Regular and Chaotic Dynamics

, Volume 15, Issue 2–3, pp 165–184 | Cite as

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

  • S. GonchenkoEmail author
  • M. -Ch. Li
L. P. Shilnikov-75 Special Issue


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.

Key words

quadratic map Smale horseshoe hyperbolic set symbolic dynamics saddle saddlefocus 

MSC2000 numbers

37C05 37D20 37B10 


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Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Research Institute of Applied Mathematics and CyberneticsNizhny Novgorod State UniversityNizhny NovgorodRussia
  2. 2.Department of Applied Mathematics & Center of Mathematical Modelling and Scientific ComputingNational Chiao Tung UniversityHsinchuTaiwan

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