Periodic orbits for a class of C 1 three-dimensional systems

We studyC ¹ perturbations of a reversible polynomial differential system of degree 4 in. We introduce the concept of strongly reversible vector field. If the perturbation is strongly reversible, the dynamics of the perturbed system does not change. For non-strongly reversible perturbations we prove the existence of an arbitrary number of symmetric periodic orbits. Additionally, we provide a polynomial vector field of degree 4 in with infinitely many limit cycles in a bounded domain if a generic assumption is satisfied. The first two authors are partially supported by a MCYT grant number MTM2005-06098-C02-01, and by a CICYT grant number 2005SGR 00550. The second author is partially supported by a FAPESP-BRAZIL grant 10246-2. All authors are also supported by the joint project CAPES-MECD grant HBP2003-0017.