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.

In the skew Hopf bifurcation a quasi-periodic attractor with nontrivial normal linear dynamics loses hyperbolicity. Periodic, quasi-periodic 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 KAM-theory, deals with the persistence of quasi-periodic circles. Other results concern the bifurcations of periodic attractors in the case of resonance.

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 KAM-theory) 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.

We discuss the dynamics of a piecewise holonomic mechanical system: a discrete sister to the classical non-holonomically 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 constant-speed straight-line 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.

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 two-form 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 exponentially-long 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 quasi-integrable) and if it satisfies a convexity condition, then a Nekhoroshev-like 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 long-time stability in the Asteroid Main Belt [9,13].

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 one-dimensional 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 Bethe-type 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.

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.