### Article

## GLOBAL BIFURCATIONS IN GENERIC ONE-PARAMETER FAMILIES WITH A PARABOLIC CYCLE ON S^2

We classify global bifurcations in generic one-parameter local families of vector elds on S2 with a parabolic cycle. The classication is quite dierent from the classical results presented in monographs on the bifurcation theory. As a by product we prove that generic families described above are structurally stable.

In the survey, we consider bifurcations of attracting (or repelling) invariant sets of some classical dynamical systems with a discrete time.

This article discusses examples of nonlinear models of economic dynamics and possibilities of their research by numerical procedures in MATLAB. Demonstrated specific effects of these models, in particular, the possibility of forming a chaotic behavior

Even six years after the acute phase of Great Recession 2007-2009, euro area economy does not show strong growth, which is indicating a severe structural and cyclical imbalances in the European economy. Empirical data evident that Euro area economy as the US, since 2009 are located in an unstable equilibrium that is prone to buckling under the influence of small internal or external price shocks. For the detection of the bifurcation process, i.e. transition to a state of metamorphosis, we have specially developed models of nonlinear dynamics, which describe five possible state of the economic system and, in particular, show that the Eurozone economy is entering a very important stage of bifurcation and the consequences of which are fundamental to determine the nature of the future economic development of both European and global economy.

In this paper, we suggest an approach to the study of the financial instability based on the model of evolutionary processes. In the first place, we present some empirical facts that confirm that the stock’s price dynamics is better described by the Markov switching model rather than by the pure random walk. Further, using the equilibrium model of price formation, we show that the temporary price trends on stock market are evolutionary processes that occur in the conditions of a duality of the equilibrium between the market price and the fair value. Then, within the framework of the constructed model, we analyze the causes of the financial market instability and its impact on the real sector, and show how the financial markets create a destructive impulse under the economic growth slowdown, and therefore adversely affect the process of innovations diffusion into the market. The conducted study shows that the causes of the financial instability are the capital concentration in the narrow circles of society and the lack of investment opportunities, as compared with the available financial resources, whereas the symptoms are frequently recurring financial bubbles and crises.

We consider dynamics of a space elevator on an asteroid, i. e., spacecraft attached to a rotating celestial body with a light inextensible tether. We study the domains attainable for the spacecraft depending on such problem parameters as the angular velocity of the asteroid, the tether length, the position of the anchor at the surface, etc. We develop a method based on Routh procedure that allows one to identify the relative equilibria of the system in study and to analyze its stability and bifurcations. Some non-trivial classes of the solutions are found and their relations to the libration points are examined.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.