Poincaré Index and Periodic Solutions of Perturbed Autonomous Systems

Classical conditions for the bifurcation of periodic solutions in perturbed auto-oscillating and conservative systems go back to Malkin and Mel’nikov, respectively. These authors’ papers were based on the Lyapunov–Schmidt reduction and the implicit function theorem, which lead to the requirement that both the cycles and the zeros of the bifurcation functions be simple. In this paper a geometric approach is put forward which does not assume these requirements, but imposes a certain condition on the Poincaré index of a generating cycle with respect to some auxiliary vector field. The approach is based on calculating the topological degree of the Poincaré operator of the perturbed system with respect to interior and exterior neighbourhoods of a generating cycle, as a consequence of which the conclusion of the main theorem guarantees bifurcation of a certain number of periodic solutions towards the interior of the cycle, and of a certain number of periodic solutions towards the exterior of the cycle. Concrete examples are given, where this approach either establishes bifurcation of a greater number of periodic solutions compared with the known classical results, or provides additional information on the location of these solutions.