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.     

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.     

Poincaré–Treshchev Mechanism in Multi-scale,Nearly Integrable Hamiltonian Systems

This paper is a continuation to our work (Xu et al. in Ann Henri Poincaré 18(1):53–83, 2017) concerning the persistence of lower-dimensional tori on resonant surfaces of a multi-scale, nearly integrable Hamiltonian system. This type of systems, being properly degenerate, arise naturally in planar and spatial lunar problems of celestial mechanics for which the persistence problem ties closely to the stability of the systems. For such a system, under certain non-degenerate conditions of Rüssmann type, the majority persistence of non-resonant tori and the existence of a nearly full measure set of Poincaré non-degenerate, lower-dimensional, quasi-periodic invariant tori on a resonant surface corresponding to the highest order of scale is proved in Han et al. (Ann Henri Poincaré 10(8):1419–1436, 2010) and Xu et al. (2017), respectively. In this work, we consider a resonant surface corresponding to any intermediate order of scale and show the existence of a nearly full measure set of Poincaré non-degenerate, lower-dimensional, quasi-periodic invariant tori on the resonant surface. The proof is based on a normal form reduction which consists of a finite step of KAM iterations in pushing the non-integrable perturbation to a sufficiently high order and the splitting of resonant tori on the resonant surface according to the Poincaré–Treshchev mechanism.     

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.     

A point-wise criterion for quasi-periodic motions in the KAM theory

We consider initial value problems for nearly integrable Hamiltonian systems. We formulate a sufficient condition for each initial value to admit the quasi-periodic solution with a Diophantine frequency vector, without any nondegeneracy of the integrable part. We reconstruct the KAM theorem under Rüssmann’s nondegeneracy by the measure estimate for the set of initial values satisfying this sufficient condition. Our point-wise version is of the form analogous to the corresponding problems for the integrable case. We compare our framework with the standard KAM theorem through a brief review of the KAM theory.     

A New Approach to the Parameterization Method for Lagrangian Tori of Hamiltonian Systems

We compute invariant Lagrangian tori of analytic Hamiltonian systems by the parameterization method. Under Kolmogorov’s non-degeneracy condition, we look for an invariant torus of the system carrying quasi-periodic motion with fixed frequencies. Our approach consists in replacing the invariance equation of the parameterization of the torus by three conditions which are altogether equivalent to invariance. We construct a quasi-Newton method by solving, approximately, the linearization of the functional equations defined by these three conditions around an approximate solution. Instead of dealing with the invariance error as a single source of error, we consider three different errors that take account of the Lagrangian character of the torus and the preservation of both energy and frequency. The condition of convergence reflects at which level contributes each of these errors to the total error of the parameterization. We do not require the system to be nearly integrable or to be written in action-angle variables. For nearly integrable Hamiltonians, the Lebesgue measure of the holes between invariant tori predicted by this parameterization result is of \({\mathcal {O}}(\varepsilon ^{1/2})\), where \(\varepsilon \) is the size of the perturbation. This estimate coincides with the one provided by the KAM theorem.     

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.     

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.     

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.     

Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann's non-degeneracy condition

In this paper we prove Gevrey smoothness of the persisting invariant tori for small perturbations of an analytic integrable Hamiltonian system with Rüssmann's non-degeneracy condition by an improved KAM iteration method with parameters.     

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.     

Persistence of hyperbolic tori in Hamiltonian systems

We generalize the well-known result of Graff and Zehnder on the persistence of hyperbolic invariant tori in Hamiltonian systems by considering non-Floquet, frequency varying normal forms and allowing the degeneracy of the unperturbed frequencies. The preservation of part or full frequency components associated to the degree of non-degeneracy is considered. As applications, we consider the persistence problem of hyperbolic tori on a submanifold of a nearly integrable Hamiltonian system and the persistence problem of a fixed invariant hyperbolic torus in a non-integrable Hamiltonian system.     

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     

Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems

In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Rssmann's non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus,numerical solutions are quasi-periodic with a diophantine frequency vector of time step size dependence. These results generalize Shang's previous ones(1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov.     

General KAM theorems and their applications to invariant tori with prescribed frequencies

In this paper we develop a new KAM technique to prove two general KAM theorems for nearly integrable Hamiltonian systems without assuming any nondegeneracy condition. Many of KAM-type results (including the classical KAM theorem) are special cases of our theorems under some nondegeneracy condition and some smoothness condition. Moreover, we can obtain some interesting results about KAM tori with prescribed frequencies.     

Completely degenerate lower-dimensional invariant tori for Hamiltonian system

We study the persistence of lower-dimensional invariant tori for a nearly integrable completely degenerate Hamiltonian system. It is shown that the majority of unperturbed invariant tori can survive from the perturbations which are only assumed the smallness and smoothness.     

A quasi-periodic Poincaré's theorem

We study the persistence of invariant tori on resonant surfaces of a nearly integrable Hamiltonian system under the usual Kolmogorov non-degenerate condition. By introducing a quasi-linear iterative scheme to deal with small divisors, we generalize the Poincaré theorem on the maximal resonance case (i.e., the periodic case) to the general resonance case (i.e., the quasi-periodic case) by showing the persistence of majority of invariant tori associated to non-degenerate relative equilibria on any resonant surface.The first author was partially supported by NSFC grant 19971042, the National 973 Project of China: Nonlinearity, and the outstanding young's project of the Ministry of Education of China.The second author was partially supported by NSF grant DMS9803581.Mathematics Subject Classification (2000): Primary 58F05, 58F27, 58F30     

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.     

Persistence of lower dimensional invariant tori on sub-manifolds in Hamiltonian systems

Chow et al. (J. Non. Sci. 12 (2002) 585) proved that the majority of the unperturbed tori on sub-manifolds will persist for standard Hamiltonian systems. Motivated by their work, in this paper, we study the persistence and tangent frequencies preservation of lower dimensional invariant tori on smooth sub-manifolds for real analytic, nearly integrable Hamiltonian systems. The surviving tori might be elliptic, hyperbolic, or of mixed type.     

The classical KAM theorem for Hamiltonian systems via rational approximations

In this paper, we give a new proof of the classical KAM theorem on the persistence of an invariant quasi-periodic torus, whose frequency vector satisfies the Bruno-Rüssmann condition, in real-analytic non-degenerate Hamiltonian systems close to integrable. The proof, which uses rational approximations instead of small divisors estimates, is an adaptation to the Hamiltonian setting of the method we introduced in [4] for perturbations of constant vector fields on the torus.