Slow–fast systems on algebraic varieties bordering piecewise-smooth dynamical systems

This article extends results contained in Buzzi et al. (2006) [4], , Llibre et al. (2007, 2008) [12,13] concerning the dynamics of non-smooth systems. In those papers a piecewise Ck discontinuous vector field Z on Rn is considered when the discontinuities are concentrated on a codimension one submanifold. In this paper our aim is to study the dynamics of a discontinuous system when its discontinuity set belongs to a general class of algebraic sets. In order to do this we first consider F:U→R a polynomial function defined on the open subset U⊂Rn. The set F−1(0) divides U into subdomains U1,U2,…,Uk, with border F−1(0). These subdomains provide a Whitney stratification on U. We consider Zi:Ui→Rn smooth vector fields and we get Z=(Z1,…,Zk) a discontinuous vector field with discontinuities in F−1(0). Our approach combines several techniques such as ε-regularization process, blowing-up method and singular perturbation theory. Recall that an approximation of a discontinuous vector field Z by a one parameter family of continuous vector fields is called an ε-regularization of Z (see Sotomayor and Teixeira, 1996 [18], ; Llibre and Teixeira, 1997 [15], ). Systems as discussed in this paper turn out to be relevant for problems in control theory (Minorsky, 1969 [16], ), in systems with hysteresis (Seidman, 2006 [17], ) and in mechanical systems with impacts (di Bernardo et al., 2008 [5]).