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2013
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# Dmitrii Treschev

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

## Publications:

 Saulin S. M., Treschev D. V. On the Inclusion of a Map Into a Flow 2016, vol. 21, no. 5, pp.  538-547 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. 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 DOI:10.1134/S1560354716050051
 Burlakov D., Treschev D. V. A Rigid Body on a Surface with Random Roughness 2014, vol. 19, no. 3, pp.  296-309 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. 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 DOI:10.1134/S1560354714030034
 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 2012, vol. 17, no. 6, pp.  547-558 Abstract 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 DOI:10.1134/S1560354712060068
 Piftankin G. N., Treschev D. V. Coarse-grained entropy in dynamical systems 2010, vol. 15, no. 4-5, pp.  575-597 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 (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, no. 4-5, pp. 575-597 DOI:10.1134/S156035471004012X
 Bolotin S. V., Treschev D. V. Remarks on the Definition of Hyperbolic Tori of Hamiltonian Systems 2000, vol. 5, no. 4, pp.  401-412 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 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 DOI:10.1070/RD2000v005n04ABEH000156
 Pronin A. V., Treschev D. V. Continuous Averaging in Multi-frequency Slow-fast Systems 2000, vol. 5, no. 2, pp.  157-170 Abstract 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 DOI:10.1070/RD2000v005n02ABEH000138
 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.  73-81 Abstract 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 DOI:10.1070/RD1998v003n03ABEH000081
 Treschev D. V. The method of continuous averaging in the problem of separation of fast and slow motions 1997, vol. 2, nos. 3-4, pp.  9-20 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. 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 DOI:10.1070/RD1997v002n03ABEH000043
 Pronin A. V., Treschev D. V. On the Inclusion of Analytic Maps into Analytic Flows 1997, vol. 2, no. 2, pp.  14-24 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. 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 DOI:10.1070/RD1997v002n02ABEH000033