0
2013
Impact Factor

# Michail Sevryuk

Leninskii pr. 38, Bldg. 2, Moscow 119334, Russia
Institute of Energy Problems of Chemical Physics RAS

## Publications:

 Sevryuk M. B. Families of Invariant Tori in KAM Theory: Interplay of Integer Characteristics 2017, vol. 22, no. 6, pp.  603–615 Abstract The purpose of this brief note is twofold. First, we summarize in a very concise form the principal information on Whitney smooth families of quasi-periodic invariant tori in various contexts of KAM theory. Our second goal is to attract (via an informal discussion and a simple example) the experts’ attention to the peculiarities of the so-called excitation of elliptic normal modes in the reversible context 2. Keywords: KAM theory, quasi-periodic invariant tori, Whitney smooth families, proper destruction of resonant tori, excitation of elliptic normal modes, reversible context 2 Citation: Sevryuk M. B.,  Families of Invariant Tori in KAM Theory: Interplay of Integer Characteristics, Regular and Chaotic Dynamics, 2017, vol. 22, no. 6, pp. 603–615 DOI:10.1134/S156035471706003X
 Sevryuk M. B. Whitney Smooth Families of Invariant Tori within the Reversible Context 2 of KAM Theory 2016, vol. 21, no. 6, pp.  599-620 Abstract We prove a general theorem on the persistence of Whitney $C^\infty$-smooth families of invariant tori in the reversible context 2 of KAM theory. This context refers to the situation where $\dim \text{Fix}\,G < (\text{codim}\,\mathcal{T})/2$, where $\text{Fix}\,G$ is the fixed point manifold of the reversing involution $G$ and $\mathcal{T}$ is the invariant torus in question. Our result is obtained as a corollary of the theorem by H. W. Broer, M.-C. Ciocci, H. Hansmann, and A. Vanderbauwhede (2009) concerning quasi-periodic stability of invariant tori with singular “normal” matrices in reversible systems. Keywords: KAM theory, reversible systems, BCHV theorem, reversible context 2, invariant tori, Whitney smooth families Citation: Sevryuk M. B.,  Whitney Smooth Families of Invariant Tori within the Reversible Context 2 of KAM Theory, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 599-620 DOI:10.1134/S1560354716060022
 Sevryuk M. B. Translation of the V. I. Arnold Paper "From Superpositions to KAM Theory" (Vladimir Igorevich Arnold. Selected–60, Moscow: PHASIS, 1997, pp. 727–740) 2014, vol. 19, no. 6, pp.  734-744 Abstract V.I.Arnold (12 June 1937 – 3 June 2010) published several papers where he described, in the form of recollections, his two earliest research problems (superpositions of continuous functions and quasi-periodic motions in dynamical systems), the main results and their interrelations: [A1], then [A2] (reprinted as [A4, A6]), and [A3] (translated into English by the author as [A5]). The first exposition [A1] has never been translated into English; however, it contains many details absent in the subsequent articles. It seems therefore that publishing the English translation of the paper [A1] would not be superfluous. What follows is this translation. In many cases, the translator gives complete bibliographic descriptions of various papers mentioned briefly in the original Russian text. The English translations of papers in Russian are also pointed out where possible. A related material is contained also in Arnold’s recollections “On A.N.Kolmogorov”. Slightly different versions of these reminiscences were published several times in Russian and English [A7–A12]. The early history of KAM theory is also discussed in detail in the recent brilliant semi-popular book [A13]. Keywords: Hilbert’s 13th problem, superpositions of continuous functions, invariant tori carrying quasi-periodic motions, KAM theory and its applications, Kolmogorov as a supervisor Citation: Sevryuk M. B.,  Translation of the V. I. Arnold Paper "From Superpositions to KAM Theory" (Vladimir Igorevich Arnold. Selected–60, Moscow: PHASIS, 1997, pp. 727–740), Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 734-744 DOI:10.1134/S1560354714060100
 Aquilanti V., Lombardi A., Sevryuk M. B. Statistics of Energy Partitions for Many-Particle Systems in Arbitrary Dimension 2014, vol. 19, no. 3, pp.  318-347 Abstract In some previous articles, we defined several partitions of the total kinetic energy $T$ of a system of $N$ classical particles in $\mathbb{R}^d$ into components corresponding to various modes of motion. In the present paper, we propose formulas for the mean values of these components in the normalization $T=1$ (for any $d$ and $N$) under the assumption that the masses of all the particles are equal. These formulas are proven at the “physical level” of rigor and numerically confirmed for planar systems $(d=2)$ at $3\leqslant N \leqslant 100$. The case where the masses of the particles are chosen at random is also considered. The paper complements our article of 2008 [Russian J. Phys. Chem. B, 2(6):947–963] where similar numerical experiments were carried out for spatial systems $(d=3)$ at $3\leqslant N \leqslant 100$. Keywords: multidimensional systems of classical particles, instantaneous phase-space invariants, kinetic energy partitions, formulas for the mean values, hyperangular momenta Citation: Aquilanti V., Lombardi A., Sevryuk M. B.,  Statistics of Energy Partitions for Many-Particle Systems in Arbitrary Dimension, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 318-347 DOI:10.1134/S1560354714030058
 Sevryuk M. B. The reversible context 2 in KAM theory: the first steps 2011, vol. 16, no. 1-2, pp.  24-38 Abstract The reversible context 2 in KAM theory refers to the situation where dim Fix $G < \frac{1}{2}$ codim $\mathcal{T}$, here Fix $G$ is the fixed point manifold of the reversing involution $G$ and $\mathcal{T}$ is the invariant torus one deals with. Up to now, this context has been entirely unexplored. We obtain a first result on the persistence of invariant tori in the reversible context 2 (for the particular case where dim Fix $G = 0$) using J. Moser’s modifying terms theorem of 1967. Keywords: KAM theory, Moser’s modifying terms theorem, reversible systems, reversible contexts, fixed point manifold, invariant torus Citation: Sevryuk M. B.,  The reversible context 2 in KAM theory: the first steps, Regular and Chaotic Dynamics, 2011, vol. 16, no. 1-2, pp. 24-38 DOI:10.1134/S1560354710520035
 Sevryuk M. B. On the Convergence of Coordinate Transformations in the KAM Procedure 2000, vol. 5, no. 2, pp.  181-188 Abstract We study the $C^r$-convergence of the compositions $W_n=U_1U_2\cdots U_n$ where mappings $U_k$ tend to the identity transformation in the $C^r$-topology as $k \to\infty$. The cases $r = 0$ and $1 \leqslant r < +\infty$ turn out to be drastically different. Citation: Sevryuk M. B.,  On the Convergence of Coordinate Transformations in the KAM Procedure, Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, pp. 181-188 DOI:10.1070/RD2000v005n02ABEH000140
 Sevryuk M. B. Invariant sets of degenerate Hamiltonian systems near equilibria 1998, vol. 3, no. 3, pp.  82-92 Abstract For any collection of $n \geqslant 2$ numbers $\omega_1,\ldots,\omega_n$, we prove the existence of an infinitely differentiable Hamiltonian system of differential equations $X$ with $n$ degrees of freedom that possesses the following properties: 1) $0$ is an elliptic (provided that all the $\omega_i$ are different from zero) equilibrium of system $X$ with eigenfrequencies $\omega_1,\ldots,\omega_n$; 2) system $X$ is linear up to a remainder flat at $0$; 3) the measure of the union of the invariant $n$-tori of system $X$ that lie in the $\varepsilon$-neighborhood of $0$ tends to zero as $\varepsilon\to 0$ faster than any prescribed function. Analogous statements hold for symplectic diffeomorphisms, reversible flows, and reversible diffeomorphisms. The results obtained are discussed in the context of the standard theorems in the KAM theory, the well-known Russmann and Anosov–Katok theorems, and a recent theorem by Herman. Citation: Sevryuk M. B.,  Invariant sets of degenerate Hamiltonian systems near equilibria, Regular and Chaotic Dynamics, 1998, vol. 3, no. 3, pp. 82-92 DOI:10.1070/RD1998v003n03ABEH000083
 Sevryuk M. B. Invariant tori of intermediate dimensions in hamiltonian systems 1998, vol. 3, no. 1, pp.  39-48 Abstract In the present paper, we survey recent results on the existence and the structure of Cantor families of invariant tori of dimensions $p>n$ in a neighborhood of families of invariant n-tori in Hamiltonian systems with $d \geqslant p$ degrees of freedom. Citation: Sevryuk M. B.,  Invariant tori of intermediate dimensions in hamiltonian systems, Regular and Chaotic Dynamics, 1998, vol. 3, no. 1, pp. 39-48
 Sevryuk M. B. Invariant tori of intermediate dimensions in Hamiltonian systems 1997, vol. 2, nos. 3-4, pp.  30-40 Abstract In the present paper, we survey recent results on the existence and the structure of Cantor families of invariant tori of dimensions $p>n$ in a neighborhood of families of invariant $n$-tori in Hamiltonian systems with $d \geqslant p$ degrees of freedom. Citation: Sevryuk M. B.,  Invariant tori of intermediate dimensions in Hamiltonian systems, Regular and Chaotic Dynamics, 1997, vol. 2, nos. 3-4, pp. 30-40 DOI:10.1070/RD1997v002n03ABEH000045