Dmitry Treschev
Gubkina 8, Moscow, 119991 Russia
Steklov Mathematical Institute, Russian Academy of Sciences
Publications:
Treschev D. V.
Normalization Flow
2023, vol. 28, nos. 45, pp. 781804
Abstract
We propose a new approach to the theory of normal forms for Hamiltonian systems
near a nonresonant elliptic singular point. We consider the space of all Hamiltonian functions
with such an equilibrium position at the origin and construct a differential equation in this space.
Solutions of this equation move Hamiltonian functions towards their normal forms. Shifts along
the flow of this equation correspond to canonical coordinate changes. So, we have a continuous
normalization procedure. The formal aspect of the theory presents no difficulties. As usual, the
analytic aspect and the problems of convergence of series are nontrivial.

Bolotin S. V., Treschev D. V.
Quasiperiodic Version of Gordon’s Theorem
2023, vol. 28, no. 1, pp. 513
Abstract
We consider Hamiltonian systems possessing families of nonresonant invariant tori
whose frequencies are all collinear. Then under certain conditions the frequencies depend
on energy only. This is a generalization of the wellknown Gordon’s theorem about periodic
solutions of Hamiltonian systems. While the proof of Gordon’s theorem uses Hamilton’s
principle, our result is based on Percival’s variational principle. This work was motivated by
the problem of isochronicity in Hamiltonian systems.

Treschev D. V.
Isochronicity in 1 DOF
2022, vol. 27, no. 2, pp. 123131
Abstract
Our main result is the complete set of explicit conditions necessary and sufficient for
isochronicity of a Hamiltonian system with one degree of freedom. The conditions are presented
in terms of Taylor coefficients of the Hamiltonian function.

Saulin S. M., Treschev D. V.
On the Inclusion of a Map Into a Flow
2016, vol. 21, no. 5, pp. 538547
Abstract
We consider the problem of the inclusion of a diffeomorphism into a flow generated by an autonomous or time periodic vector field. We discuss various aspects of the problem, present a series of results (both known and new ones) and point out some unsolved problems.

Burlakov D., Treschev D. V.
A Rigid Body on a Surface with Random Roughness
2014, vol. 19, no. 3, pp. 296309
Abstract
Consider an interval on a horizontal line with random roughness. With probability one it is supported at two points: one on the left, and another on the right from its center. We compute the probability distribution of the support points provided the roughness is fine grained. We also solve an analogous problem where a circle or a disk lies on a rough plane. Some applications in static are given.

Ramodanov S. M., Tenenev V. A., Treschev D. V.
Selfpropulsion of a Body with Rigid Surface and Variable Coefficient of Lift in a Perfect Fluid
2012, vol. 17, no. 6, pp. 547558
Abstract
We study the system of a 2D rigid body moving in an unbounded volume of incompressible, vortexfree perfect fluid which is at rest at infinity. The body is equipped with a gyrostat and a socalled 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 firstorder differential equations on the configuration space. Then an optimal control problem for several types of the inputs is solved using genetic algorithms.

Piftankin G. N., Treschev D. V.
Coarsegrained entropy in dynamical systems
2010, vol. 15, nos. 45, pp. 575597
Abstract
Let $M$ be the phase space of a physical system. Consider the dynamics, determined by the invertible map $T: M \to M$, preserving the measure $\mu$ on $M$. Let $\nu$ be another measure on $M$, $d\nu = \rho d\mu$. Gibbs introduced the quantity $s(\rho) = − \int \rho \log \rho d \mu$ as an analog of the thermodynamical entropy. We consider a modification of the Gibbs (finegrained) entropy the so called coarsegrained entropy.
First we obtain a formula for the difference between the coarsegrained and Gibbs entropy. The main term of the difference is expressed by a functional usually referenced to as the Fisher information. Then we consider the behavior of the coarsegrained entropy as a function of time. The dynamics transforms $\nu$ in the following way: $\nu \mapsto \nu_n$, $d\nu_n = \rho \circ T^{n} d\mu$. Hence, we obtain the sequence of densities $\rho_n = \rho \circ T^{n}$ and the corresponding values of the Gibbs and the coarsegrained entropy. We show that while the Gibbs entropy remains constant, the coarsegrained entropy has a tendency to a growth and this growth is determined by dynamical properties of the map $T$. Finally, we give numerical calculation of the coarsegrained entropy as a function of time for systems with various dynamical properties: integrable, chaotic and with mixed dynamics and compare these calculation with theoretical statements. 
Bolotin S. V., Treschev D. V.
Remarks on the Definition of Hyperbolic Tori of Hamiltonian Systems
2000, vol. 5, no. 4, pp. 401412
Abstract
We show that under certain natural conditions the definition of a hyperbolic torus conventional for the general theory of dynamical systems is quite suitable for needs of the KAMtheory.

Pronin A. V., Treschev D. V.
Continuous Averaging in Multifrequency Slowfast Systems
2000, vol. 5, no. 2, pp. 157170
Abstract
It is wellknown that in realanalytic multifrequency slowfast ODE systems the dependence of the righthand sides on fast angular variables can be reduced to an exponentially small order by a nearidentical change of the variables. Realistic constructive estimates for the corresponding exponentially small terms are obtained.

Treschev D. V., Zubelevich O. E.
Invariant tori in Hamiltonian systems with two degrees of freedom in a neighborhood of a resonance
1998, vol. 3, no. 3, pp. 7381
Abstract
An estimate for the difference of the frequencies on two invariant curves, bounding a resonance zone of an areapreserving close to integrable map, is obtained. Analogous results for Hamiltonian systems are presented.

Treschev D. V.
The method of continuous averaging in the problem of separation of fast and slow motions
1997, vol. 2, nos. 34, pp. 920
Abstract
Neishtadt has proved that in the problem of fast phase averaging in an analytic system of ODE the dependence on the fast variable can be reduced to the terms which are exponentially small in the small parameter. The paper contains realistic estimates for these terms. These estimates essentially depend on properties of the first order averaged system.

Pronin A. V., Treschev D. V.
On the Inclusion of Analytic Maps into Analytic Flows
1997, vol. 2, no. 2, pp. 1424
Abstract
We prove a general theorem on the representation of an analytic map isotopic to the identity as the Poincare map in a nonautonomous periodic in time analytic system of ODE. If the map belongs to some Lie group of diffeomorphisms, the vector field determining the ODE can be taken from the corresponding Lie algebra of vector fields. The proof uses a specific averaging procedure.
