Periodic Orbits For A Class Of C 1 Three-dimensional Systems

We study C 1 perturbations of a reversible polynomial differential system of degree 4 in ℝ 3. 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 ℝ 3 with infinitely many limit cycles in a bounded domain if a generic assumption is satisfied. © 2007 Springer.561101115Arnold, V.I., (1989) Mathematical Methods of Classical Mechanics, 60. , second editionth edn., Graduate Texts in Mathematics, New York: Springer-VerlagArnold, V.I., Avez, A., (1967) Problemes Ergodiques De La mécanique Classique, , Paris: Gauthier-VillarsBirkhoff, G.D., Proof of the Poincaré's last geometric theorem (1913) Trans. Amer. Math. Soc., 14, pp. 14-22Birkhoff, G.D., An extension of the Poincaré's last geometric theorem (1925) Acta Math., 47, pp. 297-311Brown, M., von Newmann, W.D., Proof of the Poincaré-Birkhoff fixed point theorem (1977) Michigan Math. J., 24, pp. 21-31Buzzi, C.A., Llibre, J., Medrado, J.C., Periodic orbits for a class of reversible quadratic vector field R 3, , to appear in J. Math. Anal. and ApplFranks, J., Recurrence and fixed points in surface homeomorphisms (1988) Erg. Theorey Dyn. Sys., 8, pp. 99-108Guckenheimer, J., Holmes, P., (1990) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 42. , Applied Mathematical Sciences, New York: Springer-VerlagMeyer, K.R., Hall, G.R., Introduction to Hamiltonian Dynamical Systems and the N-Body Problem (1992) Applied Mathematical Sciences 90, , Springer-VerlagPoincaré, H., Sur un theoreme de geometrie (1912) Rend. Circ. Math. Palermo, 33, pp. 375-40