Some Aspects of Conservative and Dissipative KAM Theorems

We discuss some aspects of conservative and dissipative KAM theorems, with particular reference to a comparison between the main assumptions needed to develop KAM theory in the two settings. After analyzing the qualitative behavior of a paradigmatic model (the standard mapping), we study the existence of quasi?Cperiodic tori in the two frameworks, paying special attention to the occurrence of small divisors and to the non?Cdegeneracy (twist) condition in the conservative and in the dissipative case. These conditions are the main requirements for the applicability of KAM theorem, which is then stated for invariant tori as well as for invariant attractors. We proceed to discuss a criterion for the determination of the breakdown threshold of invariant tori and invariant attractors through approximating periodic orbits. These results can be applied to a wide set of physical problems; concrete applications to Celestial Mechanics are discussed with particular reference to the rotational and orbital motion of celestial bodies.     

Quasi-periodic solutions of completely resonant nonlinear wave equations

It is shown that there are many elliptic invariant tori, and thus quasi-periodic solutions, for the completely resonant nonlinear wave equation subject to periodic boundary conditions via KAM theory.     

Rigorous Computer-Assisted Application of KAM Theory: A Modern Approach

In this paper, we present and illustrate a general methodology to apply KAM theory in particular problems, based on an a posteriori approach. We focus on the existence of real analytic quasi-periodic Lagrangian invariant tori for symplectic maps. The purpose is to verify the hypotheses of a KAM theorem in an a posteriori format: Given a parameterization of an approximately invariant torus, we have to check non-resonance (Diophantine) conditions, non-degeneracy conditions and certain inequalities to hold. To check such inequalities, we require to control the analytic norm of some functions that depend on the map, the ambient structure and the parameterization. To this end, we propose an efficient computer-assisted methodology, using fast Fourier transform, having the same asymptotic cost of using the parameterization method for obtaining numerical approximations of invariant tori. We illustrate our methodology by proving the existence of invariant curves for the standard map (up to \(\varepsilon =0.9716\)), meandering curves for the non-twist standard map and 2-dimensional tori for the Froeschlé map.     

KAM theorems and open problems for infinite-dimensional Hamiltonian with short range

We introduce several KAM theorems for infinite-dimensional Hamiltonian with short range and discuss the relationship between spectra of linearized operator and invariant tori.Especially,we introduce a KAM theorem by Yuan published in CMP(2002),which shows that there are rich KAM tori for a class of Hamiltonian with short range and with linearized operator of pure point spectra.We also present several open problems.     

From the restricted to the full three-body problem

The three-body problem with all the classical integrals fixed and all the symmetries removed is called the reduced three-body problem. We use the methods of symplectic scaling and reduction to show that the reduced planar or spatial three-body problem with one small mass is to the first approximation the product of the restricted three-body problem and a harmonic oscillator. This allows us to prove that many of the known results for the restricted problem have generalizations for the reduced three-body problem.

For example, all the non-degenerate periodic solutions, generic bifurcations, Hamiltonian-Hopf bifurcations, bridges and natural centers known to exist in the restricted problem can be continued into the reduced three-body problem. The classic normalization calculations of Deprit and Deprit-Bartholomé show that there are two-dimensional KAM invariant tori near the Lagrange point in the restricted problem. With the above result this proves that there are three-dimensional KAM invariant tori near the Lagrange point in the reduced three-body problem.     

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.     

Persistence of Invariant Tori on Submanifolds in Hamiltonian Systems

Summary. Generalizing the degenerate KAM theorem under the Rüssmann nondegeneracy and the isoenergetic KAM theorem, we employ a quasilinear iterative scheme to study the persistence and frequency preservation of invariant tori on a smooth submanifold for a real analytic, nearly integrable Hamiltonian system. Under a nondegenerate condition of Rüssmann type on the submanifold, we shall show the following: (a) the majority of the unperturbed tori on the submanifold will persist; (b) the perturbed toral frequencies can be partially preserved according to the maximal degeneracy of the Hessian of the unperturbed system and be fully preserved if the Hessian is nondegenerate; (c) the Hamiltonian admits normal forms near the perturbed tori of arbitrarily prescribed high order. Under a subisoenergetic nondegenerate condition on an energy surface, we shall show that the majority of unperturbed tori give rise to invariant tori of the perturbed system of the same energy which preserve the ratio of certain components of the respective frequencies.     

Invariant Tori in Hamiltonian Systems with High Order Proper Degeneracy

We study the existence of quasi-periodic, invariant tori in a nearly integrable Hamiltonian system of high order proper degeneracy, i.e., the integrable part of the Hamiltonian involves several time scales and at each time scale the corresponding Hamiltonian depends on only part of the action variables. Such a Hamiltonian system arises frequently in problems of celestial mechanics, for instance, in perturbed Kepler problems like the restricted and non-restricted 3-body problems and spatial lunar problems in which several bodies with very small masses are coupled with two massive bodies and the nearly integrable Hamiltonian systems naturally involve different time scales. Using KAM method, we will show under certain higher order non-degenerate conditions of Bruno–Rüssmann type that the majority of quasi-periodic, invariant tori associated with the integrable part will persist after the non-integrable perturbation. This actually concludes the KAM metric stability for such a properly degenerate Hamiltonian system.     

A KAM Theorem for Reversible Systems of Infinite Dimension

For reversible systems of infinite dimension we prove an infinitely dimensional KAM theorem with an application to the network of weakly coupled oscillators of friction. The KAM theorem shows that there are many invariant tori of infinite dimension, and thus many almost periodic solutions, for the reversible systems.     

Direction-Reversing Traveling Waves in the Even Fermi-Pasta-Ulam Lattice

Summary. This paper considers the famous Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions and quartic nonlinearities. Due to special resonances and discrete symmetries, the Birkhoff normal form of this Hamiltonian system is Liouville integrable. The normal form equations can easily be solved if the number of particles in the lattice is odd, but if the number of particles is even, several nontrivial phenomena occur. In the latter case we observe that the phase space of the normal form is decomposed in invariant subspaces that describe the interaction between the Fourier modes with wave number j and the Fourier modes with wave number n / 2-j . We study how the level sets of the integrals of the normal form foliate these invariant subspaces. The integrable foliations turn out to be singular and the method of singular reduction shows that the normal form has invariant pinched tori and monodromy. Monodromy is an obstruction to the existence of global action-angle variables. The pinched tori are interpreted as homoclinic and heteroclinic connections between traveling waves. Thus we discover a class of solutions of the normal form which can be described as direction-reversing traveling waves. The relation between the FPU lattice and its Birkhoff normal form can be understood from KAM theory and approximation theory. This explains why we observe the impact of the direction-reversing traveling waves numerically as a relaxation oscillation in the original FPU system.     

哈密顿系统的低维环面的保存性

本文给出了关于哈密顿系统低维环面的一个推广的KAM定理,它适用于同时存在法向频率和双曲法向分量的情况.其证明基于尤建功的一个定理的光滑性表述及法向双曲不变流形理论的应用.文中还给出了另外两种情况下的推广.     

Invariant hyperbolic tori for Gevrey-smooth Hamiltonian systems under Rüssmann’s non-degeneracy condition

In this paper, we prove the persistence of hyperbolic lower dimensional invariant tori for Gevrey-smooth perturbations of partially integrable Hamiltonian systems under Riissmann＇s nondegeneracy condition by an improved KAM iteration, and the persisting invariant tori are Gevrey smooth, with the same Gevrey index as the Hamiltonian.     

Persistence of Floquet invariant tori for a class of non-conservative dynamical systems

In this paper we consider a class of non-conservative dynamical system with small perturbation. By the KAM method we prove existence of Floquet invariant tori under the weakest non-resonant conditions.

     

The hyperbolic invariant tori of symplectic mappings

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for nearly integrable twist symplectic mappings. Under a Rüssmann-type non-degenerate condition, by introducing a modified KAM iteration scheme, we proved that nearly integrable twist symplectic mappings admit a family of lower dimensional hyperbolic invariant tori as long as the symplectic perturbation is small enough.     

BOUNDEDNESS OF SOLUTIONS FOR SUPERLINEAR REVERSIBLE SYSTEMS

51. IntroductionThe boundedness problem of solutions of the following nonlinear scalar differelltial equa-nons of second order..x f(x, t)ic g(x, t) = 0 (1'1)has been widely investigated by many authors since the 1940's, where f(x, t 1) = j(x, t),g(x, t 1) = g(x, t).When j(x, t) = 0, Equation (1.1) is a conservative system and takes the following form..x g(x, t) ~ 0. (1'2)The first contribution of the boundedness of all the solutions of Equation (1.2) is due toMorrislll, whO proved …     

Multiscale KAM theorem for Hamiltonian systems

In this paper, we study the persistence of invariant tori in nearly integrable multiscale Hamiltonian systems with highorder degeneracy in the integrable part. Such Hamiltonian systems arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem connects closely to the stability of the systems. We introduce multiscale nondegenerate condition and multiscale Diophantine condition, comparable to the usual Diophantine condition. Using quasilinear KAM method, we prove a multiscale KAM theorem.     

KAM theorem of symplectic algorithms for Hamiltonian systems

Summary. In this paper we prove that an analog of the celebrated KAM theorem holds for symplectic algorithms, which Channel and Scovel (1990), Feng Kang (1991) and Sanz-Serna and Calvo (1994) suggested a few years ago. The main results consist of the existence of invariant tori, with a smooth foliation structure, of a symplectic numerical algorithm when it applies to a generic integrable Hamiltonian system if the system is analytic and the time-step size of the algorithm is s ufficiently small. This existence result also implies that the algorithm, when it is applied to a generic integrable system, possesses n independent smooth invariant functions which are in involution and well-defined on the set filled by the invariant tori in the sense of Whitney. The invariant tori are just the level sets of these functions. Some quantitative results about the numerical invariant tori of the algorithm approximating the exact ones of the system are also given. Received December 27, 1997 / Revised version received July 15, 1998 / Published online: July 7, 1999     

Families of invariant tori in KAM theory: Interplay of integer characteristics

The purpose of this brief note is twofold. First, we summarize in a very concise form the principal information on Whitney smooth families of quasi-periodic invariant tori in various contexts of KAM theory. Our second goal is to attract (via an informal discussion and a simple example) the experts’ attention to the peculiarities of the so-called excitation of elliptic normal modes in the reversible context 2.     

Gevrey-smoothness of elliptic lower-dimensional invariant tori in Hamiltonian systems under Rüssmann's non-degeneracy condition

In this paper we prove Gevrey-smoothness of elliptic lower-dimensional invariant tori for nearly integrable analytic Hamiltonian systems under Rüssmann's non-degeneracy condition by an improved KAM iteration.     

Whitney smooth families of invariant tori within the reversible context 2 of KAM theory

We prove a general theorem on the persistence of Whitney C ^∞-smooth families of invariant tori in the reversible context 2 of KAM theory. This context refers to the situation where dim FixG < (codim T)/2, where FixG is the fixed point manifold of the reversing involution G and T is the invariant torus in question. Our result is obtained as a corollary of the theorem by H. W.Broer, M.-C.Ciocci, H.Hanßmann, and A.Vanderbauwhede (2009) concerning quasi-periodic stability of invariant tori with singular “normal” matrices in reversible systems.