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# Volume 13, Number 3, 2008

 Kozlov V. V. Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat Abstract The paper develops an approach to the proof of the "zeroth" law of thermodynamics. The approach is based on the analysis of weak limits of solutions to the Liouville equation as time grows infinitely. A class of linear oscillating systems is indicated for which the average energy becomes eventually uniformly distributed among the degrees of freedom for any initial probability density functions. An example of such systems are sympathetic pendulums. Conditions are found for nonlinear Hamiltonian systems with finite number of degrees of freedom to converge in a weak sense to the state where the mean energies of the interacting subsystems are the same. Some issues related to statistical models of the thermostat are discussed. Keywords: Hamiltonian system, sympathetic oscillators, weak convergence, thermostat Citation: Kozlov V. V., Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 141-154 DOI:10.1134/S1560354708030015
 Pokorny P. Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum Abstract Equations of motion for 3-dim heavy spring elastic pendulum are derived and rescaled to contain a single parameter. Condition for the stability of vertical large amplitude oscillations is derived analytically relating the parameter of the system and the amplitude of the vertical oscillation. Numerical continuation is used to find the border of the stability region in parameter space with high precision. The stability condition is approximated by a simple formula valid for a large range of the parameter and of the amplitude of oscillation. The bifurcation responsible for the loss of stability is identified. Keywords: elastic pendulum, stability condition Citation: Pokorny P., Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 155-165 DOI:10.1134/S1560354708030027
 Dos Santos F.,  Vidal C. Stability of Equilibrium Solutions of Hamiltonian Systems Under the Presence of a Single Resonance in the Non-Diagonalizable Case Abstract The problem of knowing the stability of one equilibrium solution of an analytic autonomous Hamiltonian system in a neighborhood of the equilibrium point in the case where all eigenvalues are pure imaginary and the matrix of the linearized system is non-diagonalizable is considered.We give information about the stability of the equilibrium solution of Hamiltonian systems with two degrees of freedom in the critical case. We make a partial generalization of the results to Hamiltonian systems with $n$ degrees of freedom, in particular, this generalization includes those in [1]. Keywords: Hamiltonian system, stability, normal form, resonances Citation: Dos Santos F.,  Vidal C., Stability of Equilibrium Solutions of Hamiltonian Systems Under the Presence of a Single Resonance in the Non-Diagonalizable Case, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 166-177 DOI:10.1134/S1560354708030039
 Tsiganov A. V. On Maximally Superintegrable Systems Abstract Locally any completely integrable system is maximally superintegrable system since we have the necessary number of the action-angle variables. The main problem is the construction of the single-valued additional integrals of motion on the whole phase space by using these multi-valued action-angle variables. Some constructions of the additional integrals of motion for the Stäckel systems and for the integrable systems related with two different quadratic $r$-matrix algebras are discussed. Among these system there are the open Heisenberg magnet and the open Toda lattices associated with the different root systems. Keywords: superintegrable systems, Toda lattices, Stackel systems Citation: Tsiganov A. V., On Maximally Superintegrable Systems, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 178-190 DOI:10.1134/S1560354708030040
 Tsiganov A. V. The Poisson Bracket Compatible with the Classical Reflection Equation Algebra Abstract We introduce a family of compatible Poisson brackets on the space of $2 \times 2$ polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes the $XXX$ Heisenberg magnet with boundary conditions, the generalized Toda lattices and the Kowalevski top. Keywords: Poisson bracket, bi-hamiltonian structure, reflection equation algebra Citation: Tsiganov A. V., The Poisson Bracket Compatible with the Classical Reflection Equation Algebra, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 191-203 DOI:10.1134/S1560354708030052
 Borisov A. V.,  Kilin A. A.,  Mamaev I. S. Absolute and Relative Choreographies in Rigid Body Dynamics Abstract For the classical problem of motion of a rigid body about a fixed point with zero area integral, we present a family of solutions that are periodic in the absolute space. Such solutions are known as choreographies. The family includes the well-known Delone solutions (for the Kovalevskaya case), some particular solutions for the Goryachev–Chaplygin case, and the Steklov solution. The "genealogy" of solutions of the family naturally appearing from the energy continuation and their connection with the Staude rotations are considered. It is shown that if the integral of areas is zero, the solutions are periodic with respect to a coordinate frame that rotates uniformly about the vertical (relative choreographies). Keywords: rigid-body dynamics, periodic solutions, continuation by a parameter, bifurcation Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., Absolute and Relative Choreographies in Rigid Body Dynamics, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 204-220 DOI:10.1134/S1560354708030064
 Borisov A. V.,  Kilin A. A.,  Mamaev I. S. Chaos in a Restricted Problem of Rotation of a Rigid Body with a Fixed Point Abstract In this paper, we consider the transition to chaos in the phase portrait of a restricted problem of rotation of a rigid body with a fixed point. Two interrelated mechanisms responsible for chaotization are indicated: (1) the growth of the homoclinic structure and (2) the development of cascades of period doubling bifurcations. On the zero level of the area integral, an adiabatic behavior of the system (as the energy tends to zero) is noted. Meander tori induced by the break of the torsion property of the mapping are found. Keywords: motion of a rigid body, phase portrait, mechanism of chaotization, bifurcations Citation: Borisov A. V.,  Kilin A. A.,  Mamaev I. S., Chaos in a Restricted Problem of Rotation of a Rigid Body with a Fixed Point, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 221-233 DOI:10.1134/S1560354708030076
 Kudryashov N. A. Solitary and Periodic Solutions of the Generalized Kuramoto–Sivashinsky Equation Abstract The generalized Kuramoto–Sivashinsky equation in the case of the power nonlinearity with arbitrary degree is considered. New exact solutions of this equation are presented. Keywords: exact solution, nonlinear differential equation, Kuramoto–Sivashinsky equation Citation: Kudryashov N. A., Solitary and Periodic Solutions of the Generalized Kuramoto–Sivashinsky Equation, Regular and Chaotic Dynamics, 2008, vol. 13, no. 3, pp. 234-238 DOI:10.1134/S1560354708030088

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