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Abstract
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We study the complicated dynamics of quasi-periodically perturbed ordinary differential equations with a homoclinic orbit to a dissipative saddle point. We show that there are four regions of parameters in which the equations have respectively: (1) attracting quasi-periodic integral manifolds of Levinson type; (2) transition to chaos; (3) strange attractors; (4) homoclinic tangles. In the case of homoclinic tangles, we not only obtain the results on horseshoes similar to the existing ones, but also give a comprehensive geometric description of the structures of tangles. This is a joint work with Wen Huang and Qiudong Wang.
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