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Volume 24, Number 6

Volume 24, Number 6, 2019

Chierchia L.,  Koudjinan C.
V. I. Arnold's “Pointwise” KAM Theorem
Abstract
We review V.I. Arnold's 1963 celebrated paper [1] Proof of A.N. Kolmogorov's Theorem on the Conservation of Conditionally Periodic Motions with a Small Variation in the Hamiltonian, and prove that, optimising Arnold's scheme, one can get ''sharp'' asymptotic quantitative conditions (as $\varepsilon \to 0$, $\varepsilon$ being the strength of the perturbation). All constants involved are explicitly computed.
Keywords: Nearly-integrable Hamiltonian systems, KAM theory, Arnold's Theorem, small divisors, perturbation theory, symplectic transformations
Citation: Chierchia L.,  Koudjinan C., V. I. Arnold's “Pointwise” KAM Theorem, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 583-606
DOI:10.1134/S1560354719060017
Kudryashov N. A.,  Safonova D. V.,  Biswas A.
Painlevé Analysis and a Solution to the Traveling Wave Reduction of the Radhakrishnan – Kundu – Lakshmanan Equation
Abstract
This paper considers the Radhakrishnan – Kundu – Laksmanan (RKL) equation to analyze dispersive nonlinear waves in polarization-preserving fibers. The Cauchy problem for this equation cannot be solved by the inverse scattering transform (IST) and we look for exact solutions of this equation using the traveling wave reduction. The Painlevé analysis for the traveling wave reduction of the RKL equation is discussed. A first integral of traveling wave reduction for the RKL equation is recovered. Using this first integral, we secure a general solution along with additional conditions on the parameters of the mathematical model. The final solution is expressed in terms of the Weierstrass elliptic function. Periodic and solitary wave solutions of the RKL equation in the form of the traveling wave reduction are presented and illustrated.
Keywords: Radhakrishnan – Kundu – Laksmanan equation, integrability, traveling waves, general solution, exact solution
Citation: Kudryashov N. A.,  Safonova D. V.,  Biswas A., Painlevé Analysis and a Solution to the Traveling Wave Reduction of the Radhakrishnan – Kundu – Lakshmanan Equation, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 607-614
DOI:10.1134/S1560354719060029
Krajňák V.,  Ezra G. S.,  Wiggins S.
Roaming at Constant Kinetic Energy: Chesnavich's Model and the Hamiltonian Isokinetic Thermostat
Abstract
We consider the roaming mechanism for chemical reactions under the nonholonomic constraint of constant kinetic energy. Our study is carried out in the context of the Hamiltonian isokinetic thermostat applied to Chesnavich’s model for an ion-molecule reaction. Through an analysis of phase space structures we show that imposing the nonholonomic constraint does not prevent the system from exhibiting roaming dynamics, and that the origin of the roaming mechanism turns out to be analogous to that found in the previously studied Hamiltonian case.
Keywords: nonholonomic constraint, phase space structures, invariant manifolds, chemical reaction, roaming
Citation: Krajňák V.,  Ezra G. S.,  Wiggins S., Roaming at Constant Kinetic Energy: Chesnavich's Model and the Hamiltonian Isokinetic Thermostat, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 615-627
DOI:10.1134/S1560354719060030
Karabanov A. A.,  Morozov A. D.
On Resonances in Hamiltonian Systems with Three Degrees of Freedom
Abstract
We address the dynamics of near-integrable Hamiltonian systems with 3 degrees of freedom in extended vicinities of unperturbed resonant invariant Liouville tori. The main attention is paid to the case where the unperturbed torus satisfies two independent resonance conditions. In this case the average dynamics is 4-dimensional, reduced to a generalised motion under a conservative force on the 2-torus and is generically non-integrable. Methods of differential topology are applied to full description of equilibrium states and phase foliations of the average system. The results are illustrated by a simple model combining the non-degeneracy and non-integrability of the isoenergetically reduced system.
Keywords: Hamiltonian systems, resonances, topological structures
Citation: Karabanov A. A.,  Morozov A. D., On Resonances in Hamiltonian Systems with Three Degrees of Freedom, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 628-648
DOI:10.1134/S1560354719060042
Gebhard B.,  Ortega R.
Stability of Periodic Solutions of the $N$-vortex Problem in General Domains
Abstract
We investigate stability properties of a type of periodic solutions of the $N$-vortex problem on general domains $\Omega\subset \mathbb{R}^2$. The solutions in question bifurcate from rigidly rotating configurations of the whole-plane vortex system and a critical point $a_0\in\Omega$ of the Robin function associated to the Dirichlet Laplacian of $\Omega$. Under a linear stability condition on the initial rotating configuration, which can be verified for examples consisting of up to 4 vortices, we show that the linear stability of the induced solutions is solely determined by the type of the critical point $a_0$. If $a_0$ is a saddle, they are unstable. If $a_0$ is a nondegenerate maximum or minimum, they are stable in a certain linear sense. Since nondegenerate minima exist generically, our results apply to most domains $\Omega$. The influence of the general domain $\Omega$ can be seen as a perturbation breaking the symmetries of the $N$-vortex system on $\mathbb{R}^2$. Symplectic reduction is not applicable and our analysis on linearized stability relies on the notion of approximate eigenvectors. Beyond linear stability, Herman's last geometric theorem allows us to prove the existence of isoenergetically orbitally stable solutions in the case of $N=2$ vortices.
Keywords: vortex dynamics, periodic solutions, stability, Floquet multipliers, bifurcation, Poincaré section
Citation: Gebhard B.,  Ortega R., Stability of Periodic Solutions of the $N$-vortex Problem in General Domains, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 649-670
DOI:10.1134/S1560354719060054
Allilueva A. I.,  Shafarevich A. I.
Conic Lagrangian Varieties and Localized Asymptotic Solutions of Linearized Equations of Relativistic Gas Dynamics
Abstract
We study asymptotic solution of the Cauchy problem for the linearized system of relativistic gas dynamics. We assume that initial condiditiopns are strongly localized near a space-like surface in the Minkowsky space. We prove that the solution can be decomposed into three modes, corresponding to different routsb of the equations of characteristics. One of these roots is twice degenerate and the there are no focal points in the corresponding miode. The other two roots are simple; in order to describe the corresponding modes, we use the modificication of the Maslov’s canonical operator which was obtained recently.
Keywords: Conic Lagrangian varieties, Maslov’s canonical operator, relativistic gas dynamics
Citation: Allilueva A. I.,  Shafarevich A. I., Conic Lagrangian Varieties and Localized Asymptotic Solutions of Linearized Equations of Relativistic Gas Dynamics, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 671-681
DOI:10.1134/S1560354719060066
Bolotin S. V.
Jumps of Energy Near a Homoclinic Set of a Slowly Time Dependent Hamiltonian System
Abstract
We consider a Hamiltonian system depending on a parameter which slowly changes with rate $\varepsilon \ll 1$. If trajectories of the frozen autonomous system are periodic, then the system has adiabatic invariant which changes much slower than energy. For a system with 1 degree of freedom and a figure 8 separatrix, Anatoly Neishtadt [18] showed that for trajectories crossing the separatrix, the adiabatic invariant, and hence the energy, have quasirandom jumps of order $\varepsilon$. We prove a partial analog of Neishtadt's result for a system with $n$ degrees of freedom such that the frozen system has a hyperbolic equilibrium possessing several homoclinic orbits. We construct trajectories staying near the homoclinic set with energy having jumps of order $\varepsilon$ at time intervals of order $|\ln\varepsilon|$, so the energy may grow with rate $\varepsilon/|\ln\varepsilon|$. Away from the homoclinic set faster energy growth is possible: if the frozen system has chaotic behavior, Gelfreich and Turaev [16] constructed trajectories with energy growth rate of order $\varepsilon$.
Keywords: Hamiltonian system, homoclinic orbit, action functional, Poincare function, symplectic relation, separatrix map, adiabatic invariant
Citation: Bolotin S. V., Jumps of Energy Near a Homoclinic Set of a Slowly Time Dependent Hamiltonian System, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 682-703
DOI:10.1134/S1560354719060078
Dobrokhotov S. Y.,  Minenkov D. S.,  Neishtadt A. I.,  Shlosman S. B.
Classical and Quantum Dynamics of a Particle in a Narrow Angle
Abstract
We consider the 2D Schrödinger equation with variable potential in the narrow domain diffeomorphic to the wedge with the Dirichlet boundary condition. The corresponding classical problem is the billiard in this domain. In general, the corresponding dynamical system is not integrable. The small angle is a small parameter which allows one to make the averaging and reduce the classical dynamical system to an integrable one modulo exponential small correction. We use the quantum adiabatic approximation (operator separation of variables) to construct the asymptotic eigenfunctions (quasi-modes) of the Schr¨odinger operator. We discuss the relation between classical averaging and constructed quasi-modes. The behavior of quasi-modes in the neighborhood of the cusp is studied. We also discuss the relation between Bessel and Airy functions that follows from different representations of asymptotics near the cusp.
Keywords: potential well, stationary Schrödinger equation, KAM theory, operator separation of variables, semiclassical asymptotics, Airy function, Bessel function
Citation: Dobrokhotov S. Y.,  Minenkov D. S.,  Neishtadt A. I.,  Shlosman S. B., Classical and Quantum Dynamics of a Particle in a Narrow Angle, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 704-716
DOI:10.1134/S156035471906008X
Bolotov D. I.,  Bolotov M. I.,  Smirnov L. A.,  Osipov G. V.,  Pikovsky A.
Twisted States in a System of Nonlinearly Coupled Phase Oscillators
Abstract
We study the dynamics of the ring of identical phase oscillators with nonlinear nonlocal coupling. Using the Ott – Antonsen approach, the problem is formulated as a system of partial derivative equations for the local complex order parameter. In this framework, we investigate the existence and stability of twisted states. Both fully coherent and partially coherent stable twisted states were found (the latter ones for the first time for identical oscillators). We show that twisted states can be stable starting from a certain critical value of the medium length, or on a length segment. The analytical results are confirmed with direct numerical simulations in finite ensembles.
Keywords: twisted state, phase oscillators, nonlocal coupling, Ott – Antonsen reduction, stability analysis
Citation: Bolotov D. I.,  Bolotov M. I.,  Smirnov L. A.,  Osipov G. V.,  Pikovsky A., Twisted States in a System of Nonlinearly Coupled Phase Oscillators, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 717-724
DOI:10.1134/S1560354719060091
Kruglov V. P.,  Kuznetsov S. P.
Topaj – Pikovsky Involution in the Hamiltonian Lattice of Locally Coupled Oscillators
Abstract
We discuss the Hamiltonian model of an oscillator lattice with local coupling. The Hamiltonian model describes localized spatial modes of nonlinear the Schrödinger equation with periodic tilted potential. The Hamiltonian system manifests reversibility of the Topaj – Pikovsky phase oscillator lattice. Furthermore, the Hamiltonian system has invariant manifolds with asymptotic dynamics exactly equivalent to the Topaj – Pikovsky model. We examine the stability of trajectories belonging to invariant manifolds by means of numerical evaluation of Lyapunov exponents. We show that there is no contradiction between asymptotic dynamics on invariant manifolds and conservation of phase volume of the Hamiltonian system. We demonstrate the complexity of dynamics with results of numerical simulations.
Keywords: reversibility, involution, Hamiltonian system, Topaj – Pikovsky model, phase oscillator lattice
Citation: Kruglov V. P.,  Kuznetsov S. P., Topaj – Pikovsky Involution in the Hamiltonian Lattice of Locally Coupled Oscillators, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 725-738
DOI:10.1134/S1560354719060108
Borisov A. V.,  Tsiganov A. V.
On the Chaplygin Sphere in a Magnetic Field
Abstract
We consider the possibility of using Dirac’s ideas of the deformation of Poisson brackets in nonholonomic mechanics. As an example, we analyze the composition of external forces that do no work and reaction forces of nonintegrable constraints in the model of a nonholonomic Chaplygin sphere on a plane. We prove that, when a solenoidal field is applied, the general mechanical energy, the invariant measure and the conformally Hamiltonian representation of the equations of motion are preserved. In addition, we consider the case of motion of the nonholonomic Chaplygin sphere in a constant magnetic field taking dielectric and ferromagnetic (superconducting) properties of the sphere into account. As a by-product we also obtain two new integrable cases of the Hamiltonian rigid body dynamics in a constant magnetic field taking the magnetization by rotation effect into account.
Keywords: nonholonomic mechanics, magnetic field, deformation of Poisson brackets, Grioli problem, Barnett – London moment
Citation: Borisov A. V.,  Tsiganov A. V., On the Chaplygin Sphere in a Magnetic Field, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 739-754
DOI:10.1134/S156035471906011X

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