Homoclinic orbits in degenerate reversible-equivariant systems in R6

Abstract We study the dynamics near an equilibrium point p 0 of a Z 2 x Z 2 -reversible vector field in R 6 with the reversing symmetry or symmetry φ satisfying φ 2 = I and d i m F i x ( φ ) = 3 . We deal with systems such that X presents at p 0 a degenerate resonance of type 0 : p : q or 0 -non-resonant. We are assuming that the linearized system of X (at p 0 ) has as eigenvalues: λ 1 = 0 λ j = ± i α j , j = 2 , 3 . Our main concern is to find conditions for the existence of families of homoclinic orbits associated to periodic orbits near the equilibrium.