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Expanding Lorenz attractors through resonant double homoclinic loops. (English) Zbl 1093.37022
Summary: We study the existence of Lorenz attractors in the unfolding of resonant double homoclinic loops in dimension three. Our results generalize the ones obtained by C. Robinson [SIAM J. Math. Anal. 32, 119–141 (2000; Zbl 0978.37013)] 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 of J. Palis and F. Takens [Hyperbolicity and sensitive choatic dynamics at homoclinic bifurcations. Fractal dimensions and infinitely many attractors. Cambridge Studies in Advanced Mathematics. 35. Cambridge: Cambridge University Press (1993; Zbl 0790.58014)] to obtain convergence to noncontinuous maps.
MSC:
37G20Hyperbolic singular points with homoclinic trajectories
37C70Attractors and repellers, topological structure
37D45Strange attractors, chaotic dynamics
37D99Dynamical systems with hyperbolic behavior