Reducible variational equations and the perturbation of invariant tori of Hamiltonian systems

Translated from Matematicheskie Zametki, Vol. 45, No. 5, pp. 38–49, May, 1989.     

On some singular perturbation problems in the theory of lubrication

Some singular perturbation problems occurring in the theory of hydrodynamic lubrication are studied. We examine in detail the case of the very long or very short journal bearing; the asymptotic problem of small eccentricity ratio is also considered.     

Phase-locking for dynamical systems on the torus and perturbation theory for mathieu-type problems

Summary When several oscillators are coupled together and the parameters of their coupling are varied, the oscillators pass through so-called phase-locked regimes. In physical terms this means that the oscillators tend to synchronize their motion. To describe this phenomenon, we frame the concepts ofpartial phase andphase-locking. A partial phase of a toral flow puts emphasis on how orbits of the flow drift around the torus in some fixed direction. The partial phase is locked if it grows in time along some orbit slower than any linear function. When a toral flow is given by a trigonometric polynomial, its phase-locked regions are quite narrow. With the coupling amplitude increasing, each region grows in width as some power of the amplitude. That power can be calculated in terms of both the partial phase and degree of the trigonometric polynomial.     

On some representations of solutions of dynamical problems in elasticity theory

The divergence and rotation of a translation vector field, which are solutions of wave equations, are used as auxiliary potentials for numerical solution of dynamical problems in elasticity theory. In a particular case, the problems of longitudinal translation are tested in two variants of numerical algorithms with their programming realization, some suggestions are given regarding the choice of the optimal algorithm for achieving highly precise calculations with a small number of discretization points.Translated from Dinamicheskie Sistemy, No. 9, pp. 27–33, 1990.     

On invariant tori of stochastic systems on a plane

We study invariant tori of stochastic systems of the ltd type on a plane and present conditions for stability of such sets in probability.     

Whitney smooth families of invariant tori within the reversible context 2 of KAM theory

We prove a general theorem on the persistence of Whitney C ^∞-smooth families of invariant tori in the reversible context 2 of KAM theory. This context refers to the situation where dim FixG < (codim T)/2, where FixG is the fixed point manifold of the reversing involution G and T is the invariant torus in question. Our result is obtained as a corollary of the theorem by H. W.Broer, M.-C.Ciocci, H.Hanßmann, and A.Vanderbauwhede (2009) concerning quasi-periodic stability of invariant tori with singular “normal” matrices in reversible systems.     

On compact invariant hypersurfaces of discrete dynamical systems

We obtain an analog of the Bendixson-Dulac criterion for discrete dynamical systems.     

Localizing sets for invariant compact sets of continuous dynamical systems with a perturbation

A functional method of localization of invariant compact sets, which was earlier developed for autonomous continuous and discrete systems, is generalized to continuous dynamical systems with perturbations. We describe properties of the corresponding localizing sets. By using that method, we construct localizing sets for positively invariant compact sets of the Lorenz system with a perturbation.     

On invariant tori of countable systems of differential equations with delay

We obtain conditions for the existence and smoothness of invariant tori of countable systems of differential equations with delay. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1292–1295, September, 1999.     

Localizing sets for invariant compact sets of discrete dynamical systems with perturbation and control

We consider the localization problem for the invariant compact sets of a discrete dynamical system with perturbation and control, that is, the problem of constructing domains in the system state space that contain all invariant compact sets of the system. The problem is solved on the basis of a functional method used earlier in localization problems for time-invariant continuous and discrete systems and also for control systems. The properties of the corresponding localizing sets are described.     

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.     

Classical perturbation theory and resonances in some rigid body systems

We consider the system of a rigid body in a weak gravitational field on the zero level set of the area integral and study its Poincaré sets in integrable and nonintegrable cases. For the integrable cases of Kovalevskaya and Goryachev–Chaplygin we investigate the structure of the Poincaré sets analytically and for nonintegrable cases we study these sets by means of symbolic calculations. Based on these results, we also prove the existence of periodic solutions in the perturbed nonintegrable system. The Chaplygin integrable case of Kirchhoff’s equations is also briefly considered, for which it is shown that its Poincaré sets are similar to the ones of the Kovalevskaya case.     

On the existence of invariant tori in nearly-integrable Hamiltonian systems with finitely differentiable perturbations

We prove the existence of invariant tori in Hamiltonian systems, which are analytic and integrable except a 2n-times continuously differentiable perturbation (n denotes the number of the degrees of freedom), provided that the moduli of continuity of the 2n-th partial derivatives of the perturbation satisfy a condition of finiteness (condition on an integral), which is more general than a Hölder condition. So far the existence of invariant tori could be proven only under the condition that the 2n-th partial derivatives of the perturbation are Hölder continuous.     

On some singular perturbation problems arising in stochastic control

We discuss the problem of the existence of periodic and almost periodic solutions in distribution of semilinear stochastic equations on a separable Hilbert space.

Under a dissipativity condition we prove that the translation of the mean square bounded solution is periodic or almost periodic. Similar results hold in the affine case under mean square stability of the linear part of the equation.     

Coisotropic invariant tori of hamiltonian systems of the quasiclassical theory of motion of a conduction electron

We investigate quasiclassical equations of motion of a conduction electron in an electric and a magnetic field. It is shown that for homogeneous fields the motion can proceed along four-dimensional coisotropic invariant tori in six-dimensional phase space. By KAM-theory methods, the motion in weakly nonhomogeneous fields is investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 346–351, March, 1990.