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.
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)
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
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\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
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Linking options:
https://www.mathnet.ru/eng/rcd176
https://www.mathnet.ru/eng/rcd/v19/i4/p495
This publication is cited in the following 12 articles:
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
Gonchenko S. Kazakov A. Turaev D., “Wild Pseudohyperbolic Attractor in a Four-Dimensional Lorenz System”, Nonlinearity, 34:4 (2021), 2018–2047
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
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
A. S. Conchenko, S. V. Conchenko, O. V. 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
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
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
Hidear Talirongan, Ariel M. Sison, Ruji P. Medina, 2018 IEEE 10th International Conference on Humanoid, Nanotechnology, Information Technology,Communication and Control, Environment and Management (HNICEM), 2018, 1
S. Gonchenko, I. Ovsyannikov, “Homoclinic tangencies to resonant saddles and discrete Lorenz attractors”, Discret. Contin. Dyn. Syst.-Ser. S, 10:2 (2017), 273–288
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
A. S. Gonchenko, S. V. Gonchenko, “Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps”, Physica D, 337 (2016), 43–57
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