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 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)

Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps

We show that maps with infinitely many homoclinic tangencies of arbitrarily high orders are dense among real-analytic area-preserving diffeomorphisms in the Newhouse regions

On dynamical properties of diffeomorphisms with homoclinic tangencies

We study bifurcations of a homoclinic tangency to a saddle fixed point without non-leading multipliers. We give criteria for the birth of an infinite set of stable periodic orbits, an infinite set of coexisting saddle periodic orbits with different instability indices, non-hyperbolic periodic orbits with more than one multiplier on the unit circle, and an infinite set of stable closed invariant curves (invariant tori). The results are based on the rescaling of the first-return map near the orbit of homoclinic tangency, which is shown to bring the map close to one of four standard quadratic maps, and on the analysis of the bifurcations in these maps

Richness of dynamics and global bifurcations in systems with a homoclinic figure-eight

We consider 2D flows with a homoclinic figure-eight to a dissipative saddle. We study the rich dynamics that such a system exhibits under a periodic forcing. First, we derive the bifurcation diagram using topological techniques. In particular, there is a homoclinic zone in the parameter space with a non-smooth boundary. We provide a complete explanation of this phenomenon relating it to primary quadratic homoclinic tangency curves which end up at some cubic tangency (cusp) points. We also describe the possible attractors that exist (and may coexist) in the system. A main goal of this work is to show how the previous qualitative description can be complemented with quantitative global information. To this end, we introduce a return map model which can be seen as the simplest one which is 'universal' in some sense. We carry out several numerical experiments on the model, to check that all the objects predicted to exist by the theory are found in the model, and also to investigate new properties of the system

Quasiperiodic regimes in multisection semiconductor lasers

We consider a mode approximation model for the longitudinal dynamics of a multisection semiconductor laser which represents a slow-fast system of ordinary differential equations for the electromagnetic field and the carrier densities. Under the condition that the number of active sections $q$ coincides with the number of critical eigenvalues we introduce a normal form which admits to establish the existence of invariant tori. The case $q=2$ is investigated in more detail where we also derive conditions for the stability of the quasiperiodic regime

Reversible perturbations of conservative Henon-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

On dynamics and bifurcations of area-preserving maps with homoclinic tangencies

We study bifurcations of area-preserving maps, both orientable (symplectic) and non-orientable, with quadratic homoclinic tangencies. We consider one and two parameter general unfoldings and establish results related to the appearance of elliptic periodic orbits. In particular, we find conditions for such maps to have infinitely many generic (KAM-stable) elliptic periodic orbits of all successive periods starting at some number.Peer Reviewe

On dynamic properties of diffeomorphisms with homoclinic tangency

We study dynamic properties of systems in Newhouse domains near a diffeomorphism having a saddle fixed point with a homoclinic tangency in the following cases: one-dimensional, two-dimensional, where a fixed point is a saddle-focus with one real and two complex-conjugate multipliers, and four-dimensional saddle-focus with two pairs of complex-conjugate multipliers