Neishtadt has proved that in the problem of fast phase averaging in an analytic system of ODE the dependence on the fast variable can be reduced to the terms which are exponentially small in the small parameter. The paper contains realistic estimates for these terms. These estimates essentially depend on properties of the first order averaged system.

An array of four globally phase-coupled oscillators with slightly different eigenfrequencies is considered. In the case of equal frequencies the system is reduced to an integrable one with almost all phase trajectories being closed. In the case of different but close to each other eigenfrequencies the system is treated with the use of the averaging method. It is shown that probabilistic phenomena take place in the system: the phase flow is divided quasi-randomly between various regions of the phase space when passing through separatrices of the unperturbed problem. Formulas describing probabilities of the phase point transition to different regions are obtained.

In the present paper, we survey recent results on the existence and the structure of Cantor families of invariant tori of dimensions $p>n$ in a neighborhood of families of invariant $n$-tori in Hamiltonian systems with $d \geqslant p$ degrees of freedom.

We analyse the operation of averaging of smooth functions along exact trajectories of dynamic systems in a neighborhood of stable nonresonance invariant tori. It is shown that there exists the first integral after the averaging; however in the typical situation the mean value is discontinuous or even not everywhere defind. If the temporal mean were a smooth function it would take its stationary values in the points of nondegenerate invariant tori. We demonstrate that this result can be properly derived if we change the operations of averaging and differentiating with respect to the initial data by their places. However, in general case for nonstable tori this property is no longer preserved. We also discuss the role of the reducibility condition of the invariant tori and the possibility of the generalization for the case of arbitrary compact invariant manifolds on which the initial dynamic system is ergodic.

We present a simple method which displays a hyperbolic structure in the phase space of an area preserving map. The method is illustrated for the case of the Carleson standard map. As it follows from our experiments, the structure of the chaotic zone for the standard map is different from the one found for the systems of Anosov type.

We consider free manifolds of dynamic billiards that allow constructing mathematical billiards equivalent to original dynamic billiards. It is shown that free manifolds of dynamic billiards in constant and Newtonian force field are surfaces of rotation in 3D Euclidean space. It is demonstrated that parabolic billiards in Newtonian attracting force field are equivalent to plane mathematical billiards.

In the paper the equations of motion of a rigid body in the Hamiltonian form on the subalgebra of algebra $e(4)$ are written. With the help of the algebraic methods a number of new isomorphisms in dynamics is established. We consider the lowering of the order as the process of decreasing rank of the Poisson structure with the algebraic point of view and indicate the possibility of arising the nonlinear Poisson brackets at this reduction as well.

While considering multiparameter families of nonlinear systems, types of behavior at the onset of chaos may appear which are distinct from Feigenbaum's universality. We present a review of such situations which can be met in families of one-dimensional maps and discuss a possibility of their realization and observation in nonlinear dissipative systems of more general form.

We consider two-dimensional analitical area-preserving diffeomorphisms that have structurally unstable symplest heteroclinic cycles. We find the conditions when diffeomorphisms under consideration possess a countable set of periodic elliptic points of stable type.

We obtained the upper limit for the number of functionally independent rational first integrals for a subgroup of the group of affine transformations of a complex affine space of finite dimension. This value does not exceed the difference between the power of the maximal set of functionally independent rational first integrals for the corresponding linear group and the maximal rank of the system consisted of the differentials of these first integrals restricted to the subspace generated by those elements of this subgroup that represent shifts.

4D-Hamiltonian systems with discrete symmetries are studied. The symmetries under consideration are such that a system possesses two invariant sub-planes which intersect each other transversally at an equilibrium state. The equilibrium state is supposed to to be of saddle type; moreover, in each invariant sub-plane there are two homoclinic loops to the saddle. We establish the existence of stable and unstable invariant manifolds for the bouquet comprised by the four homoclinic trajectories at the Hamiltonian level corresponding to the saddle. These manifolds may intersect transversely along some orbit. We call such a trajectory a super-homoclinic one. We prove that the existence of a super-homoclinic orbit implies the existence of a countable set of multi-pulse homoclinic trajectories to the saddle.

We study a $1$-parametric family of the Hamiltonian systems with $2$ hyperbolic fixed points and analyze the structure and bifurcations of homoclinic and heteroclinic trajectories under the variation of the parameter and energy values.

The article deals with discrete dynamical systems defined by iterations of a certain smooth map being a local diffeomorphism of a neighbourhood of the coordinate origin, for which the origin is a fixed point. Criteria to the existence of locally invariant curves adherent to the fixed point are obtained and the formulae for the expansion of the above curves into generalised power series are deduced. The author shows also the connection between the existence of those curves and the existence of the so-called asymptotic trajectories, going to the fixed point as the number of iterations infinitely increases or decreases. Of partitucular interest is the problem of the existence of the invariant curves the asymptotic of the motion along which is generalised power one. As an illustration, the author obtainsed criteria to the existence of asymptotic trajectories in several critical cases, when eigen values of the linear approximation matrix lie on a unit circle. The author considers also the problem of the existence of the motions of a material particle in the homogeneous gravity field, which are asymptotic to periodic skips over a critical point of a certain smooth curve.

Analysis of evolution and scale properties of subharmonic motions of dissipative and conservative nonlinear oscillators with one degree of freedom at transition from regular to chaotic regimes of motion through sequence of bifurcations is carried out. The numerical technique of research is based on a combination of methods of continuation of a solution by parameter, stability criterions, theory of branching, theory of scaling and precise methods of numerical integration. A number of universal scaling regularities, qualitatively and quantitatively describing transformation of the system phase space on a threshold of chaos, is revealed.