Jiangong You

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 Melnikovs 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 Melnikovs first non-resonance condition for the persistence of lower dimensional tori.

Existence of quasiperiodic solutions and Littlewood's boundedness problem of Duffing equations with subquadratic potentials

where p(t) is periodic has been extensively investigated due to its relevance in applications. Much work has been carried out concerning the existence of periodic solutions (see [3, 8] also for further references). In this paper we pay attention to more complicated solutions and the Lagrangian stability problem proposed by Littlewood [5]. As remarked in [5], it has been conjectured that all solutions of Eq. (1.1) are bounded if either

Invariant tori and Lagrange stability of pendulum-type equations

Abstract In this paper we prove that the pendulum-type equation x″ + g(t, x) = 0 possesses infinitely many invariant tori whenever g(t, x) has zero mean value on the torus T2, where g(t, x) belongs to C∞(T2). This yields the boundedness for solutions of the considered pendulum-type equation and thus leads to an answer to J. Mosers boundedness problem (1973 , Ann. of Math. Stud. 77).

Quasi-Periodic Solutions for 1D Schrödinger Equations with Higher Order Nonlinearity

In this paper, one-dimensional (1D) nonlinear Schrodinger equations

KAM-Type Theorem on Resonant Surfaces for Nearly Integrable Hamiltonian Systems

iu_{t}-u_{xx}+mu+\nu|u|^{4}u=0,

Persistence of the non-twist torus in nearly integrable hamiltonian systems

with Dirichlet boundary conditions are considered. It is proved that for all real parameters m, the above equation admits small-amplitude quasi-periodic solutions corresponding to b-dimensional invariant tori of an associated infinite-dimensional dynamical system. The proof is based on infinite-dimensional KAM theory, partial normal form, and scaling skills.

The rigidity of reducibility of cocycles on SO(N,\mathbb{R})

We consider perturbations of integrable Hamiltonian systems in the neighbourhood of normally parabolic invariant tori. Under appropriate transversality conditions the tori in the unperturbed system bifurcate according to a (generalized) cuspoid catastrophe. Combining techniques of KAM theory and singularity theory, we show that such bifurcation scenarios survive the perturbation on large Cantor sets. Applications to rigid body dynamics and forced oscillators are pointed out.

Point spectrum for quasi-periodic long range operators

Summary. In this paper, we consider analytic perturbations of an integrable Hamiltonian system in a given resonant surface. It is proved that, for most frequencies on the resonant surface, the resonant torus foliated by nonresonant lower dimensional tori is not destroyed completely and that there are some lower dimensional tori which survive the perturbation if the Hamiltonian satisfies a certain nondegenerate condition. The surviving tori might be elliptic, hyperbolic, or of mixed type. This shows that there are many orbits in the resonant zone which are regular as in the case of integrable systems. This behavior might serve as an obstacle to Arnold diffusion. The persistence of hyperbolic lower dimensional tori has been considered by many authors [5], [6], [15], [16], mainly for multiplicity one resonant case. To deal with the mechanisms of the destruction of the resonant tori of higher multiplicity into nonhyperbolic lower dimensional tori, we have to deal with some small coefficient matrices that are the generalization of small divisors.