On the continuation of degenerate periodic orbits via normal form : full dimensional resonant tori

We reconsider the classical problem of the continuation of degenerate periodic orbits in Hamiltonian systems. In particular we focus on periodic orbits that arise from the breaking of a completely resonant maximal torus. We here propose a suitable normal form construction that allows to identify and approximate the periodic orbits which survive to the breaking of the resonant torus. Our algorithm allows to treat the continuation of approximate orbits which are at leading order degenerate, hence not covered by classical averaging methods. We discuss possible future extensions and applications to localized periodic orbits in chains of weakly coupled oscillators

Improved convergence estimates for the Schröder-Siegel problem

We reconsider the Schröder–Siegel problem of conjugating an analytic map in ℂ in the neighborhood of a fixed point to its linear part, extending it to the case of dimension n>1 . Assuming a condition which is equivalent to Bruno’s one on the eigenvalues λ1,…,λn of the linear part, we show that the convergence radius ρ of the conjugating transformation satisfies lnρ(λ)≥−CΓ(λ)+C′ with Γ(λ) characterizing the eigenvalues λ , a constant C′ not depending on λ and C=1 . This improves the previous results for n>1 , where the known proofs give C=2 . We also recall that C=1 is known to be the optimal value for n=1

EFFECTIVE STABILITY OF HAMILTONIAN PLANETARY SYSTEMS

The stability of the Solar System is a classical, long standing and challenging problem, already pointed out by Newton. In this thesis we revisit the problem in the light of Kolmogorov and Nekhoroshev theorems, with the aim of proving that they apply to realistic approximations of the Sun-Jupiter-Saturn and Sun-Jupiter-Saturn-Uranus systems. The present thesis is devoted to the study of three main problems, namely: (i) the applicability of Kolmogorov and Nekhoroshev theories to the problem of three bodies; (ii) the stability of the secular evolution of the planar Sun-Jupiter-Saturn-Uranus system; (iii) the explicit construction of the normal form for elliptic tori in planetary systems. This work gives an original contribution on the methods for studying the stability of planetary systems. It contains an analytical contribution, namely the proof of the existence of low dimensional elliptic tori for the planetary system obtained via an algorithmic constructive method, and the explicit calculation of the stability time for the Sun-Jupiter-Saturn system and for the planar secular Sun-Jupiter-Saturn-Uranus system. We emphasize that we obtain the first realistic estimates for these problems based on a well established theoretical framework

On the nonexistence of degenerate phase-shift multibreathers in Klein-Gordon models with interactions beyond nearest neighbors

In this work, we study the existence of, low amplitude, phase-shift multibreathers for small values of the linear coupling in KleinGordon chains with interactions beyond the classical nearest-neighbor (NN) ones. In the proper parameter regimes, the considered lattices bear connections to models beyond one spatial dimension, namely the so-called zigzag lattice, as well as the two-dimensional square lattice or coupled chains. We examine initially the necessary persistence conditions of the system derived by the so-called Effective Hamiltonian Method, in order to seek for unperturbed solutions whose continuation is feasible. Although this approach provides useful insights, in the presence of degeneracy, it does not allow us to determine if they constitute true solutions of our system. In order to overcome this obstacle, we follow a different route. By means of a Lyapunov-Schmidt decomposition, we are able to establish that the bifurcation equation for our models can be considered, in the small energy and small coupling regime, as a perturbation of a corresponding, beyond nearest-neighbor, discrete nonlinear Schr\ua8odinger equation. There, nonexistence results of degenerate phase-shift discrete solitons can be demonstrated by an additional Lyapunov-Schmidt decomposition, and translated to our original problem on the Klein-Gordon system. In this way, among other results, we can prove nonexistence of four-sites vortex-like waveforms in the zigzag Klein-Gordon model. Finally, briefly considering a one-dimensional model bearing similarities to the square lattice, we conclude that the above strategy is not efficient for the proof of the existence or nonexistence of vortices due to the higher degeneracy of this configuration

On the nonexistence of degenerate phase-shift discrete solitons in a dNLS nonlocal lattice

We consider a one-dimensional discrete nonlinear Schr\uf6dinger (dNLS) model featuring interactions beyond nearest neighbors. We are interested in the existence (or nonexistence) of phase-shift discrete solitons, which correspond to four-sites vortex solutions in the standard two-dimensional dNLS model (square lattice), of which this is a simpler variant. Due to the specific choice of lengths of the inter-site interactions, the vortex configurations considered present a degeneracy which causes the standard continuation techniques to be non-applicable. In the present one-dimensional case, the existence of a conserved quantity for the soliton profile (the so-called density current), together with a perturbative construction, leads to the nonexistence of any phase-shift discrete soliton which is at least C2 with respect to the small coupling \u3f5, in the limit of vanishing \u3f5. If we assume the solution to be only C0 in the same limit of \u3f5, nonexistence is instead proved by studying the bifurcation equation of a Lyapunov-Schmidt reduction, expanded to suitably high orders. Specifically, we produce a nonexistence criterion whose efficiency we reveal in the cases of partial and full degeneracy of approximate solutions obtained via a leading order expansion

Effective resonant stability of Mercury

Mercury is the unique known planet that is situated in a 3:2 spin-orbit resonance nowadays. Observations and models converge to the same conclusion: the planet is presently deeply trapped in the resonance and situated at the Cassini state 1, or very close to it. We investigate the complete non-linear stability of this equilibrium, with respect to several physical parameters, in the framework of Birkhoffnormal form and Nekhoroshev stability theory. We use the same approach we have adopted for the 1:1 spin-orbit case with a peculiar attention to the role of Mercury's non-negligible eccentricity. The selected parameters are the polar moment of inertia, the Mercury's inclination and eccentricity and the precession rates of the perihelion and node. Our study produces a bound to both the latitudinal and longitudinal librations (of 0.1 rad) for a long but finite time (greatly exceeding the age of the Solar system). This is the so-called effective stability time. Our conclusion is that Mercury, placed inside the 3:2 spin-orbit resonance, occupies a very stable position in the space of these physical parameters, but not the most stable possible one

Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability into the light of Kolmogorov and Nekhoroshev theories

We investigate the long-time stability of the Sun-Jupiter-Saturn-Uranus system by considering a planar secular model, that can be regarded as a major refinement of the approach first introduced by Lagrange. Indeed, concerning the planetary orbital revolutions, we improve the classical circular approximation by replacing it with a solution that is invariant up to order two in the masses; therefore, we investigate the stability of the secular system for rather small values of the eccentricities. First, we explicitly construct a Kolmogorov normal form, so as to find an invariant KAM torus which approximates very well the secular orbits. Finally, we adapt the approach that is at basis of the analytic part of the Nekhoroshev's theorem, so as to show that there is a neighborhood of that torus for which the estimated stability time is larger than the lifetime of the Solar System. The size of such a neighborhood, compared with the uncertainties of the astronomical observations, is about ten times smaller.Comment: 31 pages, 2 figures. arXiv admin note: text overlap with arXiv:1010.260

Aspects of the planetary Birkhoff normal form

The discovery in [G. Pinzari. PhD thesis. Univ. Roma Tre. 2009], [L. Chierchia and G. Pinzari, Invent. Math. 2011] of the Birkhoff normal form for the planetary many--body problem opened new insights and hopes for the comprehension of the dynamics of this problem. Remarkably, it allowed to give a {\sl direct} proof of the celebrated Arnold's Theorem [V. I. Arnold. Uspehi Math. Nauk. 1963] on the stability of planetary motions. In this paper, using a "ad hoc" set of symplectic variables, we develop an asymptotic formula for this normal form that may turn to be useful in applications. As an example, we provide two very simple applications to the three-body problem: we prove a conjecture by [V. I. Arnold. cit] on the "Kolmogorov set"of this problem and, using Nehoro{\v{s}}ev Theory [Nehoro{\v{s}}ev. Uspehi Math. Nauk. 1977], we prove, in the planar case, stability of all planetary actions over exponentially-long times, provided mean--motion resonances are excluded. We also briefly discuss perspectives and problems for full generalization of the results in the paper.Comment: 44 pages. Keywords: Averaging Theory, Birkhoff normal form, Nehoro{\v{s}}ev Theory, Planetary many--body problem, Arnold's Theorem on the stability of planetary motions, Properly--degenerate kam Theory, steepness. Revised version, including Reviewer's comments. Typos correcte