A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Rigorous results

In this paper we prove rigorous results on persistence of invariant tori and their whiskers. The proofs are based on the parameterization method of [X. Cabré, E. Fontich, R. de la Llave, The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J. 52 (2) (2003) 283-328; X. Cabré, E. Fontich, R. de la Llave, The parameterization method for invariant manifolds. II. Regularity with respect to parameters, Indiana Univ. Math. J. 52 (2) (2003) 329-360]. The invariant manifolds results proved here include as particular cases of the usual (strong) stable and (strong) unstable manifolds, but also include other non-resonant manifolds. The method lends itself to numerical implementations whose analysis and implementation is studied in [A. Haro, R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms, preprint, 2005; A. Haro, R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical implementation and examples, preprint, 2005]. The results are stated as a posteriori results. Namely, that if one has an approximate solution which is not degenerate, then, one has a true solution not too far from the approximate one. This can be used to validate the results of numerical computations.     

On invariant tori of full dimension for 1D periodic NLS

Consider the NLS with periodic boundary conditions in 1D

(0.1)     

Minimal periods of holomorphic maps on complex tori

We study the set of minimal periods of holomorphic self-maps of one- and two-dimensional complex tori. In particular, we characterize when the set of minimal periods of such maps is finite. In fact, we have an algorithm for doing this characterization for holomorphic self-maps of an arbitrary dimensional complex torus.     

Persistence of lower-dimensional tori under the first Melnikov's non-resonance condition

In this paper we prove the persistence of lower-dimensional invariant tori of integrable equations after Hamiltonian perturbations under the first Melnikov's non-resonance condition. The proof is based on an improved KAM machinery which works for the angle variable dependent normal form. By an example, we also show the necessity of the Melnikov's first non-resonance condition for the persistence of lower dimensional tori.     

Lower dimensional invariant tori with prescribed frequency for nonlinear wave equation

In this paper, one-dimensional (1D) nonlinear wave equation utt−uxx+mu+u³=0, subject to Dirichlet boundary conditions is considered. We show that for each given m>0, and each prescribed integer b>1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with b-dimensional Diophantine frequencies, which correspond to b-dimensional invariant tori of an associated infinite-dimensional dynamical system. In particular, these Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.     

Invariant tori of nonlinear Schrödinger equation

In this paper, we consider one-dimensional nonlinear Schrödinger equation iut−uxx+V(x)u+f(²|u|)u=0 on [0,π]×R under the boundary conditions a₁u(t,0)−b₁ux(t,0)=0, a₂u(t,π)+b₂ux(t,π)=0, , for i=1,2. It is proved that for a prescribed and analytic positive potential V(x), the above equation admits small-amplitude quasi-periodic solutions corresponding to d-dimensional invariant tori of the associated infinite-dimensional dynamical system.     

Analytic smoothing of geometric maps with applications to KAM theory

We show that finitely differentiable diffeomorphisms which are either symplectic, volume-preserving, or contact can be approximated with analytic diffeomorphisms that are, respectively, symplectic, volume-preserving or contact. We prove that the approximating functions are uniformly bounded on some complex domains and that the rate of convergence, in Cr-norms, of the approximation can be estimated in terms of the size of such complex domains and the order of differentiability of the approximated function. As an application to this result, we give a proof of the existence, the local uniqueness and the bootstrap of regularity of KAM tori for finitely differentiable symplectic maps. The symplectic maps considered here are not assumed either to be written in action-angle variables or to be perturbations of integrable systems. Our main assumption is the existence of a finitely differentiable parameterization of a maximal dimensional torus that satisfies a non-degeneracy condition and that is approximately invariant. The symplectic, volume-preserving and contact forms are assumed to be analytic.     

Invariant tori in nonlinear oscillations

The boundedness of all the solutions for semilinear Duffing equationx″ + ω² x + φ(x) =p(t), ω ∈ ℝ⁺ℕ is proved, wherep (t) is a smooth 2π-periodic function and the perturbation ⌽(x) is bounded.     

Conjugations between circle maps with a single break point

Consider two circle homeomorphisms fi∈C2+α(S?{bi}), α>0, i=1,2 with a single break point bi i.e. a discontinuity in the derivative Dfi, and identical irrational rotation number ρ. Suppose the jump ratios and do not coincide. Then the map ψ conjugating f₁ and f₂ is a singular function i.e. it is continuous on S¹ and Dψ(x)=0 a.e. with respect to Lebesgue measure.     

Essential tori in 3-string free tangle decompositions of links

The author characterizes those link types which have 3-string essential free tangle decompositions and have essential tori in their exteriors.     

Total destruction of Lagrangian tori

For an integrable Tonelli Hamiltonian with d   (d?2$d?2$) degrees of freedom, we show that all of the Lagrangian tori can be destroyed by analytic perturbations which are arbitrarily small in the C^d−δ$C^{d-\delta }$ topology.     

Persistence of lower-dimensional hyperbolic invariant tori for generalized Hamiltonian systems

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables. In particular, systems under consideration can be odd-dimensional. Under Rüssmann-type non-degenerate condition, by introducing a modified linear KAM iterative scheme, we proved that the majority of the lower-dimensional hyperbolic invariant tori persist under small perturbations for generalized Hamiltonian systems.     

A symplectic map and its application to the persistence of lower dimensional invariant tori

We give a nonlinear symplectic coordinator transformation, which can move the normal frequencies of the lower dimensional torus up to (k,w) where ω is the frequency vector of the torus. That means the normal frequencies with a difference (k,w) may be regarded as the same. As an application, we derive a persistence result on lower dimensional tori of nearly integrable Hamiltonian systems when the second Melnikov’s condition is partially violated.     

Two-dimensional invariant tori in the neighborhood of an elliptic equilibrium of Hamiltonian systems

In this paper, we prove that a Hamiltonian system possesses either a four-dimensional invaxiant disc or an invariant Cantor set with positive （n ＋ 2）-dimensional Lebesgue measure in the neighborhood of an elliptic equilibrium provided that its lineaxized system at the equilibrium satisfies some small divisor conditions. Both of the invariant sets are foliated by two-dimensional invaxiant tori carrying quasi-oeriodic solutions.     

Existence of Invariant Tori for Certain Weakly Reversible Mappings

In this paper, we consider certain mappings, ${\mathcal {M}}In this paper, we consider certain mappings, ℳ , sufficiently close to an integrable one, which is weakly reversible with respct to the mappings ? sufficiently close to an involution of tye (m, n), where m, n∈Z ₊ are arbitrary. Under some weak non-degeneracy condition, we construct a uniform KAM iteration for providing the existence of a Cantor family of m-tori invariant under the reversible mappings ℳ and the reversing mapping ?. Received September 29, 1999, Accepted July 18, 2000     

First stability eigenvalue characterization of CMC Hopf tori into Riemannian Killing submersions

We find out upper bounds for the first eigenvalue of the stability operator for compact constant mean curvature orientable surfaces immersed in a Riemannian Killing submersion. As a consequence, the strong stability of such surfaces is studied. We also characterize constant mean curvature Hopf tori as the only ones attaining the bound in certain cases.     

Numerical calculation of invariant manifolds for maps

Many processes in the sciences and in engineering are modelled by dynamical systems and—in discretized version—by nonlinear maps. To understand the often complicated dynamical behaviour it is a well established tool to use the concept of invariant manifolds of the system. In this way it is often possible to reduce the dimension of the system considerably. In this paper we propose a new method to calculate numerically invariant manifolds near fixed points of maps. We prove convergence of our procedure and provide an error estimation. Finally, the application of the method is illustrated by examples.     

Parametrization of tamely ramified maximal tori using bounded subgroups

Let be a reductive group defined over a local complete field F with discrete valuation, and split over some unramified extension of F, and let G be its group of F-points. In this paper, we define a class of abelian “torus-like” subgroups in nonreductive groups, called pseudo-tori, which generalizes the notion of torus, and we establish a correspondence between conjugacy classes of tamely ramified maximal tori of G and association classes of maximal pseudo-tori of the quotients of parahorics of G by their second congruence subgroup, viewed as groups of k-points of algebraic groups defined over the residual field k of F.      

A connection between operator topologies, polynomial interpolation, and synthesis of diagonal operators

In this paper we extend the work done by Deters and Seubert in [I. Deters, S. Seubert, Cyclic vectors of diagonal operators on the space of functions analytic on a disk, J. Math. Anal. Appl. 334 (2007) 1209-1219]. In particular, we shall improve upon a theorem in the aforementioned paper and make connections between the strong operator topology on H(B(0,1)), polynomial behavior on a sequence of points, and the synthesis of diagonal operators on H(B(0,1)).     

Lyapunov exponents of partially hyperbolic volume-preserving maps with 2-dimensional center bundle

We consider the set of partially hyperbolic symplectic diffeomorphisms which are accessible, have 2-dimensional center bundle and satisfy some pinching and bunching conditions. In this set, we prove that the non-uniformly hyperbolic maps are $C^r$ open and there exists a $C^r$ open and dense subset of continuity points for the center Lyapunov exponents. We also generalize these results to volume-preserving systems.