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2013
Impact Factor

Marina Gonchenko

Av. Diagonal 647, 08028 Barcelona, Spain
Departament de Matematica Aplicada I Universitat Politecnica de Catalunya

Publications:

Delshams A., Gonchenko M. S., Gutierrez P.
Continuation of the Exponentially Small Transversality for the Splitting of Separatrices to a Whiskered Torus with Silver Ratio
2014, vol. 19, no. 6, pp.  663-680
Abstract
We study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori with two fast frequencies in nearly integrable Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver number $\Omega = \sqrt{2}−1$. We show that the Poincaré–Melnikov method can be applied to establish the existence of 4 transverse homoclinic orbits to the whiskered torus, and provide asymptotic estimates for the transversality of the splitting whose dependence on the perturbation parameter ε satisfies a periodicity property. We also prove the continuation of the transversality of the homoclinic orbits for all the sufficiently small values of $\varepsilon$, generalizing the results previously known for the golden number.
Keywords: transverse homoclinic orbits, splitting of separatrices, Melnikov integrals, silver ratio
Citation: Delshams A., Gonchenko M. S., Gutierrez P.,  Continuation of the Exponentially Small Transversality for the Splitting of Separatrices to a Whiskered Torus with Silver Ratio, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 663-680
DOI:10.1134/S1560354714060057
Delshams A., Gonchenko M. S., Gonchenko S. V.
On Bifurcations of Area-preserving and Nonorientable Maps with Quadratic Homoclinic Tangencies
2014, vol. 19, no. 6, pp.  702-717
Abstract
We study bifurcations of nonorientable area-preserving maps with quadratic homoclinic tangencies. We study the case when the maps are given on nonorientable twodimensional surfaces. We consider one- and two-parameter general unfoldings and establish results related to the emergence of elliptic periodic orbits.
Keywords: area-preserving map, non-orientable surface, elliptic point, homoclinic tangency, bifurcation
Citation: Delshams A., Gonchenko M. S., Gonchenko S. V.,  On Bifurcations of Area-preserving and Nonorientable Maps with Quadratic Homoclinic Tangencies, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 702-717
DOI:10.1134/S1560354714060082
Gonchenko M. S., Gonchenko S. V.
On Cascades of Elliptic Periodic Points in Two-Dimensional Symplectic Maps with Homoclinic Tangencies
2009, vol. 14, no. 1, pp.  116-136
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: Gonchenko M. S., Gonchenko S. V.,  On Cascades of Elliptic Periodic Points in Two-Dimensional Symplectic Maps with Homoclinic Tangencies, Regular and Chaotic Dynamics, 2009, vol. 14, no. 1, pp. 116-136
DOI:10.1134/S1560354709010080

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