Hamiltonian Systems: Stable or Unstable?

This is a review concerning some topics in the field of Hamiltonian dynamics, with emphasis on the problem of Arnold diffusion. Lecture held in the Seminario Matematico e Fisico on January 16, 2006 Received: May 2006     

A KAM-type Theorem for Generalized Hamiltonian Systems

In this paper we mainly concern the persistence of invariant tori in 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, system under consideration can be odd dimensional. Under the Riissmann type non-degenerate condition, we proved that the majority of the lower-dimension invariant tori of the integrable systems in generalized Hamiltonian system are persistent under small perturbation. The surviving lower-dimensional tori might be elliptic, hyperbolic, or of mixed type.     

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.     

Dynamical systems — Past and present

Plenary lecture at the International Congress of Mathematicians (Berlin 1998, August 18–27). Reprinted from Doc. Math. J., DMV, Extra Volume ICM I, 1998, pp. 381–402. ©J.Moser, 1998.     

On Reducibility of Beam Equation with Quasi-p erio dic Forcing Potential

In this paper, the Dirichlet boundary value problems of the nonlinear beam equation u_(tt) + ?_u~2 + αu + ∈Φ(t)(u + u~3) = 0, α 0 in the dimension one is considered, where u(t, x) and Φ(t) are analytic quasi-periodic functions in t, and∈ is a small positive real-number parameter. It is proved that the above equation admits a small-amplitude quasi-periodic solution. The proof is based on an infinite dimensional KAM iteration procedure.     

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.     

Numerical estimates for stability domains of Lagrangian solutions to the restricted three-body problem

The geometric parameters of stability domains of Lagrangian solutions to the classical restricted three-body problem are quantitatively estimated. It is shown that these domains are ellipsoid-like plane figures stretched along the tangent to the circle that passes through the Lagrangian triangle solutions. A heuristic algorithm is proposed for determining the maximum size of these domains of attraction.     

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.     

Asymptotic Solutions of Hamilton–Jacobi Equations with Semi-Periodic Hamiltonians

We study the long time behavior of viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations in ? ⁿ . We prove that if the Hamiltonian H(x, p) is coercive and strictly convex in a mild sense in p and upper semi-periodic in x, then any solution of the Cauchy problem “converges” to an asymptotic solution for any lower semi-almost periodic initial function.     

The rate of convergence of the Lax-Oleinik semigroup-degenerate fixed point case

For the standard Lagrangian in classical mechanics, which is defined as the kinetic energy of the system minus its potential energy, we study the rate of convergence of the corresponding Lax-Oleinik semigroup. Under the assumption that the unique global minimum point of the Lagrangian is a degenerate fixed point, we provide an upper bound estimate of the rate of convergence of the semigroup.     

Persistence of Hyperbolic Tori in Generalized Hamiltonian Systems

In this paper we prove the persistence of hyperbolic invariant tori in generalized Hamiltonian systems, which may admit a distinct number of action and angle variables. The systems under consideration can be odd dimensional in tangent direction. Our results generalize the well-known results of Graff and Zehnder in standard Hamiltonians. In our case the unperturbed Hamiltonian systems may be degenerate. We also consider the persistence problem of hyperbolic tori on sub manifolds.     

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.     

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.     

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.