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Nelineinaya Dinamika [Russian Journal of Nonlinear Dynamics], 2011, Volume 7, Number 1, Pages 3–24
(Mi nd206)
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This article is cited in 2 scientific papers (total in 2 papers)
Homoclinic $\Omega$-explosion: hyperbolicity intervals and their boundaries
S. V. Gonchenko, O. V. Sten'kin Research Institute of Applied Mathematics and Cybernetics, Nizhny Novgorod State University
Abstract:
It has been established by Gavrilov and Shilnikov in [1] that, at the bifurcation boundary separating Morse–Smale systems from systems with complicated dynamics, there are systems with homoclinic tangencies. Moreover, when crossing this boundary, infinitely many periodic orbits appear immediately, just by “explosion”. Newhouse and Palis have shown in [2] that in this case there are infinitely many intervals of values of the splitting parameter corresponding to hyperbolic systems. In the present paper, we show that such hyperbolicity intervals have natural bifurcation boundaries, so that the phenomenon of homoclinic $\Omega$-explosion gains, in a sense, complete description in the case of two-dimensional diffeomorphisms.
Keywords:
homoclinic tangency; heteroclinic cycle; $\Omega$-explosion; hyperbolic set.
Received: 03.10.2010 Revised: 02.02.2011
Citation:
S. V. Gonchenko, O. V. Sten'kin, “Homoclinic $\Omega$-explosion: hyperbolicity intervals and their boundaries”, Nelin. Dinam., 7:1 (2011), 3–24
Linking options:
https://www.mathnet.ru/eng/nd206 https://www.mathnet.ru/eng/nd/v7/i1/p3
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