229 research outputs found
Growth of Sobolev norms for the quintic NLS on
We study the quintic Non Linear Schr\"odinger equation on a two dimensional
torus and exhibit orbits whose Sobolev norms grow with time. The main point is
to reduce to a sufficiently simple toy model, similar in many ways to the one
used in the case of the cubic NLS. This requires an accurate combinatorial
analysis.Comment: 41 pages, 5 figures. arXiv admin note: text overlap with
arXiv:0808.1742 by other author
Quasiperiodic spin-orbit motion and spin tunes in storage rings
We present an in-depth analysis of the concept of spin precession frequency
for integrable orbital motion in storage rings. Spin motion on the periodic
closed orbit of a storage ring can be analyzed in terms of the Floquet theorem
for equations of motion with periodic parameters and a spin precession
frequency emerges in a Floquet exponent as an additional frequency of the
system. To define a spin precession frequency on nonperiodic synchro-betatron
orbits we exploit the important concept of quasiperiodicity. This allows a
generalization of the Floquet theorem so that a spin precession frequency can
be defined in this case too. This frequency appears in a Floquet-like exponent
as an additional frequency in the system in analogy with the case of motion on
the closed orbit. These circumstances lead naturally to the definition of the
uniform precession rate and a definition of spin tune. A spin tune is a uniform
precession rate obtained when certain conditions are fulfilled. Having defined
spin tune we define spin-orbit resonance on synchro--betatron orbits and
examine its consequences. We give conditions for the existence of uniform
precession rates and spin tunes (e.g. where small divisors are controlled by
applying a Diophantine condition) and illustrate the various aspects of our
description with several examples. The formalism also suggests the use of
spectral analysis to ``measure'' spin tune during computer simulations of spin
motion on synchro-betatron orbits.Comment: 62 pages, 1 figure. A slight extension of the published versio
Singular Bohr-Sommerfeld Rules for 2D Integrable Systems
In this paper, we describe Bohr-Sommerfeld rules for semi-classical
completely integrable systems with 2 degrees of freedom with non degenerate
singularities (Morse-Bott singularities) under the assumption that the energy
level of the first Hamiltonian is non singular. The more singular case of {\it
focus-focus} singularities is studied in [Vu Ngoc San, CPAM 2000] and [Vu Ngoc
San, PhD 1998] The case of 1 degree of freedom has been studied in [Colin de
Verdiere-Parisse, CMP 1999] Our theory is applied to some famous examples: the
geodesics of the ellipsoid, the -resonance, and Schroedinger operators on
the sphere . A numerical test shows that the semiclassical Bohr-Sommerfeld
rules match very accurately the ``purely quantum'' computations.Comment: postscript, 61 pages, figures best seen in color. Preprint Institut
Fourie
Homoclinic phenomena in conservative systems
The goal of this thesis is the study of homoclinic orbits in conservative systems (area-preserving maps and Hamiltonian systems). We consider homoclinic (bi-asymptotic) orbits either to saddle periodic orbits or to whiskered tori. Such type orbits, called homoclinic by Poincaré, are of great interest in the theory of dynamical systems since their presence implies complicated dynamics.
The thesis is divided in two parts according to two quite different topics considered. In the first part, we study area-preserving maps (APMs) with a nontransversal homoclinic orbit (homoclinic tangency) to a saddle fixed point in order to know the behavior of orbits near the given homoclinic trajectory. To this end, we construct first return maps, for which we use finitely-smooth normal forms of the saddle maps and introduce cross-coordinates. The fixed points of the first return maps correspond to single-round periodic orbits of the maps under consideration. Applying rescaling methods we derive the first return maps to the HĂ©non-like maps whose bifurcations are well known. Thus, translating the results obtained for the fixed points of the return maps to the periodic orbits, we prove the existence of cascades of elliptic periodic points. We also study the phenomenon of the coexistence of infinitely many single-round periodic orbits of different large periods (called global resonance). We consider the related problems in different types of APMs (symplectic maps and non-orientable APMs) with quadratic or cubic tangencies. We also establish the structure of 1:4 resonance for some conservative HĂ©non-like maps.
The second part of the thesis is dedicated to the study of exponentially small splitting of separatrices arising from a perturbation of a Hamiltonian system with a homoclinic connection (separatrix). We consider a perturbation of an integrable Hamiltonian system having whiskered tori with coincident stable and unstable whiskers. Generally, in the perturbed system, the whiskers do not coincide anymore and our goal is to detect the transverse homoclinic orbits associated to the persistent whiskered tori. The perturbed system turns out to be not integrable due to the presence of these homoclinic trajectories and, consequently, there is chaotic dynamics near them. We give a suitable parametrization to the whiskers to determine the distance between them. This distance is given by the splitting function, and the simple zeros of this function give rise to transverse homoclinic orbits. We use the classical Poincaré-Melnikov approach to measure the splitting, although in the case of exponential smallness we have to ensure that the first order approximation overcome the error term. We consider Hamiltonian systems possessing two-dimensional whiskered tori with quadratic frequencies and three-dimensional whiskered tori with cubic golden frequency. In the two-dimensional case, we find 23 new quadratic numbers for which the Poincaré-Melnikov method can be applied and establish the existence of 4 transverse homoclinic orbits. We also study the continuation of the homoclinic orbits for all values of the parameter of perturbation in the case of the silver number sqrt(2)-1. For the three-dimensional whiskered torus with frequency vector given by the so-called "cubic golden number", we establish the existence of exponentially small splitting of separatrices and detect the transversality of 8 homoclinic orbits
Periodic and quasi-periodic attractors for the spin-orbit evolution of Mercury with a realistic tidal torque
Mercury is entrapped in a 3:2 resonance: it rotates on its axis three times
for every two revolutions it makes around the Sun. It is generally accepted
that this is due to the large value of the eccentricity of its orbit. However,
the mathematical model originally introduced to study its spin-orbit evolution
proved not to be entirely convincing, because of the expression commonly used
for the tidal torque. Only recently, in a series of papers mainly by Efroimsky
and Makarov, a different model for the tidal torque has been proposed, which
has the advantages of being more realistic, and of providing a higher
probability of capture in the 3:2 resonance with respect to the previous
models. On the other hand, a drawback of the model is that the function
describing the tidal torque is not smooth and consists of a superposition of
kinks, so that both analytical and numerical computations turn out to be rather
delicate: indeed, standard perturbation theory based on power series expansion
cannot be applied and the implementation of a fast algorithm to integrate the
equations of motion numerically requires a high degree of care. In this paper,
we make a detailed study of the spin-orbit dynamics of Mercury, as predicted by
the realistic model: In particular, we present numerical and analytical results
about the nature of the librations of Mercury's spin in the 3:2 resonance. The
results provide evidence that the librations are quasi-periodic in time.Comment: 32 pages, 8 figures, 5 table
Torsional solutions of convection in rotating fluid spheres
A numerical study of the nonlinear torsional solutions of convection in rotating, internally heated, self-gravitating fluid spheres is presented. Their dependence on the Rayleigh number has been found for two pairs of Ekman (E) and small Prandtl (Pr) numbers in the region of parameters where, according to Zhang et al. [J. Fluid Mech. 813, R2 (2017)], the linear stability of the conduction state predicts that they can be preferred at the onset of convection. The bifurcation to periodic torsional solutions is supercritical for sufficiently small Pr. They are not rotating waves, unlike the nonaxisymmetric case. Therefore they have been computed by using continuation methods for periodic orbits. Their stability with respect to axisymmetric perturbations and physical characteristics have been analyzed. It was found that the time- and space-averaged equatorially antisymmetric part of the kinetic energy of the stable orbits splits into equal poloidal and toroidal parts, while the symmetric part is much smaller. Direct numerical simulations for E=10-4 at higher Rayleigh numbers (Ra) show that this trend is also valid for the nonperiodic flows and that the mean values of the energies remain almost constant with Ra. However, the modulated oscillations bifurcated from the quasiperiodic torsional solutions reach a high amplitude, compared with that of the periodic, increasing slowly and decaying very fast. This repeated behavior is interpreted as trajectories near heteroclinic chains connecting unstable periodic solutions. The torsional flows give rise to a meridional propagation of the kinetic energy near the outer surface and an axial oscillation of the hot nucleus of the metallic fluid sphere.Postprint (published version
Symplectic Techniques for Semiclassical Completely Integrable Systems
This article is a survey of classical and quantum completely integrable
systems from the viewpoint of local ``phase space'' analysis. It advocates the
use of normal forms and shows how to get global information from glueing local
pieces. Many crucial phenomena such as monodromy or eigenvalue concentration
are shown to arise from the presence of non-degenerate critical points.Comment: 32 pages, 7 figures. Review articl
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