Lower Dimensional Tori Research Articles

Degenerate KAM theory for partial differential equations

This paper deals with degenerate KAM theory for lower dimensional elliptic tori of infinite dimensional Hamiltonian systems, depending on one parameter only. We assume that the linear frequencies are analytic functions of the parameter, satisfy a weak non-degeneracy condition of Rüssmann type and an asymptotic behavior. An application to nonlinear wave equations is given.

A renormalization approach to lower-dimensional tori with Brjuno frequency vectors

We extend the renormalization scheme for vector fields on T d × R m in order to construct lower-dimensional invariant tori with Brjuno frequency vectors for near-integrable Hamiltonian flows. For every Brjuno frequency vector ω ∈ R d and every vector Ω ∈ R D satisfying a Diophantine condition with respect to ω, there exists an analytic manifold W of infinitely renormalizable Hamiltonian vector fields; each vector field on W is shown to have an analytic invariant torus with frequency vector ω.

Algebraic geometry codes from polyhedral divisors

A description of complete normal varieties with lower-dimensional torus action has been given by Altmann et al. (2008), generalizing the theory of toric varieties. Considering the case where the acting torus T has codimension one, we describe T -invariant Weil and Cartier divisors and provide formulae for calculating global sections, intersection numbers, and Euler characteristics. As an application, we use divisors on these so-called T -varieties to define new evaluation codes called T -codes. We find estimates on their minimum distance using intersection theory. This generalizes the theory of toric codes and combines it with AG codes on curves. As the simplest application of our general techniques we look at codes on ruled surfaces coming from decomposable vector bundles. Already this construction gives codes that are better than the related product code. Further examples show that we can improve these codes by constructing more sophisticated T -varieties. These results suggest looking further for good codes on T -varieties.

Energy localization onq-tori, long-term stability, and the interpretation of Fermi-Pasta-Ulam recurrences

We focus on two approaches that have been proposed in recent years for the explanation of the so-called Fermi-Pasta-Ulam (FPU) paradox, i.e., the persistence of energy localization in the "low-q " Fourier modes of Fermi-Pasta-Ulam nonlinear lattices, preventing equipartition among all modes at low energies. In the first approach, a low-frequency fraction of the spectrum is initially excited leading to the formation of "natural packets" exhibiting exponential stability, while in the second, emphasis is placed on the existence of "q breathers," i.e., periodic continuations of the linear modes of the lattice, which are exponentially localized in Fourier space. Following ideas of the latter, we introduce in this paper the concept of " q-tori" representing exponentially localized solutions on low-dimensional tori and use their stability properties to reconcile these two approaches and provide a more complete explanation of the FPU paradox.

Minimal homeomorphisms on low-dimensional tori

AbstractWe study minimal homeomorphisms (all orbits are dense) of the tori Tn, n≤4. The linear part of a homeomorphism φ of Tn is the linear mapping L induced by φ on the first homology group of Tn. It follows from the Lefschetz fixed point theorem that 1 is an eigenvalue of L if φ minimal. We show that if φ is minimal and n≤4, then L is quasi-unipontent, that is, all of the eigenvalues of L are roots of unity and conversely if L∈GL(n,ℤ) is quasi-unipotent and 1 is an eigenvalue of L, then there exists a C∞ minimal skew-product diffeomorphism φ of Tn whose linear part is precisely L. We do not know whether these results are true for n≥5. We give a sufficient condition for a smooth skew-product diffeomorphism of a torus of arbitrary dimension to be smoothly conjugate to an affine transformation.

Detecting chaos, determining the dimensions of tori and predicting slow diffusion in Fermi–Pasta–Ulam lattices by the Generalized Alignment Index method

The recently introduced GALI method is used for rapidly detecting chaos, determining the dimensionality of regular motion and predicting slow diffusion in multi-dimensional Hamiltonian systems. We propose an efficient computation of the GALIk indices, which represent volume elements of k randomly chosen deviation vectors from a given orbit, based on the Singular Value Decomposition (SVD) algorithm. We obtain theoretically and verify numerically asymptotic estimates of GALIs long-time behavior in the case of regular orbits lying on low-dimensional tori. The GALIk indices are applied to rapidly detect chaotic oscillations, identify low-dimensional tori of Fermi–Pasta–Ulam (FPU) lattices at low energies and predict weak diffusion away from quasiperiodic motion, long before it is actually observed in the oscillations.

Quasi-periodic stability of normally resonant tori

We study quasi-periodic tori under a normal-internal resonance, possibly with multiple eigenvalues. Two non-degeneracy conditions play a role. The first of these generalizes invertibility of the Floquet matrix and prevents drift of the lower dimensional torus. The second condition involves a Kolmogorov-like variation of the internal frequencies and simultaneously versality of the Floquet matrix unfolding. We focus on the reversible setting, but our results carry over to the Hamiltonian and dissipative contexts.

Kolmogorov–Arnold–Moser aspects of the periodic Hamiltonian Hopf bifurcation

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, we prove the branching off of a two-parameter family of two-dimensional invariant tori of the normalized 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 from the resonant periodic orbit. Both patterns are mentioned in the literature as the direct and inverse, respectively, periodic Hopf bifurcation. In this paper we focus on the direct case, which has many applications in several fields of science. Our target is to prove, in the framework of Kolmogorov–Arnold–Moser (KAM) theory, 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 characterization of the parameters labelling them, which can be selected with a very clear dynamical meaning. Furthermore, we give sharp quantitative estimates on the ‘density’ of surviving tori, when the distance to the resonant periodic orbit goes to zero, and show that the four-dimensional invariant Cantor manifold holding them admits a Whitney-C∞ extension. Due to the strong degeneracy of the problem, some standard KAM methods for elliptic low-dimensional tori of Hamiltonian systems do not apply directly, so one needs to properly suit these techniques to the context.

Dynamical Bottlenecks to Intramolecular Energy Flow

Vibrational energy flows unevenly in molecules, repeatedly going back and forth between trapping and roaming. We identify bottlenecks between diffusive and chaotic behavior, and describe generic mechanisms of these transitions, taking the carbonyl sulfide molecule OCS as a case study. The bottlenecks are found to be lower-dimensional tori; their bifurcations and unstable manifolds govern the transition mechanisms.

A KAM Theorem with Applications to Partial Differential Equations of Higher Dimensions

The existence of lower dimensional KAM tori is shown for a class of nearly integrable Hamiltonian systems where the second Melnikov's conditions are eliminated. As a consequence, it is proved that there exist many invariant tori and thus quasi-periodic solutions for nonlinear wave equations, Schr\"odinger equations and other equations of higher spatial dimension.

Degenerate lower-dimensional tori under the Bryuno condition

We study the problem of conservation of maximal and lower-dimensional invariant tori for analytic convex quasi-integrable Hamiltonian systems. In the absence of perturbation the lower-dimensional tori are degenerate, in the sense that the normal frequencies vanish, so that the tori are neither elliptic nor hyperbolic. We show that if the perturbation parameter is small enough, for a large measure subset of any resonant submanifold of the action variable space, under some generic non-degeneracy conditions on the perturbation function, there are lower-dimensional tori which are conserved. They are characterized by rotation vectors satisfying some generalized Bryuno conditions involving also the normal frequencies. We also show that, again under some generic assumptions on the perturbation, any torus with fixed rotation vector satisfying the Bryuno condition is conserved for most values of the perturbation parameter in an interval small enough around the origin. According to the sign of the normal frequencies and of the perturbation parameter the torus becomes either hyperbolic or elliptic or of mixed type.

Analysis on discrete cocompact subgroups of the generic filiform Lie groups

Let n, n 1, denote the sequence of generic filiform (connected, simply connected) Lie groups. Here we study, for each n, the infinite dimensional simple quotients of the group C ⁄ -algebra of (the most obvious) one of its discrete cocompact subgroups n. For n, the most attractive concrete faithful represen- tations are given in terms of Anzai flows, in analogy with the representations of the discrete Heisenberg group H3 ‰ G3 on L 2 (T) that result from the irrational rotation flows on ; the representations of n generate infinite-dimensional sim- ple quotients An;µ of the group C ⁄ -algebra C ⁄ ( n). For n > 1, there are other infinite-dimensional simple quotients of C ⁄ ( n) arising from non-faithful repre- sentations of n. Flows for these are determined, and they are also characterized and represented as matrix algebras over simple ane Furstenberg transformation group C ⁄ -algebras of the lower dimensional tori.

On Hamiltonian bifurcations of invariant tori with a Floquet multiplier −1

Nearly integrable Hamiltonian systems are considered, for which the unperturbed system has a lower dimensional torus not satisfying the second Mel'nikov condition; on a 2 : 1 covering space a suitable choice of the toral angles yields a vanishing Floquet exponent. A nilpotent Floquet matrix leads to the quasi-periodic analogue of the period-doubling bifurcation, so particular emphasis is given to the case of vanishing normal linear behaviour. The actions conjugate to the toral angles unfold the various ways in which the degenerate torus becomes normally elliptic, hyperbolic or parabolic. With a KAM-theoretic approach it is then shown that this bifurcation scenario survives a non-integrable perturbation, parametrized by pertinent large Cantor sets. The bifurcation scenario is governed by , the ‘first’ unimodal planar singularity, which has co-dimension 7 with respect to all planar singularities. In the present context this high number is reduced to co-dimension 3 since the π-rotation on the 2 : 1 covering space has to be respected, and in case the Hamiltonian system is reversible there is a further reduction by 1 and the co-dimension becomes 2. In such low co-dimensions it becomes more transparent why the modulus μ – although playing a prominent role during the KAM iteration – is of limited influence on the dynamical implications.

On the persistence of elliptic lower-dimensional tori in Hamiltonian systems under the first Melnikov condition and Rüssmann’s non-degeneracy condition

In this paper we formulate a theorem on the persistence of elliptic lower-dimensional invariant tori for nearly integrable analytic Hamiltonian systems under the first Melnikov condition and Rüssmann’s non-degeneracy condition, and give the measure estimates of parameters for the non-resonance conditions under Rüssmann’s non-degeneracy condition, which is essential for the proof of our result.

Partial preservation of frequencies in KAM theory

We consider perturbations of moderately degenerate integrable or partially integrable Hamiltonian systems, so that unperturbed invariant n-tori with prescribed frequencies or frequency ratios do not persist, but there is preservation of, say, the first d < n frequencies or their ratios. Lagrangian and lower dimensional tori are treated in a unified way. The proofs are very simple and follow Herman's idea of 1990: we introduce external parameters to remove degeneracies and then eliminate these parameters making use of a suitable number-theoretical lemma concerning Diophantine approximations of dependent quantities. Parallel results for reversible, volume preserving and dissipative systems are also presented.

Degenerate lower-dimensional tori in Hamiltonian systems

We study the persistence of lower-dimensional tori in Hamiltonian systems of the form H ( x , y , z ) = 〈 ω , y 〉 + 1 2 〈 z , M ( ω ) z 〉 + ε P ( x , y , z , ω ) , where ( x , y , z ) ∈ T n × R n × R 2 m , ε is a small parameter, and M ( ω ) can be singular. We show under a weak Melnikov nonresonant condition and certain singularity-removing conditions on the perturbation that the majority of unperturbed n-tori can still survive from the small perturbation. As an application, we will consider the persistence of invariant tori on certain resonant surfaces of a nearly integrable, properly degenerate Hamiltonian system for which neither the Kolmogorov nor the g-nondegenerate condition is satisfied.

On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition

In this paper we prove the persistence of elliptic lower dimensional invariant tori for nearly integrable Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition by an improved KAM iteration, and the persisting invariant tori are Gevrey smooth with respect to parameters in the sense of Whitney, with a Gevrey index depending on the Gevrey class of Hamiltonian systems and on the exponent in the Diophantine condition. Moreover the Gevrey index should be optimal for the Diophantine condition in the proof of our theorem.

Fractional Lindstedt series

The parametric equations of the surfaces on which highly resonant quasiperiodic motions develop (lower-dimensional tori) cannot be analytically continued, in general, in the perturbation parameter ε, i.e., they are not analytic functions of ε. However rather generally quasiperiodic motions whose frequencies satisfy only one rational relation (“resonances of order 1”) admit formal perturbation expansions in terms of a fractional power of ε depending on the degeneration of the resonance. We find conditions for this to happen, and in such a case we prove that the formal expansion is convergent after suitable resummation.

Umbilical torus bifurcations in Hamiltonian systems

We consider perturbations of integrable Hamiltonian systems in the neighbourhood of normally umbilic invariant tori. These lower dimensional tori do not satisfy the usual non-degeneracy conditions that would yield persistence by an adaption of KAM theory, and there are indeed regions in parameter space with no surviving torus. We assume appropriate transversality conditions to hold so that the tori in the unperturbed system bifurcate according to a (generalised) umbilical catastrophe. Combining techniques of KAM theory and singularity theory we show that such bifurcation scenarios of invariant tori survive the perturbation on large Cantor sets. Applications to gyrostat dynamics are pointed out.

Persistence of lower dimensional tori of general types in Hamiltonian systems

This work is a generalization to a result of J. You (1999). We study the persistence of lower dimensional tori of general type in Hamiltonian systems of general normal forms. By introducing a modified linear KAM iterative scheme to deal with small divisors, we shall prove a persistence result, under a Melnikov type of non-resonance condition, which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori.