A generalization of the theory of algebro-geometric Poisson brackets on the space of finite-gap Schrodinger operators, developped by S.P.Novikov and A.P.Veselov, to the case of periodic zero-diagonal difference operators of second order is proposed. A necessary and sufficient condition for such a bracket to be compatible with higher Volterra flows is found.

It is proved that trajectories of any mechanical system $(M,g,V)$, where $M$ is the configuration space, $g$ the metric defined by the kinetic energy form, and $V$ a potential function, with the natural Lagrangian, are pregeodesics with respect to the Jacobi metric $g_E=2|E-V|g$, $E$ being the total energy of the system.

Fractal interpolation functions have become popular after the works of M.Barnsley and co-authors on iterated function systems (see, e.g., [5]). We consider here the following problem: given a set of values of a fractal interpolation function (FIF), determine the contractive affine mappings generating this function. The suggested solution is based on the observation that the fixed points of some of the affine mappings in question are among the points where the FIF has its strongest singularity. These points may be detected with the aid of wavelet-based techniques, such as modulus maxima lines tracing. After this is done, necessary matrices are computed from a system of linear equations. The method was tested numerically on FIFs with local Holder exponent as low as 0.3, and allowed to recover the generating matrices almost precisely. When applied to segments of financial time series, this approach gave FIFs reproducing some of the apparently chaotic patterns in the series. This suggests the potential usefulness of this techniques for detection of hidden rescaling parameters in the observed data.

We suggest a simple approach for obtaining integrals of Hamiltonian systems if there is known a trajectorian map of two Hamiltonian systems. An explicite formila is given. As an example, it is proved that if on a manifold are given two Riemannian metrics which are geodesically equivalent then there is a big family of integrals. Our theorem is a generalization of the well-known Painleve–Liouville theorems.

We consider dynamical system in phase space, which has closed submanifold filled by periodic orbits. The following problem is analysed. Let us consider small perturbations of the system. What we can say about the number of survived periodic orbits, about their number and about their location in the neighborhood of a given submanifold? We obtain the solution of this problem for the perturbations of general type in terms of averaged perturbnation. The main result of the paper is as follows. Theorem: Let us consider the Hamiltonian system with Hamiltonian function $H$ on symplectic manifold $(M^{2n},\omega^2)$. Let $\Lambda \subset H^{-1}(h)$ be the closed nondegenerate submanifold filled by periodic orbits of this system. Then for the arbitrary perturbed function $\tilde{H}$, which is $C^2$-close to the initial function $H$, the system with the Hamiltonian $\tilde{H}$ has no less than two periodic orbits on the isoenergy surface $\tilde{H}^{-1}(h)$. Moreover, if either the fibration of $\Lambda$ by closed orbits is trivial, or the base $B=\Lambda /S^1$ of this fibration is locally flat, then the number of such orbits is not less than the minimal number of the critical points of smooth function on the quotient manifold $B$.

The evolution of the non-equilibrium colliding hard disk systems in the velocity space has been studied using a mathematical model based on the collision matrix. It has been shown that the main features of evolution can be determined by the property of the collision matrix. The characteristic parameter which defines the maximal possible velocity at which the system approaches to the equilibrium state was calculated. This parameter decreases as the number of collision increases. It has been found how the evolution of the hard disk systems to the equilibrium can be described in terms of the distribution function of impact parameters of colliding disks. Is shown, that the condition of inhomogeneiaty of impact parameters causes reduction with growth of number of collisions total on all colliding pairs sum of a difference of speeds of colliding disks. It is equivalent to aspiration to zero the total internal force of collisions of disks.

The motion of a round dynamically symmetrical disk within a sphere covered by smooth ice is considered. It is shown that in the absence of a gravity field equations of motion may be reduced to quadratures. Conditions for the stability of stationary motions have been found. Probability of a falling of a disk has been investigated. Regions allowed for the motion in a gravity field and characterizing by the disk orientation and the situation of a trajectory of a contact point on a sphere surface have been built. The conclusion on the non-integrability on the analysis of Poincare sections of a problem on a round dynamically symmetrical disk moving on a smooth sphere surface has been made.

In the paper, topology of energy surfaces is described and bifurcation sets is constructed for the classical Chaplygin problem and its generalization. We also describe bifurcations of Liouville tori and calculate the Fomenko invariant (for the classical case this result is obtained analytically and for the generalized case it is obtained with the help of computer modeling). Topological analysis shows that some topological characteristics (such as the form of the bifurcation set) change continuously and some of them (such as topology of energy surfaces) change drastically as $g\to0$.

The method is proposed of the explicit embedding of the some types of the singular orbits of the adjoint action of the some classical Lie groups in the corresponding (co)algebras as the level surfaces of the special polynomials. In fact, orbits of types $SO(2n)/SO(2k) \times SO(2)^{n-k}$, $SO(2n+1)/SO(2k+1) \times SO(2)^{n-k}$, $E(2n-1)/R \times SO(2k) \times SO(2)^{n-k-1}$, $E(2n)/R \times SO(2k+1) \times SO(2)^{n-k-1}$, $(S)U(n)/(S)(U(2k) \times U(2)^{n-k})$ can be embeded by the method. Particularly, the minimal-dimensional orbits can be described as intersections of quadrics.

Integrable problem of three vorteces on a plane and sphere are considered. The classification of Poisson structures is carried out. We accomplish the bifurcational analysis using the variables introduced in previous part of the work.