Volume 4, Number 2
Volume 4, Number 2, 1999
Sinai Y. G.
Navier–Stokes systems with periodic boundary conditions
Abstract
The article presents a review of geometrical proof of the results earlier obtained by Foias, Temam with respect to Gevre class regularity for the solution of the Navier–Stokes equations. The articleis based on the lectures, delivered at the Moscow Independent University in june 1999.

Broer H. W., Takens F., Wagener F. O.
Integrable and nonintegrable deformations of the skew Hopf bifurcation
Abstract
In the skew Hopf bifurcation a quasiperiodic attractor with nontrivial normal linear dynamics loses hyperbolicity. Periodic, quasiperiodic and chaotic dynamics occur, including motion with mixed spectrum. The case of $3$dimensional skew Hopf bifurcation families of diffeomorphisms near integrability is discussed, surveying some recent results in a broad perspective. One result, using KAMtheory, deals with the persistence of quasiperiodic circles. Other results concern the bifurcations of periodic attractors in the case of resonance.

Kozlov V. V.
Canonical Gibbs distribution and thermodynamics of mechanical systems with a finite number of degrees of freedom
Abstract
Traditional derivation of Gibbs canonical distribution and the justification of thermodynamics are based on the assumption concerning an isoenergetic ergodicity of a system of n weakly interacting identical subsystems and passage to the limit $n \to\infty$. In the presented work we develop another approach to these problems assuming that n is fixed and $n \geqslant 2$. The ergodic hypothesis (which frequently is not valid due to known results of the KAMtheory) is substituted by a weaker assumption that the perturbed system does not have additional first integrals independent of the energy integral. The proof of nonintegrability of perturbed Hamiltonian systems is based on the Poincare method. Moreover, we use the natural Gibbs assumption concerning a thermodynamic equilibrium of bsystems at vanishing interaction. The general results are applied to the system of the weakly connected pendula. The averaging with respect to the Gibbs measure allows to pass from usual dynamics of mechanical systems to the classical thermodynamic model.

Coleman M. J., Holmes P. J.
Motions and stability of a piecewise holonomic system: the discrete Chaplygin sleigh
Abstract
We discuss the dynamics of a piecewise holonomic mechanical system: a discrete sister to the classical nonholonomically constrained Chaplygin sleigh. A slotted rigid body moves in the plane subject to a sequence of pegs intermittently placed and sliding freely along the slot; motions are smooth and holonomic except at instants of peg insertion. We derive a return map and analyze stability of constantspeed straightline motions: they are asymptotically stable if the mass center is in front of the center of the slot, and unstable if it lies behind the slot; if it lies between center and rear of the slot, stability depends subtly on slot length and radius of gyration. As slot length vanishes, the system inherits the eigenvalues of the Chaplygin sleigh while remaining piecewise holonomic. We compare the dynamics of both systems, and observe that the discrete skate exhibits a richer range of behaviors, including coexistence of stable forward and backward motions.

Guzzo M.
Nekhoroshev stability of quasiintegrable degenerate hamiltonian systems
Abstract
A perturbation of a degenerate integrable Hamiltonian system has the form $H=h(I)+\varepsilon f(I,\varphi ,p,q)$ with $(I,\varphi )\in {\bf R}^n\times {\bf T}^n$, $(p,q)\in {\cal B} \subseteq {\bf R}^{2m}$ and the twoform is $dI\wedge d\varphi + dp\wedge dq$. In the case $h$ is convex, Nekhoroshev theorem provides the usual bound to the motion of the actions $I$, but only for a time which is the smaller between the usual exponentiallylong time and the escape time of $p,q$ from ${\cal B}$. Furthermore, the theorem does not provide any estimate for the "degenerate variables" $p,q$ better than the a priori one $\dot p,\dot q\sim \varepsilon$, and in the literature there are examples of systems with degenerate variables that perform large chaotic motions in short times. The problem of the motion of the degenerate variables is relevant to understand the long time stability of several systems, like the three body problem, the asteroid belt dynamical system and the fast rotations of the rigid body.
In this paper we show that if the "secular" Hamiltonian of $H$, i.e. the average of $H$ with respect to the fast angles $\varphi$, is integrable (or quasiintegrable) and if it satisfies a convexity condition, then a Nekhoroshevlike bound holds for the degenerate variables (actually for the actions of the secular integrable system) for all initial data with initial action $I(0)$ outside a small neighbourhood of the resonant manifolds of order lower than $\ln \dfrac{1}{\varepsilon}$. This paper generalizes a result proved in connection with the problem of the longtime stability in the Asteroid Main Belt [9,13]. 
Van Diejen J. F.
On the zeros of the KdV soliton Baker–Akhiezer function
Abstract
Recent results concerning the zeros of the KdV soliton Baker–Akhiezer function are outlined. Specifically, it is shown that the zeros of the wave function of a onedimensional Schrodinger operator with a reflectionless potential are characterized by (i) the equations of motion of a rational Ruijsenaars–Schneider particle system with harmonic term and (ii) a nonlinear algebraic system of Bethetype equations. The integration of the particle system provides us with an explicit parametrization of the solution curve of the Bethe equations. The flows corresponding to the higher integrals of the particle system encode the dynamics of the zeros of the solitonic Baker–Akhiezer function for the KdV hierarchy.

Chernoivan V. A., Mamaev I. S.
The restricted twobody problem and the kepler problem in the constant curvature spaces
Abstract
In this work we carry out the bifurcation analysis of the Kepler problem on $S^3$ and $L^3$, and construct the analogues of Delaunau variables. We consider the problem of motion of a mass point in the field of moving Newtonian center on $S^2$ and $L^2$. The perihelion deviation is derived by the method of perturbation theory under the small curvature, and a numerical investigation is made, using anology of this problem with rigid body dynamics.
