFb0095239. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

M. Kunze, Non-smooth Dynamical Systems, vol. 1744 of Lecture Notes in Mathematics,, Springer-Verlag, (2000). doi: 10.1007/BFb0103843. Google Scholar

[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. Google Scholar

[18]

R. I. Leine, Bifurcations in Discontinuous Mechanical Systems of Filippov-type,, ProQuest LLC, (2000). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[22]

J. Sijbrand, Properties of center manifolds,, Trans. Amer. Math. Soc., 289 (1985), 431. doi: 10.1090/S0002-9947-1985-0783998-8. Google Scholar

[23]

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

[24]

P. Szmolyan and M. Wechselberger, Canards in $\mathbbR^3$,, J. Differential Equations, 177 (2001), 419. doi: 10.1006/jdeq.2001.4001. Google Scholar

[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. Google Scholar

[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. Google Scholar

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