Contracting Lorenz Attractors through Resonant Double Homoclinic Loops

Abstract

A contracting Lorenz attractor of a three-dimensional vector field is an attractor with a unique singularity whose eigenvalues are real and satisfy the eigenvalue conditions $\lambda_{ss} < \lambda_s < 0 < \lambda_u$ and $\lambda_s + \lambda_u < 0$. The study of contracting Lorenz attractors started in [A. Rovella, Bol. Soc. Brasil. Mat. (N.S.), 24 (1993), pp. 233-259]. In this paper we show that certain resonant double homoclinic loops in dimension three generate contracting Lorenz attractors in a positive Lebesgue subset of the parameter space. This gives a positive answer to a question posed in [C. Robinson, SIAM J. Math. Anal., 32 (2000), pp. 119-141].