Abstract:
We study the main bifurcations of multidimensional diffeomorphisms having a nontransversal homoclinic orbit to a saddle-node fixed point. On a parameter plane we build a bifurcation diagram for single-round periodic orbits lying entirely in a small neighborhood of the homoclinic orbit. Also, a relation of our results to the well-known codimension one bifurcations of a saddle fixed point with a quadratic homoclinic tangency and a saddle-node fixed point with a transversal homoclinic orbit is discussed.
Keywords:
saddle-node, homoclinic tangency, Arnold tongues.
Sections 5 and 6 are 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, 3-01-97028–povolzhje and 14-01-00344.
SG 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). IO was supported by the Leverhulme Trust grant RPG-279 and the EPSRC
Mathematics Platform grant EP/I019111/1.
Citation:
Sergey V. Gonchenko, Оlga V. Gordeeva, Valery I. Lukyanov, Ivan I. Ovsyannikov, “On Bifurcations of Multidimensional Diffeomorphisms Having a Homoclinic Tangency to a Saddle-node”, Regul. Chaotic Dyn., 19:4 (2014), 461–473
\Bibitem{GonGorLuk14}
\by Sergey~V.~Gonchenko, Оlga~V.~Gordeeva, Valery~I.~Lukyanov, Ivan~I.~Ovsyannikov
\paper On Bifurcations of Multidimensional Diffeomorphisms Having a Homoclinic Tangency to a Saddle-node
\jour Regul. Chaotic Dyn.
\yr 2014
\vol 19
\issue 4
\pages 461--473
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\crossref{https://doi.org/10.1134/S1560354714040029}
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Linking options:
https://www.mathnet.ru/eng/rcd173
https://www.mathnet.ru/eng/rcd/v19/i4/p461
This publication is cited in the following 3 articles:
Sergey V. Gonchenko, Ol'ga V. Gordeeva, “On the Structure of Orbits from a Neighborhood of a Transversal Homoclinic Orbit to a Nonhyperbolic Fixed Point”, Regul. Chaotic Dyn., 30:1 (2025), 9–25
S. V. Gonchenko, O. V. Gordeeva, “On Two-Dimensional Diffeomorphisms with Homoclinic Orbits to Nonhyperbolic Fixed Points”, Rus. J. Nonlin. Dyn., 20:1 (2024), 151–165
Gonchenko A.S. Gonchenko V S. Turaev D., “Doubling of Invariant Curves and Chaos in Three-Dimensional Diffeomorphisms”, Chaos, 31:11 (2021), 113130
Citation in format AMSBIB
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