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Chen’s attractor exists. (English) Zbl 1129.37326
Summary: By applying the undetermined coefficient method, this paper finds homoclinic and heteroclinic orbits in the Chen system. It analytically demonstrates that the Chen system has one heteroclinic orbit of Shilnikov type that connects two nontrivial singular points. The Shilnikov criterion guarantees that the Chen system has Smale horseshoes and the horseshoe chaos. In addition, there also exists one homoclinic orbit joined to the origin. The uniform convergence of the series expansions of these two types of orbits are proved in this paper. It is shown that the heteroclinic and homoclinic orbits together determine the geometric structure of Chen’s attractor.
MSC:
37D45Strange attractors, chaotic dynamics
34C28Complex behavior, chaotic systems (ODE)