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 crossmap 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.
\Bibitem{GonLi10}
\by S. Gonchenko, M.-Ch. Li
\paper Shilnikov’s cross-map method and hyperbolic dynamics of three-dimensional Hénon-like maps
\jour Regul. Chaotic Dyn.
\yr 2010
\vol 15
\issue 2-3
\pages 165--184
\mathnet{http://mi.mathnet.ru/rcd486}
\crossref{https://doi.org/10.1134/S1560354710020061}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2644328}
\zmath{https://zbmath.org/?q=an:1203.37029}
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This publication is cited in the following 7 articles:
Xu Zhang, Guanrong Chen, “Diffeomorphisms with infinitely many Smale horseshoes”, Journal of Difference Equations and Applications, 2024, 1
Aikan Shykhmamedov, Efrosiniia Karatetskaia, Alexey Kazakov, Nataliya Stankevich, “Scenarios for the creation of hyperchaotic attractors in 3D maps”, Nonlinearity, 36:7 (2023), 3501
Sajjad Bakrani, Jeroen S.W. Lamb, Dmitry Turaev, “Invariant manifolds of homoclinic orbits and the dynamical consequences of a super-homoclinic: A case study in R4 with Z2-symmetry and integral of motion”, Journal of Differential Equations, 327 (2022), 1
Karatetskaia E., Shykhmamedov A., Kazakov A., “Shilnikov Attractors in Three-Dimensional Orientation-Reversing Maps”, Chaos, 31:1 (2021), 011102
Gonchenko S., Li M.-Ch., Malkin M., “Criteria on Existence of Horseshoes Near Homoclinic Tangencies of Arbitrary Orders”, Dynam. Syst., 33:3 (2018), 441–463
Lorenzo J Díaz, Shin Kiriki, Katsutoshi Shinohara, “Blenders in centre unstable Hénon-like families: with an application to heterodimensional bifurcations”, Nonlinearity, 27:3 (2014), 353
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