Expanding Lorenz Attractors through Resonant Double Homoclinic Loops

Abstract

In this paper we study the existence of Lorenz attractors in the unfolding of resonant double homoclinic loops in dimension three. Our results generalize the ones obtained in [C. Robinson, SIAM J. Math. Anal., 32 (2000), pp. 119--141] in two ways. First, we obtain attractors instead of weak attractors obtained there. Second, we enlarge considerably the region in the parameter space corresponding to flows presenting expanding Lorenz attractors. The proof is based on rescaling techniques [J. Palis and F. Takens, Hyperbolicity and Sensitive Choatic Dynamics at Homoclinic Bifurcations. Fractal Dimensions and Infinitely Many Attractors, Cambridge University Press, Cambridge, UK, 1993] to obtain convergence to noncontinuous maps.