Volume 6, Number 1
Volume 6, Number 1, 2001
Borisov A. V., Mamaev I. S., Kholmskaya A. G.
Kovalevskaya top and generalizations of integrable systems
Abstract
Generalizations of the Kovalevskaya, Chaplygin, Goryachev–Chaplygin and Bogoyavlensky systems on a bundle are considered in this paper. Moreover, a method of introduction of separating variables and actionangle variables is described. Another integration method for the Kovalevskaya top on the bundle is found. This method uses a coordinate transformation that reduces the Kovalevskaya system to the Neumann system. The Kolosov analogy is considered. A generalization of a recent Gaffet system to the bundle of Poisson brackets is obtained at the end of the paper.

Karabanov A. A.
Resonances in FourDimensional QuasiHamiltonian Systems: Analysis and Simulations
Abstract
The problem of qualitative behaviour of fourdimensional quasiHamiltonian system in a neighbourhood of a fixed resonance is considered. The general analytical grounds of the problem are touched upon. We turn to the global analysis of special threedimensional system averaged near the resonance. Furthermore, a checking the theory against numerical simulations is made. Two physical examples, revealing an irregular resonant dynamics, are studied.

Ramodanov S. M.
Motion of a Circular Cylinder and a Vortex in an Ideal Fluid
Abstract
The motion of a circular cylinder and a point vortex in an unbounded ideal fluid is treated here on the basis of a potential framework. The formulas for the hydrodynamic force and moment acting upon a cylinder of arbitrary cross section are obtained. The equations governing the motion of a circular cylinder are derived and partially investigated.

Szulikowska V. V.
Limit Behaviour of Independent Random Matrices' Products
Abstract
The asymptotic behavior of the products of the independent identically distributed by $\mu$ unimodular random matrices is studied, when the conditions of the spectrum simplicity do not hold. In particular, the strong law of the large numbers is proved for the given dynamical system, and the conditions on the distribution $\mu$ are written, providing the difference of some fixed Lyapunov exponents. Moreover, we assume, that the distribution $\mu$ depends on some parameter a, and investigate the continuity of their limit characteristics from a for the random matrices' products.

Polikarpov S. A.
The Split of Separatrice Loop and Birth of NonDegenerate Solutions with Long Period in the Case of NonConservative Perturbations of Hamiltonian Systems
Abstract
The work is dedicated to the investigation of the connection between separatrix split and birth of the isolated periodic solutions in the perturbated Hamiltonian system with one degree of freedom. By means of H. Poincaré [1] and V.V. Kozlov [2] methods the result of [3] is generalized to the case of nonconservative perturbation. The general theorem, obtained in chapter 2, permits to arque about system's periodic solutions by value of asymptotic surfaces split. In the final part of the work, nonconservative perturbation in Duffingtype equation serves as an example (see [4]).

Ivanov A. V.
Study of the Double Mathematical Pendulum — IV. Quantitative Bounds on Values of the System Parameters when the Homoclinic Transversal Intersections Exist
Abstract
We consider the double mathematical pendulum in the limit of small ratio of pendulum masses. Besides we assume that values of other two system parameters are close to the degenerate ones (i.e. zero or infinity). In these limit cases we prove asymptotic formulae for the homoclinic invariant of some special chosen homoclinic trajectories and obtain quantitative bounds on values of the system parameters when these formulae are valid.

Kessi A., Messaoud K. M.
First Order Equations without Mobile Critical Points
Abstract
We study in this paper the ordinary differential equations which are polynomial of order $3$ with respect to $\omega'$, whose coefficients are polynomial with respect to $\omega$ and analytical with respect to $z$. We are looking for the sufficient conditions on the coefficients as functions of $z$, in order to have the solution $\omega$ with fixed critical points.

Krawiec A., Szydlowski M.
On Nonlinear Mechanics of Business Cycle Model
Abstract
This paper covers an application of nonlinear mechanics in macrodynamic model of the business cycle. The timetobuild is introduced into the capital accumulation equation according to Kalecki's idea of delay in investment processes. The dynamics of this model is represented in terms of a time delay differential equation system. It is found that there are two causes which generate cyclic behaviour in the model. Apart from the standard Kaldor proposition of special nonlinearity in the investment function, the cycle behaviour is due to the time delay parameter. In both scenarios, cyclic behaviour emerges from the Hopf bifurcation to the periodic orbit.
In the special case of a small timetobuild parameter the general dynamics is reduced to a twodimensional autonomous dynamical system. This system is examined in detail by methods of qualitative analysis of differential equations. Then cyclic behaviour in the system is represented by a limit cycle on the plane phase. It is shown that there is a certain bifurcation value of the time delay parameter which leads to a periodic orbit. We discuss the problem of the existence of a global attractor in $2$dimensional phase space whose counterpart for the Kaldor model was considered by Chang and Smyth. It is shown that the presence of timetobuild excludes the asymptotically stable global critical point. Additionally, we analyse the question of uniqueness of the limit cycles of the model. 