Proof of the strengthened version of simultaneous attainability of central exponents in the class of Hamiltonian systems

The central exponents of a linear Hamiltonian system are moved apart through uniformly small Hamiltonian perturbations of its coefficients, and then they are simultaneously attained by the Lyapunov exponents through infinitesimally small perturbations of the obtained Hamiltonian system.     

Proof of Simultaneous Conditional Exponential Stabilization and Destabilization of Linear Hamiltonian Systems

It is proved that any linear Hamiltonian system is simultaneously conditionally (with respect to a subspace of half dimension) exponentially stabilizable and destabilizable by uniformly small Hamiltonian perturbations.     

Simultaneous attainability of the central exponents of small-dimensional linear Hamiltonian systems

We claim that the upper and lower central exponents of linear Hamiltonian systems of second and fourth orders are simultaneously attainable under uniformly small and infinitesimal Hamiltonian perturbations.     

On random perturbations of Hamiltonian systems with many degrees of freedom

We consider a class of random perturbations of Hamiltonian systems with many degrees of freedom. We assume that the perturbations consist of two components: a larger one which preserves the energy and destroys all other first integrals, and a smaller one which is a non-degenerate white noise type process. Under these assumptions, we show that the long time behavior of such a perturbed system is described by a diffusion process on a graph corresponding to the Hamiltonian of the system. The graph is homeomorphic to the set of all connected components of the level sets of the Hamiltonian. We calculate the differential operators which govern the process inside the edges of the graph and the gluing conditions at the vertices.     

Sharp bounds on mobility of the Lyapunov exponents of linear Hamiltonian systems under small average perturbations

Sharp bounds on mobility of the Lyapunov exponents of linear Hamiltonian systems are found for arbitrarily small average linear perturbations of the coefficients, with the help of the Millionshchikov turning method. The stabilizability and destabilizability of any solution by the indicated perturbations in the class of linear Hamiltonian systems are proved.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 125–139, 1989.     

Bifurcation of a Whitney-Smooth Family of Coisotropic Invariant Tori of a Hamiltonian System under Small Deformations of a Symplectic Structure

We investigate the influence of small deformations of a symplectic structure and perturbations of the Hamiltonian on the behavior of a completely integrable Hamiltonian system. We show that a Whitney-smooth family of coisotropic invariant tori of the perturbed system emerges in the neighborhood of a certain submanifold of the phase space.     

Evolution of perturbations on a weakly inhomogeneous background

The evolution of growing and decaying one-dimensional linear perturbations on a stationary, weakly inhomogeneous background is investigated studied. Attention is focused on the amplification of waves that arise from initial perturbations, localized in regions whose width is small compared with the inhomogeneity scale. A relation between the Hamiltonian formalism (with a complex dispersion equation) and the saddle-point method is established for an asymptotic representation of the integral that expresses perturbations in terms of the initial data. Model examples of the evolution of perturbations are examined.     

Perturbations of degenerate coisotropic invariant tori of Hamiltonian systems

We consider a Hamiltonian system with a one-parameter family of degenerate coisotropic invariant tori. We prove a theorem on the preservation of the majority of tori under small perturbations of the Hamiltonian. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 1, pp. 72–86, January, 1998.     

On the stability of the spectral Galerkin approximation

We study stability properties of the spectral Galerkin approximation of the solutions of semilinear problems. Assuming that the data of the problem are known within a certain error, we investigate when the solution of the Galerkin approximate equation provides a desired accuracy uniformly with respect to small perturbations of the data. We show that for certain classes of semilinear problems an additional compactness assumption is sufficient to assure that the spectral Galerkin method provides an accurate approximation to the exact solution uniformly with respect to small perturbations of the data. This revised version was published online in June 2006 with corrections to the Cover Date.     

Small-divisor equation of higher order with large variable coefficient and application to the coupled KdV equation

In this paper, we establish an estimate for the solutions of small-divisor equation of higher order with large variable coefficient. Then by formulating an infinite-dimensional KAM theorem which allows for multiple normal frequencies and unbounded perturbations, we prove that there are many periodic solutions for the coupled KdV equation subject to small Hamiltonian perturbations.     

Composition methods in the presence of small parameters

We derive numerical methods for arbitrary small perturbations of exactly solvable differential equations. The methods, based in one instance on Gaussian quadrature, are symplectic if the system is Hamiltonian and are asymptotically more accurate than previously known methods.     

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.     

Lipschitz properties of structure preserving matrix perturbations

It is proved that, under small perturbations that preserve the Jordan structure, canonical transformation matrices have the Lipschitz property. Analogous result is proved for matrices under additional structure, such as Hamiltonian, skew-Hamiltonian, and symplectic. Both real and complex cases are considered.     

Slow Evolution in Perturbed Hamiltonian Systems

For parametrized Hamiltonian systems with an arbitrary, finite number of degrees of freedom, it is shown that secularly stable families of equilibrium solutions represent approximate trajectories for small (not necessarily Hamiltonian) perturbations of the original system. This basic result is further generalized to certain conservative, but not necessarily Hamiltonian, systems of differential equations. It generalizes to the conservative case a theorem due, in the dissipative case, to Tikhonov, to Gradshtein, and to Levin and Levinson. It justifies the use of physically motivated approximation procedures without invoking the method of averaging and without requiring nonresonance conditions or the integrability of the unperturbed Hamiltonian.     

Relative equilibria of Hamiltonian systems with symmetry: Linearization,smoothness, and drift

Summary A relative equilibrium of a Hamiltonian system with symmetry is a point of phase space giving an evolution which is a one-parameter orbit of the action of the symmetry group of the system. The evolutions of sufficiently small perturbations of a formally stable relative equilibrium are arbitrarily confined to that relative equilibrium's orbit under the isotropy subgroup of its momentum. However, interesting evolution along that orbit, here called drift, does occur. In this article, linearizations of relative equilibria are used to construct a first order perturbation theory explaining drift, and also to determine when the set of relative equilibria near a given relative equilibrium is a smooth symplectic submanifold of phase space.     

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.     

Regularity theory for Hamilton-Jacobi equations

The objective of this paper is to discuss the regularity of viscosity solutions of time independent Hamilton-Jacobi Equations. We prove analogs of the KAM theorem, show stability of the viscosity solutions and Mather sets under small perturbations of the Hamiltonian.     

Invariant tori of locally Hamiltonian systems close to conditionally integrable systems

We study the problem of perturbations of quasiperiodic motions in the class of locally Hamiltonian systems. By using methods of the KAM-theory, we prove a theorem on the existence of invariant tori of locally Hamiltonian systems close to conditionally integrable systems. On the basis of this theorem, we investigate the bifurcation of a Cantor set of invariant tori in the case where a Liouville-integrable system is perturbed by a locally Hamiltonian vector field and, simultaneously, the symplectic structure of the phase space is deformed. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 71–98, January, 2007.     

Partial regularity for harmonic maps of evolution into spheres

We consider the global Cauchy problem for generalized Kirchhoff equations with small non-linear terms or small data. We solve this problem in the space of functions which are twice differentiable with respect to time coordinate and uniformly analytic with respect to other coordinates. We determine, in two different situations, estimates of lifespan of solutions for some problems with perturbations and we give stability result of the solution for small perturbations.     

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.