Generic bifurcations of low codimension of planar Filippov Systems

Journal of Differential Equations

Volumn 250, Issue 4, 2011, Pages 1967-2023

  Guardia, M ^a   Seara, T M ^a   Teixeira, M A ^b  

^a UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

^b UNIVERSITY OF CAMPINAS   (Brazil)

Author keywords

Bifurcation; Non smooth vector field; Singularity; Structural stability

Indexed keywords

EID: 78650974770     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2010.11.016     Document Type: Article

References (28)

1. Arnol'd V.I. Ordinary Differential Equations 1973, The MIT Press, Cambridge, MA, London, translated from Russian and edited by Richard A. Silverman.
2. Bautin N.N., Leontovich E.A. Metody i priemy kachestvennogo issledovaniya dinamicheskikh sistem na ploskosti. Spravochnaya Matematicheskaya Biblioteka 1976, Nauka, Moscow.
3. Broucke Mireille E., Pugh Charles C., Simić Slobodan N. Structural stability of piecewise smooth systems. Comput. Appl. Math. 2001, 20(1-2):51-89.
4. di Bernardo Mario, Budd Chris J., Champneys Alan R., Kowalczyk Piotr, Nordmark Arne B., Olivar Tost Gerard, Piiroinen Petri T. Bifurcations in nonsmooth dynamical systems. SIAM Rev. 2008, 50(4):629-701.
5. di Bernardo M., Budd C.J., Champneys A.R., Kowalczyk P. Piecewise-Smooth Dynamical Systems: Theory and Applications. Appl. Math. Sci. 2008, vol. 163. Springer-Verlag London Ltd., London.
6. di Bernardo M., Feigin M.I., Hogan S.J., Homer M.E. Local analysis of C-bifurcations in n-dimensional piecewise-smooth dynamical systems. Chaos Solitons Fractals 1999, 10(11):1881-1908.
7. di Bernardo M., Kowalczyk Piotr, Nordmark A. Bifurcations of dynamical systems with sliding: derivation of normal-form mappings. Phys. D 2002, 170(3-4):175-205.
8. di Bernardo M., Kowalczyk P., Nordmark A. Sliding bifurcations: a novel mechanism for the sudden onset of chaos in dry friction oscillators. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2003, 13(10):2935-2948.
9. di Bernardo Mario, Pagano Daniel J., Ponce Enrique Nonhyperbolic boundary equilibrium bifurcations in planar Filippov systems: a case study approach. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2008, 18(5):1377-1392.
10. Filippov A.F. Differential Equations with Discontinuous Right-Hand Sides. Math. Appl. (Sov. Ser.) 1988, vol. 18. Kluwer Academic Publishers Group, Dordrecht.
11. Guckenheimer J., Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields 1983, Springer-Verlag.
12. M.R. Jeffrey, S.J. Hogan, The geometry of generic sliding bifurcations, SIAM Rev. (2011), in press.
13. Kowalczyk P., di Bernardo M. Two-parameter degenerate sliding bifurcations in Filippov systems. Phys. D 2005, 204(3-4):204-229.
14. Kowalczyk P., di Bernardo M., Champneys A.R., Hogan S.J., Homer M., Piiroinen P.T., Kuznetsov Yu.A., Nordmark A. Two-parameter discontinuity-induced bifurcations of limit cycles: classification and open problems. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2006, 16(3):601-629.
15. Kozlova V.S. Roughness of a discontinuous system. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1984, 5:16-20.
16. Kowalczyk P., Piiroinen P.T. Two-parameter sliding bifurcations of periodic solutions in a dry-friction oscillator. Phys. D 2008, 237(8):1053-1073.
17. Kuznetsov Yu.A., Rinaldi S., Gragnani A. One-parameter bifurcations in planar Filippov systems. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2003, 13(8):2157-2188.
18. Mel'nikov V.K. On the stability of a center for time-periodic perturbations. Tr. Mosk. Mat. Obs. 1963, 12:3-52.
19. Sotomayor J. Generic one-parameter families of vector fields on two-dimensional manifolds. Inst. Hautes Études Sci. Publ. Math. 1974, 43:5-46.
20. Sotomayor J. Liç oes de equaç oes diferenciais ordinárias. Projeto Euclides 1979, vol. 11. Instituto de Matemática Pura e Aplicada, Rio de Janeiro.
21. Teixeira Marco Antonio Generic bifurcation in manifolds with boundary. J. Differential Equations 1977, 25(1):65-89.
22. Teixeira Marco Antonio Generic bifurcation of certain singularities. Boll. Un. Mat. Ital. B (5) 1979, 16(1):238-254.
23. Teixeira Marco Antonio Generic singularities of discontinuous vector fields. An. Acad. Brasil. Ciênc. 1981, 53(2).
24. Teixeira Marco Antonio Stability conditions for discontinuous vector fields. J. Differential Equations 1990, 88(1).
25. Teixeira Marco Antonio Generic bifurcation of sliding vector fields. J. Math. Anal. Appl. 1993, 176(2).
26. Teixeira Marco Antonio On attractivity of discontinuous systems. Dynam. Systems Appl. 1996, 5(1).
27. Teixeira Marco Antonio Codimension two singularities of sliding vector fields. Bull. Belg. Math. Soc. Simon Stevin 1999, 6(3).
28. Teixeira Marco Antonio Perturbation theory for non-smooth systems. Encyclopedia of Complexity and Systems Science 2009, 6697-6719. Springer. R. Meyers (Ed.).