Existence of multi-hump generalized homoclinic solutions for a class of reversible systems

In this paper, we investigate a class of reversible dynamical systems in four dimensions. The spectrums of their linear operators at the equilibria are assumed to have a pair of positive and negative real eigenvalues and a pair of purely imaginary eigenvalues for the small parameter mu > 0, where these two real eigenvalues bifurcate from a double eigenvalue 0 for mu = 0. It has been shown that this class of systems owns a generalized homoclinic solution with one hump at the center (a homoclinic solution exponentially approaching a periodic solution with a small amplitude). This paper gives a rigorous existence proof of two-hump solutions. These two humps are far away and are glued by the small oscillations in the middle if some appropriate free constants are activated. The obtained results are also applied to some classical systems. The ideas here may be used to study the existence of 2(k)-hump solutions.