A strange attractor in the unfolding of an orbit-flip homoclinic orbit

Abstract

An orbit-flip homoclinic orbit o of a vector field defined on R 3 is a homoclinic orbit to an equilibrium point for which the one-dimensional unstable manifold of the equilibrium point is connected to the one-dimensional strong stable manifold. In this paper, we show that in a generic unfolding of such a homoclinic orbit, there exists a positive Lebesgue measure set in the parameter space for which the corresponding vector field possesses a suspended strange attractor. To prove the result, we propose a rescaling in the phase space and a blowing up in the parameter space, and in the new system, we show that the Poincaré return map is close to the map (x,y) M (1 - āx2,bx) when b is close to 0. With a similar rescaling/blowing up, we also obtain a similar result in the case where o is an inclination-flip homoclinic orbit.