Birth Of Limit Cycles Bifurcating From A Nonsmooth Center

This paper is concerned with a codimension analysis of a two-fold singularity of piecewise smooth planar vector fields, when it behaves itself like a center of smooth vector fields (also called nondegenerate σ-center). We prove that any nondegenerate σ-center is σ-equivalent to a particular normal form Z0. Given a positive integer number k we explicitly construct families of piecewise smooth vector fields emerging from Z0 that have k hyperbolic limit cycles bifurcating from the nondegenerate σ-center of Z0 (the same holds for k=∞). Moreover, we also exhibit families of piecewise smooth vector fields of codimension k emerging from Z0. As a consequence we prove that Z0 has infinite codimension. © 2013 Elsevier Masson SAS.10213647Arrowsmith, D.K., Place, C.M., (1990) An Introduction to Dynamical Systems, , Cambridge University PressBuzzi, C.A., de Carvalho, T., Teixeira, M.A., On three-parameter families of Filippov systems - the fold-saddle singularity (2012) Int. J. Bifurc. Chaos, 22 (12), p. 18. , 1250291Buzzi, C.A., de Carvalho, T., Teixeira, M.A., On 3-parameter families of piecewise smooth vector fields in the plane (2012) SIAM J. Appl. Dyn. Syst., 11 (4), pp. 1402-1424Caubergh, M., (2004), Limit cycles near centers, Thesis Limburgh University, DiepenbeckCaubergh, M., Dumortier, F., Hopf-Takens bifurcations and centers (2004) J. Differ. Equ., 202, pp. 1-31Chow, S.N., Hale, J.K., (1982) Methods of Bifurcation Theory, , Springer-VerlagCeragioli, F., (1999) Discontinuous ordinary differential equations and stabilization, , http://calvino.polito.it/~ceragioli, PhD thesis, University of Firenze, Italy, Electronically available atCortés, J., Discontinuous dynamical systems: A tutorial on solutions, nonsmooth analysis, and stability, , arxiv:0901.3583, posted inDumortier, F., Singularities of Vector Fields (1978) Monografias de Matematica, 32. , Instituto de Matematica Pura e Aplicada, Rio de JaneiroEkeland, I., Discontinuits de champs hamiltoniens et existence de solutions optimales en calcul des variations (1977) Publ. Math. Inst. Hautes Études Sci., 47, pp. 5-32. , (in French)Filippov, A.F., Differential Equations with Discontinuous Right-Hand Sides (1988) Math. Appl., Sov. Ser., , Kluwer Academic Publishers, DordrechtGavrilov, L., Horozov, E., Limit cycles of perturbations of quadratic Hamiltonian vector fields (1993) J. Math. Pures Appl., 72 (2), pp. 213-238Golubitski, M., Guillemin, V., (1973) Stable Mappings and Their Singularities, , Springer-VerlagGuardia, M., Seara, T.M., Teixeira, M.A., Generic bifurcations of low codimension of planar Filippov systems (2011) J. Differ. Equ., 250, pp. 1967-2023Kuznetsov, Y.A., Rinaldi, S., Gragnani, A., One-parameter bifurcations in planar Filippov systems (2003) Int. J. Bifurc. Chaos, 13, pp. 2157-2188Takens, F., Unfoldings of certain singularities of vector fields generalised Hopf bifurcations (1973) J. Differ. Equ., 14 (3), pp. 476-493Teixeira, M.A., Perturbation theory for non-smooth systems, Meyers: encyclopedia of complexity and systems (2008) Science, 15