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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:

Gonchenko M. S., Kazakov A. O., Samylina E. A., Shykhmamedov A.
On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps
2022, vol. 27, no. 2, pp.  198-216
Abstract
We consider reversible nonconservative perturbations of the conservative cubic Hénon maps $H_3^{\pm}: \bar x = y, \bar y = -x + M_1 + M_2 y \pm y^3$ and study their influence on the 1:3 resonance, i.e., bifurcations of fixed points with eigenvalues $e^{\pm i 2\pi/3}$. It follows from [1] that this resonance is degenerate for $M_1=0$, $M_2=-1$ when the corresponding fixed point is elliptic. We show that bifurcations of this point under reversible perturbations give rise to four 3-periodic orbits, two of them are symmetric and conservative (saddles in the case of map $H_3^+$ and elliptic orbits in the case of map $H_3^-$), the other two orbits are nonsymmetric and they compose symmetric couples of dissipative orbits (attracting and repelling orbits in the case of map $H_3^+$ and saddles with the Jacobians less than 1 and greater than 1 in the case of map $H_3^-$). We show that these local symmetry-breaking bifurcations can lead to mixed dynamics due to accompanying global reversible bifurcations of symmetric nontransversal homo- and heteroclinic cycles. We also generalize the results of [1] to the case of the $p:q$ resonances with odd $q$ and show that all of them are also degenerate for the maps $H_3^{\pm}$ with $M_1=0$.
Keywords: cubic Hénon map, reversible system, 1:3 resonance, homoclinic tangencies, mixed dynamics
Citation: Gonchenko M. S., Kazakov A. O., Samylina E. A., Shykhmamedov A.,  On 1:3 Resonance Under Reversible Perturbations of Conservative Cubic Hénon Maps, Regular and Chaotic Dynamics, 2022, vol. 27, no. 2, pp. 198-216
DOI:10.1134/S1560354722020058
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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