Multiple Limit Cycles and Wintner-Perko Termination Principle

Abstract

In this chapter, we consider bifurcations of multiple limit cycles. In particular, we apply some results of the works [166, 169], obtained by L. M. Perko for two-dimensional analytic systems, to the study of global bifurcations of multiple limit cycles in polynomial systems. There is a quite definite number of field-rotation parameters determining the bifurcations of multiple limit cycles in the polynomial systems, and in some cases, for example, in the case of quadratic systems, we have got enough information on the boundary properties of global bifurcation surfaces of these cycles. Using the obtained results and applying the Wintner-Perko termination principle for multiple limit cycles, we suggest a new (global) approach to the solution of Hilbert’s Sixteenth Problem in the case of quadratic systems. This approach can be applied also to cubic and more general polynomial systems.