Homoclinic and heteroclinic orbits for the $0^2$ or $0^2i \omega$ singularity in the presence of two reversibility symmetries

This paper is devoted to the study of the dynamics of reversible vector fields close to an equilibrium when a $0^2$ or a $0^2i\omega$ resonance occurs in the presence of two symmetries of reversibility. In the presence of a unique reversibility symmetry the existence of a homoclinic connection to 0 is known for the $0^2$ resonance whereas for the $0^2i\omega$ resonance there is generically no homoclinic connection to 0 but there is always a homoclinic connection to an exponentially small periodic orbit.

In the presence of a second symmetry of reversibility, the situation is more degenerate. Indeed, because of the second symmetry the quadratic part of the normal forms vanishes, and so the dynamics of the normal forms is governed by the cubic part. For the $0^2$ resonance we prove the existence of homoclinic connections to 0 and of heteroclinic orbits. For the $0^2i\omega$ resonance we prove that in most of the cases the second symmetry induces the existence of homoclinic connections to 0 and of heteroclinic orbits whereas with a unique symmetry there is generically no homoclinic connection to 0. Such a reversible vector field with two reversibility symmetries occurs for instance after center manifold reduction when studying 2-dimensional waves in NLS type systems with one-dimensional potential or when studying localized waves in nonlinear chains of coupled oscillators. It also occurs when studying localized buckling in rods with noncircular cross section.