Dmitry Treschev

Evolution of slow variables in a priori unstable Hamiltonian systems

We study diffusion phenomena in a priori unstable (initially hyperbolic) Hamiltonian systems. These systems are perturbations of integrable ones, which have a family of hyperbolic tori. We prove that in the case of two and a half degrees of freedom the action variable generically drifts (i.e. changes on a trajectory by a quantity of order one). Moreover, there exists a trajectory such that the velocity of this drift is e/loge, where e is the parameter of the perturbation.

Unbounded growth of energy in nonautonomous Hamiltonian systems

The result of Mather on the existence of trajectories with unbounded energy for time periodic Hamiltonian systems on a torus is generalized to a class of multi-dimensional Hamiltonian systems with Hamiltonian polynomial in momenta. It is assumed that the leading homogeneous term of the Hamiltonian is autonomous and the corresponding Hamiltonian system has a hyperbolic invariant torus possessing a transversal homoclinic trajectory. Under certain Melnikov-type condition, the existence of trajectories with unbounded energy is proved. Instead of the variational methods of Mather, a geometrical approach based on KAM theory and the Poincare-Melnikov method is used. This makes it possible to study a more general class of Hamiltonian systems, but requires additional smoothness assumptions on the Hamiltonian. AMS classification scheme number: 58F05

Stable periodic motions in the problem on passage through a separatrix.

A Hamiltonian system with one degree of freedom depending on a slowly periodically varying in time parameter is considered. For every fixed value of the parameter there are separatrices on the phase portrait of the system. When parameter is changing in time, these separatrices are pulsing slowly periodically, and phase points of the system cross them repeatedly. In numeric experiments region swept by pulsing separatrices looks like a region of chaotic motion. However, it is shown in the present paper that if the system possesses some additional symmetry (like a pendulum in a slowly varying gravitational field), then typically in the region in question there are many periodic solutions surrounded by stability islands; total measure of these islands does not vanish and does not tend to 0 as rate of changing of the parameter tends to 0.(c) 1997 American Institute of Physics.

Multidimensional Symplectic Separatrix Maps

Summary. We consider an a priori unstable (initially hyperbolic) near-integrable Hamiltonian system in a neighborhood of stable and unstable asymptotic manifolds of a family of hyperbolic tori. Such a neighborhood contains the most chaotic part of the dynamics. The main result of the paper is the construction of the separatrix map as a convenient tool for the studying of such dynamics. We present evidence that the separatrix map combined with the method of anti-integrable limit can give a large class of chaotic trajectories as well as diffusion trajectories.

Width of stochastic layers in near-integrable two-dimensional symplectic maps

Abstract Width of stochastic layers around separatrix “eight” is studied for two-dimensional symplectic maps close to integrable. We prove that under rather general assumptions the relation w d ∼ 1 log μ holds, where w is the width of the stochastic layer, d is the width of a lobe domain and μ is a multiplier at the corresponding hyperbolic point.

An averaging method for Hamiltonian systems, exponentially close to integrable ones

Exponentially small separatrix splitting for a pendulum with rapidly oscillating suspension point and for the standard Chirikov map is studied by means of a new averaging method, which is a continuous version of the Neishtadt averaging procedure. An asymptotic formula for the rate of the separatrix splitting is obtained. (c) 1996 American Institute of Physics.

On stability loss delay for a periodic trajectory

Stability loss delay is an interesting, important and so far not completely clear phenomenon. Its essence is as follows. Consider a system of differential equations depending on a slowly varying parameter. Suppose that the system has an equilibrium position or a periodic trajectory for any fixed value of the parameter. Suppose also that the parameter passes through a bifurcational value: the equilibrium (periodic trajectory) loses stability but remains nondegenerate. In the case of an equilibrium a pair of conjugate eigenvalues leaves the left half-plane not passing through zero. For a periodic trajectory either a pair of conjugate multipliers leaves the unit circle not passing through the point 1, or one real multiplier goes away from the unit circle through the point —1. If the system is analytic, a delay of stability loss takes place: phase points attracted to the equilibrium (periodic trajectory) long before the moment of the bifurcation remain close to the unstable equilibrium (periodic trajectory) until the change of the parameter is of order one. The velocity of the parameter changing can be arbitrary small. In non-analytic systems (even in the C ∞case) in general there is no such a delay of stability loss.

Oscillator and thermostat

We study the problem of a potential interaction of a finite-dimensional Lagrangian system (an oscillator) with a linear infinite-dimensional one (a thermostat). In spite of the energy preservation and the Lagrangian (Hamiltonian) nature of the total system, under some natural assumptions the final dynamics of the finite-dimensional component turns out to be simple while the thermostat produces an effective dissipation.

On a conjugacy problem in billiard dynamics

We study symmetric billiard tables for which the billiard map is locally (near an elliptic periodic orbit of period 2) conjugate to a rigid rotation. In the previous paper (Physica D 255, 31–34 (2013)), we obtained an equation (called below the conjugacy equation) for such tables and proved that if α, the rotation angle, is rationally incommensurable with π, then the conjugacy equation has a solution in the category of formal series. In the same paper there is also numerical evidence that for “good” rotation angles the series have positive radii of convergence. In the present paper we carry out a further study (both analytic and numerical) of the conjugacy equation. We discuss its symmetries, dependence of the convergence radius on α, and other aspects.

Self-propulsion of a Body with Rigid Surface and Variable Coefficient of Lift in a Perfect Fluid

We study the system of a 2D rigid body moving in an unbounded volume of incompressible, vortex-free perfect fluid which is at rest at infinity. The body is equipped with a gyrostat and a so-called Flettner rotor. Due to the latter the body is subject to a lifting force (Magnus effect). The rotational velocities of the gyrostat and the rotor are assumed to be known functions of time (control inputs). The equations of motion are presented in the form of the Kirchhoff equations. The integrals of motion are given in the case of piecewise continuous control. Using these integrals we obtain a (reduced) system of first-order differential equations on the configuration space. Then an optimal control problem for several types of the inputs is solved using genetic algorithms.