Abstract:
We study bifurcations of two-dimensional symplectic maps with quadratic homoclinic tangencies and prove results on the existence of cascade of elliptic periodic points for one and two parameter general unfoldings.
Keywords:
symplectic map, homoclinic tangency, bifurcation, generic elliptic point, KAM-theory.
Citation:
M. S. Gonchenko, S. V. Gonchenko, “On Cascades of Elliptic Periodic Points in Two-Dimensional Symplectic Maps with Homoclinic Tangencies”, Regul. Chaotic Dyn., 14:1 (2009), 116–136
\Bibitem{GonGon09}
\by M. S. Gonchenko, S. V. Gonchenko
\paper On Cascades of Elliptic Periodic Points in Two-Dimensional Symplectic Maps with Homoclinic Tangencies
\jour Regul. Chaotic Dyn.
\yr 2009
\vol 14
\issue 1
\pages 116--136
\mathnet{http://mi.mathnet.ru/rcd542}
\crossref{https://doi.org/10.1134/S1560354709010080}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2480954}
\zmath{https://zbmath.org/?q=an:1229.37054}
Linking options:
https://www.mathnet.ru/eng/rcd542
https://www.mathnet.ru/eng/rcd/v14/i1/p116
This publication is cited in the following 19 articles:
Pierre Berger, Anna Florio, Daniel Peralta-Salas, “Steady Euler Flows on ${\mathbb {R}}^3$ with Wild and Universal Dynamics”, Commun. Math. Phys., 401:1 (2023), 937
Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson, “Unfolding globally resonant homoclinic tangencies”, DCDS, 42:8 (2022), 4013
Muni S.Sh. McLachlan I R. Simpson D.J.W., “Homoclinic Tangencies With Infinitely Many Asymptotically Stable Single-Round Periodic Solutions”, Discret. Contin. Dyn. Syst., 41:8 (2021), 3629–3650
Lerman L.M., Trifonov K.N., “Saddle-Center and Periodic Orbit: Dynamics Near Symmetric Heteroclinic Connection”, Chaos, 31:2 (2021), 023113
S. V. Gonchenko, M. S. Gonchenko, I. O. Sinitsky, “On mixed dynamics of two-dimensional reversible diffeomorphisms with symmetric non-transversal heteroclinic cycles”, Izv. Math., 84:1 (2020), 23–51
Simpson D.J.W., “Unfolding Codimension-Two Subsumed Homoclinic Connections in Two-Dimensional Piecewise-Linear Maps”, Int. J. Bifurcation Chaos, 30:3 (2020), 2030006
M. Gonchenko, S. V. Gonchenko, I. Ovsyannikov, A. Vieiro, “On local and global aspects of the 1:4 resonance in the conservative cubic Hénon maps”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28:4 (2018)
David J. W. Simpson, Christopher P. Tuffley, “Subsumed Homoclinic Connections and Infinitely Many Coexisting Attractors in Piecewise-Linear Maps”, Int. J. Bifurcation Chaos, 27:02 (2017), 1730010
M. Gonchenko, S. Gonchenko, I. Ovsyannikov, R. Ibragimov, V. Vougalter, “Bifurcations of Cubic Homoclinic Tangencies in Two-dimensional Symplectic Maps”, Math. Model. Nat. Phenom., 12:1 (2017), 41
Amadeu Delshams, Marina Gonchenko, Sergey Gonchenko, Springer Proceedings in Mathematics & Statistics, 180, Difference Equations, Discrete Dynamical Systems and Applications, 2016, 107
D. J. W. Simpson, “Unfolding homoclinic connections formed by corner intersections in piecewise-smooth maps”, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26:7 (2016)
Amadeu Delshams, Marina Gonchenko, Sergey Gonchenko, “On dynamics and bifurcations of area-preserving maps with homoclinic tangencies”, Nonlinearity, 28:9 (2015), 3027
Sergey Gonchenko, Alexander Gonchenko, Ming-Chia Li, Nonlinear Systems and Complexity, 12, Nonlinear Dynamics New Directions, 2015, 29
Amadeu Delshams, Marina Gonchenko, Sergey V. Gonchenko, “On Bifurcations of Area-preserving and Nonorientable Maps with Quadratic Homoclinic Tangencies”, Regul. Chaotic Dyn., 19:6 (2014), 702–717
Sergey Kryzhevich, “Dynamics near nonhyperbolic fixed points or nontransverse homoclinic points”, Mathematics and Computers in Simulation, 95 (2014), 163
S.V. Gonchenko, A.S. Gonchenko, I.I. Ovsyannikov, D.V. Turaev, L. Lerman, D. Turaev, V. Vougalter, M. Zaks, “Examples of Lorenz-like Attractors in Hénon-like Maps”, Math. Model. Nat. Phenom., 8:5 (2013), 48
L. Lerman, V. Rom-Kedar, “A Saddle in a Corner—A Model of Collinear Triatomic Chemical Reactions”, SIAM J. Appl. Dyn. Syst., 11:1 (2012), 416
S. V. Gonchenko, V. S. Gonchenko, L. P. Shilnikov, “On a homoclinic origin of Hénon-like maps”, Regul. Chaotic Dyn., 15:4 (2010), 462–481
L. Lerman, “Breaking hyperbolicity for smooth symplectic toral diffeomorphisms”, Regul. Chaotic Dyn., 15:2 (2010), 194–209
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