## Relatório de pesquisa 17/08

Bifurcation
of Periodic Solutions for $C^5$ and $C^6$ Vector Fields in $R^4$ with
Pure Imaginary Eigenvalues in Resonance 1:4 and 1:5, Jaume Llibre and Ana Cristina Mereu, submitted Sept. 04.

Abstract

In this paper we study the bifurcation of families of periodic
orbits at a singular point of a C^5 and C^6 differential
system in R^4 with pure imaginary eigenvalues with resonance 1:4
and 1:5 respectively. From the singular point of the C^5 vector
field with resonance 1:4 can bifurcate 0, 1, 2, 3, 4, 5 or 6
one-parameter families of periodic orbits. For the C^6 vector field
with resonance 1:5, the maximal number of families of periodic orbits
that bifurcate from this singular point is $40$. The tool for proving
such a result is the averaging theory.

Mathematics Subject Classifications
(2000): 34C29, 34C25, 47H11.

Keywords: limit cycle, periodic orbit, Hopf bifurcation, Liapunov center theorem, averaging theory, resonance 1:4, resonance 1:5.

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September 04, 2008

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