Bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps

We study bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps. We distinguish two types of cubic homoclinic tangencies, and each type gives different first return maps derived to diverse conservative cubic H\'enon maps with quite different bifurcation diagrams. In this way, we establish the structure of bifurcations of periodic orbits in two parameter general unfoldings generalizing to the conservative case the results previously obtained for the dissipative case. We also consider the problem of 1:4 resonance for the conservative cubic H\'enon maps.Comment: 20 pages, 12 figure

On local and global aspects of the 1:4 resonance in the conservative cubic H\'enon maps

We study the 1:4 resonance for the conservative cubic H\'enon maps $\mathbf{C}_\pm$ with positive and negative cubic term. These maps show up different bifurcation structures both for fixed points with eigenvalues $\pm i$ and for 4-periodic orbits. While for $\mathbf{C}_-$ the 1:4 resonance unfolding has the so-called Arnold degeneracy (the first Birkhoff twist coefficient equals (in absolute value) to the first resonant term coefficient), the map $\mathbf{C}_+$ has a different type of degeneracy because the resonant term can vanish. In the last case, non-symmetric points are created and destroyed at pitchfork bifurcations and, as a result of global bifurcations, the 1:4 resonant chain of islands rotates by $\pi/4$. For both maps several bifurcations are detected and illustrated.Comment: 21 pages, 13 figure

On Andronov-Hopf bifurcations of two-dimensional diffeomorphisms with homoclinic tangencies

The bifurcation of the birth of a closed invariant curve in the two-parameter unfolding of a two-dimensional diffeomorphism with a homoclinic tangency of invariant manifolds of a hyperbolic fixed point of neutral type (i.e. such that the Jacobian at the fixed point equals to 1) is studied. The existence of periodic orbits with multipliers e±iψ (0 < ψ < π) is proved and the first Lyapunov value is computed. It is shown that, generically, the first Lyapunov value is non-zero and its sign coincides with the sign of some separatrix value (i.e. a function of coefficients of the return map near the global piece of the homoclinic orbit)

Reversible perturbations of conservative Hénon-like maps

For area-preserving Hénon-like maps and their compositions, we consider smooth perturbations that keep the reversibility of the initial maps but destroy their conservativity. For constructing such perturbations, we use two methods, a new method based on reversible properties of maps written in the so-called cross-form, and the classical Quispel-Roberts method based on a variation of involutions of the initial map. We study symmetry breaking bifurcations of symmetric periodic orbits in reversible families containing quadratic conservative orientable and nonorientable Hénon maps as well as a product of two Hénon maps whose Jacobians are mutually inverse.Peer ReviewedPostprint (published version

A methodology for obtaining asymptotic estimates for the exponentially small splitting of separatrices to whiskered tori with quadratic frequencies

The aim of this work is to provide asymptotic estimates for the splitting of separatrices in a perturbed 3-degree-of-freedom Hamiltonian system, associated to a 2-dimensional whiskered torus (invariant hyperbolic torus) whose frequency ratio is a quadratic irrational number. We show that the dependence of the asymptotic estimates on the perturbation parameter is described by some functions which satisfy a periodicity property, and whose behavior depends strongly on the arithmetic properties of the frequencies.Comment: 5 pages, 1 figur