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Regular and Chaotic Dynamics, 2006, Volume 11, Issue 2, Pages 191–212
DOI: https://doi.org/10.1070/RD2006v011n02ABEH000345
(Mi rcd668)
 

This article is cited in 54 scientific papers (total in 54 papers)

On the 70th birthday of L.P. Shilnikov

Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation

S. V. Gonchenkoa, J. D. Meissb, I. I. Ovsyannikovc

a Institute for Applied Mathematics and Cybernetics, 10, Uljanova Str. 603005 Nizhny Novgorod, Russia
b Applied Mathematics, University of Colorado, Boulder, CO 80309
c Radio and Physical Department, Nizhny Novgorod State University, 23 Gagarin str., 603000 Nizhny Novgorod, Russia
Citations (54)
Abstract: We study bifurcations of a three-dimensional diffeomorphism, g0g0, that has a quadratic homoclinic tangency to a saddle-focus fixed point with multipliers (λeiφ,λeiφ,γ), where 0<λ<1<|γ| and |λ2γ|=1. We show that in a three-parameter family, gε, of diffeomorphisms close to g0, there exist infinitely many open regions near ε=0 where the corresponding normal form of the first return map to a neighborhood of a homoclinic point is a three-dimensional Hénon-like map. This map possesses, in some parameter regions, a "wild-hyperbolic" Lorenz-type strange attractor. Thus, we show that this homoclinic bifurcation leads to a strange attractor. We also discuss the place that these three-dimensional Hénon maps occupy in the class of three-dimensional quadratic maps with constant Jacobian.
Keywords: saddle-focus fixed point, three-dimensional quadratic map, homoclinic bifurcation, strange attractor.
Received: 03.10.2005
Accepted: 12.11.2005
Bibliographic databases:
Document Type: Article
MSC: 37C05, 37G25, 37G35
Language: English
Citation: S. V. Gonchenko, J. D. Meiss, I. I. Ovsyannikov, “Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation”, Regul. Chaotic Dyn., 11:2 (2006), 191–212
Citation in format AMSBIB
\Bibitem{GonMeiOvs06}
\by S. V. Gonchenko, J.~D.~Meiss, I. I. Ovsyannikov
\paper Chaotic dynamics of three-dimensional H\'{e}non maps that originate from a homoclinic bifurcation
\jour Regul. Chaotic Dyn.
\yr 2006
\vol 11
\issue 2
\pages 191--212
\mathnet{http://mi.mathnet.ru/rcd668}
\crossref{https://doi.org/10.1070/RD2006v011n02ABEH000345}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2245077}
\zmath{https://zbmath.org/?q=an:1164.37306}
Linking options:
  • https://www.mathnet.ru/eng/rcd668
  • https://www.mathnet.ru/eng/rcd/v11/i2/p191
  • This publication is cited in the following 54 articles:
    1. Vassilis M. Rothos, Stavros Anastassiou, Katerina G. Hadjifotinou, “Stationary solitons in discrete nonlinear Schrödinger with non-nearest neighbour interactions”, Proc. R. Soc. A., 481:2310 (2025)  crossref
    2. Petr Boriskov, “Chaotic discrete map of pulse oscillator dynamics with threshold nonlinear rate coding”, Nonlinear Dyn, 112:5 (2024), 3917  crossref
    3. Dana C'Julio, Bernd Krauskopf, Hinke M. Osinga, “Computing parametrised large intersection sets of 1D invariant manifolds: a tool for blender detection”, Numer Algor, 2024  crossref
    4. Amanda E. Hampton, James D. Meiss, “Connecting Anti-integrability to Attractors for Three-Dimensional Quadratic Diffeomorphisms”, SIAM J. Appl. Dyn. Syst., 23:1 (2024), 616  crossref
    5. Xu Zhang, Guanrong Chen, “Diffeomorphisms with infinitely many Smale horseshoes”, Journal of Difference Equations and Applications, 2024, 1  crossref
    6. Aikan Shykhmamedov, Efrosiniia Karatetskaia, Alexey Kazakov, Nataliya Stankevich, “Scenarios for the creation of hyperchaotic attractors in 3D maps”, Nonlinearity, 36:7 (2023), 3501  crossref
    7. Sergey Gonchenko, Aleksandr Gonchenko, “On discrete Lorenz-like attractors in three-dimensional maps with axial symmetry”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 33:12 (2023)  crossref
    8. Ivan I. Ovsyannikov, “On the Birth of Discrete Lorenz Attractors Under Bifurcations of 3D Maps with Nontransversal Heteroclinic Cycles”, Regul. Chaotic Dyn., 27:2 (2022), 217–231  mathnet  crossref  mathscinet
    9. Sergey V. Gonchenko, Klim A. Safonov, Nikita G. Zelentsov, “Antisymmetric Diffeomorphisms and Bifurcations of a Double Conservative Hénon Map”, Regul. Chaotic Dyn., 27:6 (2022), 647–667  mathnet  crossref  mathscinet
    10. Andy Hammerlindl, Bernd Krauskopf, Gemma Mason, Hinke M. Osinga, “Determining the global manifold structure of a continuous-time heterodimensional cycle”, JCD, 9:3 (2022), 393  crossref
    11. Amanda E. Hampton, James D. Meiss, “Anti-integrability for Three-Dimensional Quadratic Maps”, SIAM J. Appl. Dyn. Syst., 21:1 (2022), 650  crossref
    12. Amanda E. Hampton, James D. Meiss, “The three-dimensional generalized Hénon map: Bifurcations and attractors”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 32:11 (2022)  crossref
    13. S. Anastassiou, “Complicated behavior in cubic Hénon maps”, Theoret. and Math. Phys., 207:2 (2021), 572–578  mathnet  crossref  crossref  adsnasa  isi
    14. Gonchenko S. Kazakov A. Turaev D., “Wild Pseudohyperbolic Attractor in a Four-Dimensional Lorenz System”, Nonlinearity, 34:4 (2021), 2018–2047  crossref  mathscinet  isi  scopus
    15. Gonchenko V S. Kaynov M.N. Kazakov A.O. Turaev V D., “On Methods For Verification of the Pseudohyperbolicity of Strange Attractors”, Izv. Vyss. Uchebn. Zaved.-Prikl. Nelineynaya Din., 29:1 (2021), 160–185  mathnet  crossref  isi  scopus
    16. Zhang X., Chen G., “Polynomial Maps With Hidden Complex Dynamics”, Discrete Contin. Dyn. Syst.-Ser. B, 24:6 (2019), 2941–2954  crossref  mathscinet  zmath  isi  scopus
    17. Diaz L.J. Perez S.A., “Henon-Like Families and Blender-Horseshoes At Nontransverse Heterodimensional Cycles”, Int. J. Bifurcation Chaos, 29:3 (2019), 1930006  crossref  mathscinet  zmath  isi  scopus
    18. Lu Ch., Wu Sh., Jiang Ch., Hu J., “Weak Harmonic Signal Detection Method in Chaotic Interference Based on Extended Kalman Filter”, Digit. Commun. Netw., 5:1, SI (2019), 51–55  crossref  isi  scopus
    19. Zhang X., “Characterization of Dynamics of a Class of Polynomial Automorphisms in C-N”, Int. J. Bifurcation Chaos, 29:1 (2019), 1950007  crossref  mathscinet  zmath  isi  scopus
    20. Sergey Gonchenko, Ming-Chia Li, Mikhail Malkin, “Criteria on existence of horseshoes near homoclinic tangencies of arbitrary orders”, Dynamical Systems, 33:3 (2018), 441  crossref
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