Abstract
Lyapunov, Weinstein and Moser obtained remarkable theorems giving sufficient conditions for the existence of periodic orbits emanating from an equilibrium point of a differential system with a first integral. Using averaging theory of first order we established in [1] a similar result for a differential system without assuming the existence of a first integral. Now, using averaging theory of the second order, we extend our result to the case when the first order average is identically zero. Our result can be interpreted as a kind of degenerated Hopf bifurcation.
Highlights
Introduction and Statement of the MainResultsConsider a system of ordinary differential equations x = f (x), x = (x1, . . . , xm) (1.1)near an equilibrium point, which we assume to be the origin x = 0
Every pair of conjugated purely imaginary eigenvalues of A gives rise to periodic solutions of (1.2)
For such systems we provide sufficient conditions so that periodic orbits bifurcate from the origin when ε = 0 is sufficiently small
Summary
In 1907 Lyapunov [4] established the existence of a one-parameter family of periodic solutions under two assumptions He assumed the existence of a first integral and a nonresonance condition on the purely imaginary eigenvalues of A (see Theorem 9.2.1 of [6]). We consider vector fields f = fε depending on a real parameter ε such that when ε = 0 the origin is an equilibrium point of system (1.1) with eigenvalues ±ωi = 0 and 0 with multiplicity m − 2 For such systems we provide sufficient conditions so that periodic orbits bifurcate from the origin when ε = 0 is sufficiently small. In [1] we used averaging theory of first order to establish a similar result for a differential system without assuming the existence of a first integral.
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