Dmitry Treschev

Dmitry Treschev
Gubkina 8, Moscow, 119991 Russia
Steklov Mathematical Institute, Russian Academy of Sciences


Treschev D. V.
Normalization Flow
2023, vol. 28, nos. 4-5, pp.  781-804
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.
Keywords: normal forms, Hamiltonian systems, small divisors
Citation: Treschev D. V.,  Normalization Flow, Regular and Chaotic Dynamics, 2023, vol. 28, nos. 4-5, pp. 781-804
Bolotin S. V., Treschev D. V.
Quasiperiodic Version of Gordon’s Theorem
2023, vol. 28, no. 1, pp.  5-13
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 well-known 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.
Keywords: isochronicity, superintegrability, Hamiltonian systems, variational pronciples
Citation: Bolotin S. V., Treschev D. V.,  Quasiperiodic Version of Gordon’s Theorem, Regular and Chaotic Dynamics, 2023, vol. 28, no. 1, pp. 5-13
Treschev D. V.
Isochronicity in 1 DOF
2022, vol. 27, no. 2, pp.  123-131
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.
Keywords: isochronicity, superintegrability, normal forms, Hamiltonian dynamics
Citation: Treschev D. V.,  Isochronicity in 1 DOF, Regular and Chaotic Dynamics, 2022, vol. 27, no. 2, pp. 123-131
Saulin S. M., Treschev D. V.
On the Inclusion of a Map Into a Flow
2016, vol. 21, no. 5, pp.  538-547
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.
Keywords: Poincaré map, averaging, time periodic vector field
Citation: Saulin S. M., Treschev D. V.,  On the Inclusion of a Map Into a Flow, Regular and Chaotic Dynamics, 2016, vol. 21, no. 5, pp. 538-547
Burlakov D., Treschev D. V.
A Rigid Body on a Surface with Random Roughness
2014, vol. 19, no. 3, pp.  296-309
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.
Keywords: rigid body, support with random roughness
Citation: Burlakov D., Treschev D. V.,  A Rigid Body on a Surface with Random Roughness, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 296-309
Ramodanov S. M., Tenenev V. A., Treschev D. V.
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.
Keywords: perfect fluid, self-propulsion, Flettner rotor
Citation: Ramodanov S. M., Tenenev V. A., Treschev D. V.,  Self-propulsion of a Body with Rigid Surface and Variable Coefficient of Lift in a Perfect Fluid, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 547-558
Piftankin G. N., Treschev D. V.
Coarse-grained entropy in dynamical systems
2010, vol. 15, nos. 4-5, pp.  575-597
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 (fine-grained) entropy the so called coarse-grained entropy.
First we obtain a formula for the difference between the coarse-grained 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 coarse-grained 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 coarse-grained entropy. We show that while the Gibbs entropy remains constant, the coarse-grained 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 coarse-grained 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.
Keywords: Gibbs entropy, nonequilibrium thermodynamics, Lyapunov exponents, Gibbs ensemble
Citation: Piftankin G. N., Treschev D. V.,  Coarse-grained entropy in dynamical systems, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 4-5, pp. 575-597
Bolotin S. V., Treschev D. V.
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 KAM-theory.
Citation: Bolotin S. V., Treschev D. V.,  Remarks on the Definition of Hyperbolic Tori of Hamiltonian Systems, Regular and Chaotic Dynamics, 2000, vol. 5, no. 4, pp. 401-412
Pronin A. V., Treschev D. V.
Continuous Averaging in Multi-frequency Slow-fast Systems
2000, vol. 5, no. 2, pp.  157-170
It is well-known that in real-analytic multi-frequency slow-fast ODE systems the dependence of the right-hand sides on fast angular variables can be reduced to an exponentially small order by a near-identical change of the variables. Realistic constructive estimates for the corresponding exponentially small terms are obtained.
Citation: Pronin A. V., Treschev D. V.,  Continuous Averaging in Multi-frequency Slow-fast Systems, Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, pp. 157-170
Treschev D. V., Zubelevich O. E.
An estimate for the difference of the frequencies on two invariant curves, bounding a resonance zone of an area-preserving close to integrable map, is obtained. Analogous results for Hamiltonian systems are presented.
Citation: Treschev D. V., Zubelevich O. E.,  Invariant tori in Hamiltonian systems with two degrees of freedom in a neighborhood of a resonance, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 73-81
Treschev D. V.
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.
Citation: Treschev D. V.,  The method of continuous averaging in the problem of separation of fast and slow motions, Regular and Chaotic Dynamics, 1997, vol. 2, nos. 3-4, pp. 9-20
Pronin A. V., Treschev D. V.
On the Inclusion of Analytic Maps into Analytic Flows
1997, vol. 2, no. 2, pp.  14-24
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.
Citation: Pronin A. V., Treschev D. V.,  On the Inclusion of Analytic Maps into Analytic Flows, Regular and Chaotic Dynamics, 1997, vol. 2, no. 2, pp. 14-24

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