On the multiple zeros of a real analytic function with applications to the averaging theory of differential equations

Abstract

In this work we improve the classical averaging theory applied to -families of analytic T-periodic ordinary differential equations in standard form defined on R. First we characterize the set of points z_0 in the phase space and the parameters where T-periodic solutions can be produced when we vary a small parameter . Second we expand the displacement map in powers of the parameter whose coefficients are the averaged functions. The main contribution consists in analyzing the role that have the multiple zeros z_0 R of the first non-zero averaged function. The outcome is that these multiple zeros can be of two different classes depending on whether the points (z_0, ) belong or not to the analytic set defined by the real variety associated to the ideal generated by the averaged functions in the Noetheriang ring of all the real analytic functions at (z_0, ). Next we are able to bound the maximum number of branches of isolated T-periodic solutions that can bifurcate from each multiple zero z_0. Sometimes these bounds depend on the cardinalities of minimal bases of the former ideal. Several examples illustrate our results.