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Contracting Lorenz attractors through resonant double homoclinic loops. (English) Zbl 1107.37041
Summary: 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 λ ss <λ s <0<λ u and λ s +λ u <0. The study of contracting Lorenz attractors started by A. Rovella [Bol. Soc. Bras. Mat., Nova Sér. 24, 233–259 (1993; Zbl 0797.58051)]. 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 by C. Robinson [SIAM J. Math. Anal. 32, 119–141 (2000; Zbl 0978.37013)].
MSC:
37G35Attractors and their bifurcations
37G10Bifurcations of singular points
37C29Homoclinic and heteroclinic orbits
37G20Hyperbolic singular points with homoclinic trajectories
37D45Strange attractors, chaotic dynamics
37C70Attractors and repellers, topological structure