New results on averaging theory and applications. (English) Zbl 1364.34065
Summary: The usual averaging theory reduces the computation of some periodic solutions of a system of ordinary differential equations, to find the simple zeros of an associated averaged function. When one of these zeros is not simple, i.e., the Jacobian of the averaged function in it is zero, the classical averaging theory does not provide information about the periodic solution associated to a non-simple zero. Here we provide sufficient conditions in order that the averaging theory can be applied also to non-simple zeros for studying their associated periodic solutions. Additionally, we do two applications of this new result for studying the zero-Hopf bifurcation in the Lorenz system and in the Fitzhugh-Nagumo system.
MSC:
| 34C29 | Averaging method for ordinary differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
Keywords:
averaging theory; periodic solutions; polynomial differential system; Lorenz system; Fitzhugh-Nagumo systemReferences:
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