Bifurcation from family of periodic orbits in discontinuous autonomous systems
The authors consider perturbed piecewise-smooth nonlinear differential systems, i.e., systems of the form \[ \dot{x}=F(t,x, \chi), \tag{1} \] where \(F(t,x, \chi)\) is a smooth function on \(\mathbb{R} \times (\mathbb{R}^n \setminus \mathbb{S}) \times \mathbb{R}^n\), periodic in \(t \in \mathbb{R}\). \(\mathbb{S}\) is called the ``discontinuity set'', and in the paper it is a hypersurface of \(\mathbb{R}^n\). Moreover, it is supposed that at \(\chi = 0\) the function \(F(t,x,0)=F(x)\) is independent of \(t\) and the associated autonomous nonlinear differential system \[ \dot{x}=F(x), \tag{2} \] has a periodic orbit \(\gamma(t)\). The first natural case is when \(\gamma\) crosses \(\mathbb{S}\) transversally. This case was studied by the authors in a previous work and also by other authors. In the present paper, the authors consider the case when \(F\) is independent of \(t\) and \(\gamma\) crosses \(\mathbb{S}\) transversally and call this case ``transverse autonomous''. It is also supposed that the system (2) possesses a nondegenerate family of periodic solutions. Then, the authors determine bifurcation conditions in terms of Poincaré-Andronov-Melnikov functions that ensure the bifurcation of certain periodic solutions from the family. The local asymptotic properties of the bifurcating periodic solutions such as their stability, instability or hyperbolicity are also investigated. Finally, the paper present examples of 3- and 4-dimensional discontinuous ordinary differential systems to illustrate the theory.
scientific article; zbMATH DE number 6111600
scientific article