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Averaging methods for finding periodic orbits via Brouwer degree. (English) Zbl 1055.34086
The authors consider the problem of finding periodic solutions for a differential system whose right-hand side depends on a small parameter. Taking into account that the problem of finding periodic solutions for some differential systems is equivalent to that of finding zeros of some corresponding operator equations, they obtain different results by using the averaging method in combination with topological methods based on Brouwer degree theory and coincidence degree theory. Here, the main contribution to the averaging theory is the dropping of regularity conditions (up to second order in the parameter). Moreover, a result for the third-order averaging method is given. Finally, the authors provide a way to study bifurcations of limit cycles from the period annulus of a planar system and notice relations with the displacement function.
MSC:
34C29Averaging method
34C25Periodic solutions of ODE
47H11Degree theory (nonlinear operators)