Title: Shilnikov's Cross-map method and hyperbolic dynamics of three-dimensional H,non-like maps
Authors: Gonchenko, S.
Li, M. -Ch.
Department of Applied Mathematics
Graduate Program of Mathematical Modeling and Scientific Computing, Department of Applied Mathematics
Keywords: quadratic map;Smale horseshoe;hyperbolic set;symbolic dynamics;saddle;saddle-focus
Issue Date: 1-Jun-2010
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 H,non 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.
URI: http://dx.doi.org/10.1134/S1560354710020061
ISSN: 1560-3547
DOI: 10.1134/S1560354710020061
Volume: 15
Issue: 2-3
Begin Page: 165
End Page: 184
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