Hopf bifurcation for the hydrogen atom in a circularly polarized microwave field

We consider the Rydberg electron in a circularly polarized microwave field, whose dynamics is described by a Hamiltonian depending on one parameter, K¿>¿0. The corresponding Hamiltonian system of ODE has two equilibrium points L1 (unstable for all K and energy value h(L1)) and L2 (a center for K¿¿Kcrit, with energy value h(L2)). We study the Hamiltonian-Hopf bifurcation phenomena that take place for K close to Kcrit around L2. First, a local analysis based on the computation of the integrable normal form up to a finite order is carried out and the steps for the computation of this (resonant) normal form are explained in a constructive manner. The analysis of the normal form obtained allows: to claim the type of the Hopf bifurcation –supercritical–; to study the local behavior of the electron in a neighborhood of the equilibrium L2 for the original non integrable Hamiltonian (as a perturbative approach from the integrable normal form); to obtain (approximations for) the parametrizations of the relevant invariant objects that take place due to the bifurcation (periodic orbits and invariant manifolds of L2). We compute numerically such objects and analyse not only the local picture of the dynamics close to L2, but also a global description of the dynamics and the effect of the Hopf bifurcation as well as other objects that organize the dynamics are discussed. We conclude that, for K close to Kcrit and the energy level h(L2), the Hopf bifurcation has essentially no effect on the dynamics from a physical point of view. However, for bigger values of K¿>¿Kcrit, the Hopf bifurcation has a dramatic effect: different kind of orbits coexist, mostly chaotic. Such orbits provide a ionization mechanism with several passages far from and close to L2 before ionizing. Surprisingly enough, also robust confinement regions (where the electron remains confined for ever), exist in the middle of chaotic areasPeer ReviewedPostprint (published version

Motion near the transition to complex instability: some examples

Complex instability is a generic kind of instability in Hamiltonian systems with three degrees of freedom. In this work, some examples of such instability are shown, together with a numerical analysis of the dynamics close to the transition from stability to comlex instability for a family of periodic orbits

KAM aspects of the quasi-periodic Hamiltonian Hopf bifurcation: summary of results

In this work we consider a 1:-1 non semi-simple resonant periodic orbit of a three-degrees of freedom real analytic Hamiltonian system. From the formal analysis of the normal form, it is proved the branching off a two-parameter family of two-dimensional invariant tori of the normalised system, whose normal behaviour depends intrinsically on the coefficients of its low-order terms. Thus, only elliptic or elliptic together with parabolic and hyperbolic tori may detach form the resonant periodic orbit. Both patterns are mentioned in the literature as the direct and, respectively, inverse quasiperiodic Hopf bifurcation. In this report we focus on the direct case, which has many applications in several fields of science. Thus, we present here a summary of the results, obtained in the framework of KAM theory, concerning the persistence of most of the (normally) elliptic tori of the normal form, when the whole Hamiltonian is taken into account, and to give a very precise characterisation of the parameters labelling them, which can be selected with a very clear dynamical meaning. These results include sharp quantitative estimates on the “density” of surviving tori, when the distance to the resonant periodic orbit goes to zero, and state that the 4-dimensional Cantor manifold holding these tori admits a Whitney-C^\infty extension. In addition, an application to the Circular Spatial Three-Body Problem (CSRTBP) is reviewed.Peer Reviewe

Transversality of homoclinic orbits to hyberbolic equilibria in a Hamiltonian system, via the Hamilton-Jacobi equation

We consider a Hamiltonian system with 2 degrees of freedom, with a hyperbolic equilibrium point having a loop or homoclinic orbit (or, alternatively, two hyperbolic equilibrium points connected by a heteroclinic orbit), as a step towards understanding the behavior of nearly-integrable Hamiltonians near double resonances. We provide a constructive approach to study whether the unstable and stable invariant manifolds of the hyperbolic point intersect transversely along the loop, inside their common energy level. For the system considered, we establish a necessary and suffcient condition for the transversality, in terms of a Riccati equation whose solutions give the slope of the invariant manifolds in a direction transverse to the loop.Preprin

KAM aspects of the quasiperiodic Hamiltonian Hopf bifurcation

In this work we consider a normal form around a 1:-1 non semi-simple resonant periodic orbit of a three-degree of freedom Hamiltonian system. From the formal analysis it can be proved the branching off a two-parameter family of two-dimensional invariant tori, whose normal behavior depends intrinsically on the Hamiltonian. Thus, only elliptic or elliptic, together with parabolic and hyperbolic tori may detach form the resonant periodic orbit. Both patterns are mentioned in the literature as the direct and, respectively, inverse Hopf bifurcations. Here we focus in the direct case, which seems more interesting to us due to its immediate applications in Celestial Mechanics. More precisely, our target is to prove the persistence of the (normally) elliptic invariant tori when the whole Hamiltonian, and not only the truncated normal form is taking into account. Finally, and as far as we know, none of the standard KAM results and methods apply directly to the situation described above, so one needs to suite properly these techniques to the problem at hand

Motion close to the hopf bifurcation of the vertical family of periodic orbits of L^4

The paper deals with different kinds of invariant motions (periodic orbits, 2D and 3D invariant tori and invariant manifolds of periodic orbits) in order to analyze the Hamiltonian direct Hopf bifurcation that takes place close to the Lyapunov vertical family of periodic orbits of the triangular equilibrium point L4 in the 3D restricted three-body problem (RTBP) for the mass parameter, µ, greater than (and close to) µR (Routh’s mass parameter). Consequences of such bifurcation, concerning the confinement of the motion close to the hyperbolic orbits and the 3D nearby tori are also described

Transverse intersections between invariant manifolds of doubly hyperbolic invariant tori, via the Poincaré-Mel'nikov method

We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptic-hyperbolic type, having a homoclinic orbit. More precisely, we consider an (n+2)-degree-of-freedom near integrable Hamiltonian with n centres and 2 saddles, and assume that the homoclinic orbit is preserved under the perturbation. On the centre manifold near the equilibrium, there is a Cantorian family of hyperbolic KAM tori, and we study the homoclinic intersections between the stable and unstable manifolds associated to such tori. We establish that, in general, the manifolds intersect along transverse homoclinic orbits. In a more concrete model, such homoclinic orbits can be detected, in a first approximation, from nondegenerate critical points of a Mel'nikov potential. We provide bounds for the number of transverse homoclinic orbits using that, in general, the potential will be a Morse function (which gives a lower bound# and can be approximated by a trigonometric polynomial #which gives an upper bound)Preprin

Miniversal Deformations of Bimodal Picewise Linear Systems

Keywords: Bimodal piecewise linear system, miniversal deformations, reduced forms.Bimodal linear systems are those consisting of two linear systems on each side of a given hyperplane, having continuous dynamics along that hyperplane. In this work, we focus on the derivation of (orthogonal) miniversal deformations, by using reduced forms.Postprint (published version