122 research outputs found

    The structure and evolution of confined tori near a Hamiltonian Hopf Bifurcation

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    We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In this paper we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the corresponding orbit in a 4D surface of section. To visualize this surface of section we use the method of color and rotation [Patsis and Zachilas 1994]. We find that the consequents are contained in 2D "confined tori". Then, we investigate the structure of the phase space in the neighborhood of complex unstable periodic orbits, which are further away from the transition point. In these cases we observe clouds of points in the 4D surfaces of section. The transition between the two types of orbital behavior is abrupt.Comment: 10 pages, 14 figures, accepted for publication in the International Journal of Bifurcation and Chao

    Geometry of complex instability and escape in four-dimensional symplectic maps

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    In 4D symplectic maps complex instability of periodic orbits is possible, which cannot occur in the 2D case. We investigate the transition from stable to complex unstable dynamics of a fixed point under parameter variation. The change in the geometry of regular structures is visualized using 3D phase-space slices and in frequency space using the example of two coupled standard maps. The chaotic dynamics is studied using escape time plots and by computations of the 2D invariant manifolds associated with the complex unstable fixed point. Based on a normal-form description, we investigate the underlying transport mechanism by visualizing the escape paths and the long-time confinement in the surrounding of the complex unstable fixed point. We find that the slow escape is governed by the transport along the unstable manifold while going across the approximately invariant planes defined by the corresponding normal form

    Hamiltonian Hopf bifurcations in the discrete nonlinear Schr\"odinger trimer: oscillatory instabilities, quasiperiodic solutions and a 'new' type of self-trapping transition

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    Oscillatory instabilities in Hamiltonian anharmonic lattices are known to appear through Hamiltonian Hopf bifurcations of certain time-periodic solutions of multibreather type. Here, we analyze the basic mechanisms for this scenario by considering the simplest possible model system of this kind where they appear: the three-site discrete nonlinear Schr\"odinger model with periodic boundary conditions. The stationary solution having equal amplitude and opposite phases on two sites and zero amplitude on the third is known to be unstable for an interval of intermediate amplitudes. We numerically analyze the nature of the two bifurcations leading to this instability and find them to be of two different types. Close to the lower-amplitude threshold stable two-frequency quasiperiodic solutions exist surrounding the unstable stationary solution, and the dynamics remains trapped around the latter so that in particular the amplitude of the originally unexcited site remains small. By contrast, close to the higher-amplitude threshold all two-frequency quasiperiodic solutions are detached from the unstable stationary solution, and the resulting dynamics is of 'population-inversion' type involving also the originally unexcited site.Comment: 25 pages, 11 figures, to be published in J. Phys. A: Math. Gen. Revised and shortened version with few clarifying remarks adde

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

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    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

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

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    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

    Instabilities and stickiness in a 3D rotating galactic potential

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    We study the dynamics in the neighborhood of simple and double unstable periodic orbits in a rotating 3D autonomous Hamiltonian system of galactic type. In order to visualize the four dimensional spaces of section we use the method of color and rotation. We investigate the structure of the invariant manifolds that we found in the neighborhood of simple and double unstable periodic orbits in the 4D spaces of section. We consider orbits in the neighborhood of the families x1v2, belonging to the x1 tree, and the z-axis (the rotational axis of our system). Close to the transition points from stability to simple instability, in the neighborhood of the bifurcated simple unstable x1v2 periodic orbits we encounter the phenomenon of stickiness as the asymptotic curves of the unstable manifold surround regions of the phase space occupied by rotational tori existing in the region. For larger energies, away from the bifurcating point, the consequents of the chaotic orbits form clouds of points with mixing of color in their 4D representations. In the case of double instability, close to x1v2 orbits, we find clouds of points in the four dimensional spaces of section. However, in some cases of double unstable periodic orbits belonging to the z-axis family we can visualize the associated unstable eigensurface. Chaotic orbits close to the periodic orbit remain sticky to this surface for long times (of the order of a Hubble time or more). Among the orbits we studied we found those close to the double unstable orbits of the x1v2 family having the largest diffusion speed.Comment: 29pages, 25 figures, accepted for publication in the International Journal of Bifurcation and Chao

    Accelerator modes and anomalous diffusion in 3D volume-preserving maps

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    Angle-action maps that have a periodicity in the action direction can have accelerator modes: orbits that are periodic when projected onto the torus, but that lift to unbounded orbits in an action variable. In this paper we construct a family of volume-preserving maps, with two angles and one action, that have accelerator modes created at Hopf-one (or saddle-center-Hopf) bifurcations. Near such a bifurcation we show that there is often a bubble of invariant tori. Computations of chaotic orbits near such a bubble show that the trapping times have an algebraic decay similar to that seen around stability islands in area-preserving maps. As in the 2D case, this gives rise to anomalous diffusive properties of the action in our 3D map
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