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Shilnikov’s Cross-map method and hyperbolic dynamics of three-dimensional Hénon-like maps

  • L. P. Shilnikov-75
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Abstract

We study the hyperbolic dynamics of three-dimensional quadratic maps with constant Jacobian the inverse of which are again quadratic maps (the so-called 3D Hénon maps). We consider two classes of such maps having applications to the nonlinear dynamics and find certain sufficient conditions under which the maps possess hyperbolic nonwandering sets topologically conjugating to the Smale horseshoe. We apply the so-called Shilnikov’s cross-map for proving the existence of the horseshoes and show the existence of horseshoes of various types: (2,1)- and (1,2)-horseshoes (where the first (second) index denotes the dimension of stable (unstable) manifolds of horseshoe orbits) as well as horseshoes of saddle and saddle-focus types.

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Authors and Affiliations

  1. Research Institute of Applied Mathematics and Cybernetics, Nizhny Novgorod State University, Nizhny Novgorod, Russia

    S. Gonchenko

  2. Department of Applied Mathematics & Center of Mathematical Modelling and Scientific Computing, National Chiao Tung University, Hsinchu, Taiwan

    M. -Ch. Li

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  1. S. Gonchenko
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  2. M. -Ch. Li
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Correspondence to S. Gonchenko.

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Gonchenko, S., Li, M.C. Shilnikov’s Cross-map method and hyperbolic dynamics of three-dimensional Hénon-like maps. Regul. Chaot. Dyn. 15, 165–184 (2010). https://doi.org/10.1134/S1560354710020061

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  • DOI: https://doi.org/10.1134/S1560354710020061

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