Persistence Of Lower Dimensional Tori Research Articles

An infinite dimensional KAM theorem with normal degeneracy

In this paper, we consider a classical Hamiltonian normal form with degeneracy in the normal direction. In previous results, one needs to assume that the perturbation satisfies certain non-degenerate conditions in order to remove the degeneracy in the normal form. Instead of that, we introduce a topological degree condition and a weak convexity condition, which are easy to verify, and we prove the persistence of lower dimensional tori without any restriction on perturbation but only smallness and analyticity.

On the persistence of lower-dimensional tori in reversible systems with high dimensional degenerate equilibrium under small perturbations

<p style='text-indent:20px;'>This paper focuses on the persistence of lower-dimensional tori in reversible systems with high dimensional degenerate equilibrium under small perturbations. By an improved KAM iteration and Topological degree theory, we prove that the invariant torus with given frequency persists under small perturbations. Our result is a generalization of X. Wang et al [On the persistence of degenerate lower-dimensional tori in reversible systems, Ergodic Theory Dynam. Systems, 35(2015), 2311-2333].</p>

ARNOLD-TYPE THEOREM ABOUT LOWER-DIMENSIONAL INVARIANT TORI IN GENERALIZED HAMILTONIAN SYSTEMS

In this paper, we consider the lower dimensional invariant tori for generalized Hamiltonian system. A so-called generalized Hamiltonian system is defined on a Poisson manifold which can be odd dimensional and structurally degenerate. The existence of quasi-periodic invariant tori for Hamiltonian with standard symplectic structure was first shown by Arnold [?] under a degeneracy-removing condition. We prove the persistence of lower dimensional tori for Hamiltonian with Poisson structure instead of standard symplectic structure, when the tangential and normal frequencies satisfy some certain conditions.

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

This paper is a continuation to our work (Xu et al. in Ann Henri Poincare 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 Russmann type, the majority persistence of non-resonant tori and the existence of a nearly full measure set of Poincare 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 Poincare 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 Poincare 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 Poincare–Treshchev mechanism.

The persistence of elliptic lower dimensional tori with prescribed frequency for Hamiltonian systems

In this paper we consider the persistence of lower dimensional tori of a class of analytic perturbed Hamiltonian system, H =hw(x), Ii + 1 W0 (u 2 + v 2 ) + P(q, I, z, ¯ z;x) and prove that if the frequencies (w0,W0) satisfy some non-resonance condition and the Brouwer degree of the frequency mapping w(x) at w0 is nonzero, then there exists an invariant lower dimensional invariant torus, whose frequencies are a small dilation of w0.

On the persistence of degenerate lower-dimensional tori in reversible systems

This work focuses on the persistence of lower-dimensional tori with prescribed frequencies and singular normal matrices in reversible systems. By the Kolmogorov–Arnold–Moser theory and the special structure of unperturbed nonlinear terms in the differential equation, we prove that the invariant torus with given frequency persists under small perturbations. Our result is a generalization of X. Wang et al [Degenerate lower dimensional tori in reversible systems. J. Math. Anal. Appl.387 (2012), 776–790].

Persistence of Lower Dimensional Tori for a Class of Nearly Integrable Reversible Systems

In this paper we consider the persistence of lower dimensional invariant tori for a class of reversible systems, and prove that if the average of the linear terms has full rank, then the invariant torus with Diophantine frequencies persists under small perturbations.

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.

Normal Form of Reversible Systems and Persistence of Lower Dimensional Tori under Weaker Nonresonance Conditions

In this paper we give a normal form for reversible systems and then prove the persistence of lower dimensional invariant tori for integrable reversible systems under small perturbations with weaker nonresonance conditions than have previously been imposed. Our nonresonance conditions correspond to the first Melnikov's conditions in the case of Hamiltonian systems.

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.