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 Regul. Chaotic Dyn., 2014, Volume 19, Issue 4, Pages 495–505 (Mi rcd176)

Birth of Discrete Lorenz Attractors at the Bifurcations of 3D Maps with Homoclinic Tangencies to Saddle Points

Sergey V. Gonchenkoa, Ivan I. Ovsyannikovab, Joan C. Tatjerc

a Nizhny Novgorod State University, pr. Gagarina 23, Nizhny Novgorod, 603000 Russia
b Imperial College London, SW7 2AZ, London, UK
c Dept. de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain

Abstract: It was established in [1] that bifurcations of three-dimensional diffeomorphisms with a homoclinic tangency to a saddle-focus fixed point with the Jacobian equal to 1 can lead to Lorenz-like strange attractors. In the present paper we prove an analogous result for three-dimensional diffeomorphisms with a homoclinic tangency to a saddle fixed point with the Jacobian equal to 1, provided the quadratic homoclinic tangency under consideration is nonsimple.

Keywords: Homoclinic tangency, rescaling, 3D Hénon map, bifurcation, Lorenz-like attractor

 Funding Agency Grant Number Russian Science Foundation 14-12-00811 Russian Foundation for Basic Research 13-01-0058913-01-97028-povolzhje14-01-00344 Ministry of Education and Science of the Russian Federation 02.B.49.21.0003 Leverhulme Trust RPG-279 Engineering and Physical Sciences Research Council EP/I019111/1 Ministerio de Educación y Ciencia, Spain MTM2009-09723 Section 3 is carried out by the RSF-grant (project No. 14-12-00811). The paper was partially supported by the grants of RFBR No. 13-01-00589, 13-01-97028–povolzhje and 14-01-00344. The first author was supported by the grant (the agreement of August 27, 2013 No. 02.B.49.21.0003 between The Ministry of education and science of the Russian Federation and Lobachevsky State University of Nizhni Novgorod). The second author was supported by the Leverhulme Trust grant RPG-279 and the EPSRC Mathematics Platform grant EP/I019111/1. The third author was supported by the MEC grant MTM2009-09723 (Spain) and the CIRIT grant 2009 SGR 67 (Spain)

DOI: https://doi.org/10.1134/S1560354714040054

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MSC: 37C05, 37G25, 37G35
Accepted:25.04.2014
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Citation: Sergey V. Gonchenko, Ivan I. Ovsyannikov, Joan C. Tatjer, “Birth of Discrete Lorenz Attractors at the Bifurcations of 3D Maps with Homoclinic Tangencies to Saddle Points”, Regul. Chaotic Dyn., 19:4 (2014), 495–505

Citation in format AMSBIB
\Bibitem{GonOvsTat14} \by Sergey~V.~Gonchenko, Ivan~I.~Ovsyannikov, Joan~C.~Tatjer \paper Birth of Discrete Lorenz Attractors at the Bifurcations of 3D Maps with Homoclinic Tangencies to Saddle Points \jour Regul. Chaotic Dyn. \yr 2014 \vol 19 \issue 4 \pages 495--505 \mathnet{http://mi.mathnet.ru/rcd176} \crossref{https://doi.org/10.1134/S1560354714040054} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3240982} \zmath{https://zbmath.org/?q=an:1335.37031} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000340380900005} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. S. Gonchenko, S. V. Gonchenko, “Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps”, Physica D, 337 (2016), 43–57
2. Aminur Rahman, Yogesh Joshi, Denis Blackmore, “Sigma Map Dynamics and Bifurcations”, Regul. Chaotic Dyn., 22:6 (2017), 740–749
3. S. Gonchenko, I. Ovsyannikov, “Homoclinic tangencies to resonant saddles and discrete Lorenz attractors”, Discret. Contin. Dyn. Syst.-Ser. S, 10:2 (2017), 273–288
4. I. I. Ovsyannikov, V. D. Turaev, “Analytic proof of the existence of the Lorenz attractor in the extended Lorenz model”, Nonlinearity, 30:1 (2017), 115–137
5. M. J. Capinski, D. Turaev, P. Zgliczynski, “Computer assisted proof of the existence of the Lorenz attractor in the Shimizu-Morioka system”, Nonlinearity, 31:12 (2018), 5410–5440
6. A. S. Conchenko, V S. Conchenko, V O. Kazakovt, A. D. Kozlov, “Elements of contemporary theory of dynamical chaos: a tutorial. Part I. Pseudohyperbolic attractors”, Int. J. Bifurcation Chaos, 28:11 (2018), 1830036
7. J. Eilertsen, J. Magnan, “On the chaotic dynamics associated with the center manifold equations of double-diffusive convection near a codimension-four bifurcation point at moderate thermal Rayleigh number”, Int. J. Bifurcation Chaos, 28:8 (2018), 1850094
8. S. Gonchenko, M.-Ch. Li, M. Malkin, “Criteria on existence of horseshoes near homoclinic tangencies of arbitrary orders”, Dynam. Syst., 33:3 (2018), 441–463