170 research outputs found
Invariant curves near Hamiltonian-Hopf bifurcations of 4D symplectic maps
In this paper we give a numerical description of the neighbourhood of a fixed point of a symplectic map undergoing a transition from linear stability to complex instability, i.e., the so called Hamiltonian-Hopf bifurcation. We have considered both the direct and inverse cases. The study is based on the numerical computation of the Lyapunov families of invariant curves near the fixed point. We show how these families, jointly with their invariant manifolds and the invariant manifolds of the fixed point organise the phase space around the bifurcation
Invariant manifolds of L_3 and horseshoe motion in the restricted three-body problem
In this paper, we consider horseshoe motion in the planar restricted three-body
problem. On one hand, we deal with the families of horseshoe periodic orbits (which
surround three equilibrium points called L3, L4 and L5), when the mass parameter
µ is positive and small; we describe the structure of such families from the two-body
problem (µ = 0). On the other hand, the region of existence of horseshoe periodic
orbits for any value of µ ∈ (0, 1/2] implies the understanding of the behaviour of the
invariant manifolds of L3. So, a systematic analysis of such manifolds is carried out.
As well the implications on the number of homoclinic connections to L3, and on the
simple infinite and double infinite period homoclinic phenomena are also analysed.
Finally, the relationship between the horseshoe homoclinic orbits and the horseshoe
periodic orbits are considered in detail
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
Pseudo-heteroclinic connections between bicircular restricted four-body problems
In this paper, we show a mechanism to explain transport from the outer to the inner Solar system. Such a mechanism is based on dynamical systems theory. More concretely, we consider a sequence of uncoupled bicircular restricted four-body problems –BR4BP –(involving the Sun, Jupiter, a planet and an infinitesimal mass), being the planet Neptune, Uranus and Saturn. For each BR4BP, we compute the dynamical substitutes of the collinear equilibrium points of the corresponding restricted three-body problem (Sun, planet and infinitesimal mass), which become periodic orbits. These periodic orbits are unstable, and the role that their invariant manifolds play in relation with transport from exterior planets to the inner ones is discussed.Peer ReviewedPostprint (published version
Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP
We consider the planar restricted three-body problem and the collinear equilibrium
point L3, as an example of a center×saddle equilibrium point in a Hamiltonian with two
degrees of freedom.We explore numerically the existence of symmetric and non-symmetric
homoclinic orbits to L3, when varying the mass parameter μ. Concerning the symmetric homoclinic
orbits (SHO), we study the multi-round, m-round, SHO for m ≥ 2. More precisely,
given a transversal value of μ for which there is a 1-round SHO, say μ1, we show that for
any m ≥ 2, there are countable sets of values of μ, tending to μ1, corresponding to m-round
SHO. Some comments on related analytical results are also made.Peer ReviewedPostprint (published version
Dinà mica de varietats espacials: les autopistes de l’univers
En aquest article volem il.lustrar com la comprensió de la dinà mica d’alguns models de la mecà nica celeste permet explicar alguns fenòmens astronòmics i dissenyar missions realistes a l’espai. El model paradigmà tic usat és el problema restringit de tres cossos, en el qual els objectes que tenen un paper essencial són les varietats invariants de les anomenades òrbites de libració, és a dir, òrbites periòdiques i quasi periòdiques al voltant dels anomenats punts d’equilibri col.lineals del model.Descriurem alguns d’aquests fenòmens i esmentarem algunes missions concretes.Finalment, comentarem altres models també útils (i més sofisticats) a l’astrodinà mica i acabarem amb algun comentari de com les eines de sistemes dinà mics es poden traslladar del món macroscòpic (celeste) al microscòpic, com per exemple el de la fÃsica atòmica clà ssica.Postprint (published version
Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP
We consider the planar restricted three-body problem and the collinear equilibrium
point L3, as an example of a center×saddle equilibrium point in a Hamiltonian with two
degrees of freedom.We explore numerically the existence of symmetric and non-symmetric
homoclinic orbits to L3, when varying the mass parameter μ. Concerning the symmetric homoclinic
orbits (SHO), we study the multi-round, m-round, SHO for m ≥ 2. More precisely,
given a transversal value of μ for which there is a 1-round SHO, say μ1, we show that for
any m ≥ 2, there are countable sets of values of μ, tending to μ1, corresponding to m-round
SHO. Some comments on related analytical results are also made.Peer ReviewedPostprint (published version
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