Marina Gonchenko

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


Gonchenko M. S., Kazakov A. O., Samylina E. A., Shykhmamedov A.
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
Delshams A., Gonchenko M. S., Gutierrez P.
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
Delshams A., Gonchenko M. S., Gonchenko S. V.
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
Gonchenko M. S., Gonchenko S. V.
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

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