Search

In modern natural sciences, the term of a dynamic system plays an important role and is a common type of mathematical models. Dynamical systems are rarely come to simple functional dependencies. Therefore, qualitative analysis methods of dynamical systems are crucial. In the paper, we consider the simplest type of dynamic systems | continuous dynamical systems described by the systems of ordinary differential equations.Qualitative analysis of differential equations systems usually starts with a search for equilibrium points and a study of the behaviour of a dynamic system in the neighborhood of each equilibrium points. The main attention is paid to the stability of equilibrium, as well as their behaviour type classification. Effective qualitative analysis of differential equations systems is best approached through the bifurcation theory which explains modification of quality in the behaviour of a dynamic system if its parameters are changed.In the behavior of dynamic systems, in addition to the equilibrium points, other bounded trajectories (for example, boundary cycles or separatrix) and their certain conglomerates (such as attractors, invariant tori) play an important role. Investigation of bounded trajectories, in particular, attractors is a difficult task and a lot of scientific articles deal with this problem.In this paper, we study a continuous Lorenz-type system. For this system, all of the equilibrium points are defined and the analysis of equilibrium points types are performed in accordance with the system parameters. The analysis of some bifurcations of equilibrium points are carried out. In particular, the Andronov | Hopf bifurcation is determined and it is shown that it leads to a bifurcation of boundary cycles.DOI: 10.7463/mathm.0315.0789497

Hardy spaces play an important role in the complex analysis and its numerous applications. However, unlike a one-dimensional case, the Hardy-type spaces in a polydisc are investigated a little. In this paper, the integral representations of the classes of n-harmonic in a polydisc Un functions are received, in particular the characterization of n-harmonic functions in a polydisc which can be represented as a multiple Poisson integral from functions, measurable on a skeleton of a polydisc, from a class where 1 < pi < + ∞, is given. At the proof of the main result the general methods of the complex and functional analysis, theory of Hardy classes is used.

The concept of crossing or non-crossing sets with objects of different nature. The principles, which give the transformed arrays properties that increase the efficiency of information technology. As the main characteristic defining the relationship on the sets, forming the arrays, we consider the system of indexing of elements of the compared sets. The terms an index of non-crossing array components are. We propose a new class of arrays which components are indexed by disjoint sets, the notion of « non-crossing index sets»; provides methods of forming arrays of different classes. It is proved that any permutation of the elements of arrays are implemented with some regularity, associated with changes in indexing. The developed method of formation of arrays based on cyclic shifts of the rows and columns of the original array for the proposed algorithms. The task of forming arrays with disjoint sets of three types has been solved. The paper analizez the proposed options for building arrays with disjoint components - sets, proving that the properties of non-crossing components are stored for the first type arrays in all permutations of columns.

The paper deals with the inverse problem for the Sobolev type equation of the second order in Banach spaces. The introduction contains a problem statement and the historiography of Sobolev type equations. The second part includes preliminary information based on the results of the theory of higher-order Sobolev type equations. In the third part the initial problem is reduced to the inverse regular and singular problems. A theorem of unique solvability of regular problem is formulated and proved. Using the results of the third part, the solution for the singular problem is obtained in the fourth part. The sum of regular and singular solutions is a solution to the original problem, thus a theorem on the unique solvability of the inverse problem for Sobolev type equation of the second order is stated and proved.

В статье приводится обзор некоторых методов решения класса интегро- дифференциальных обратных задач. А именно, представлены методы решения интегро-дифференциальных обратных задач параболического типа, основанные на теории аналитических полугрупп, результатах по оптимальной регулярности и теореме о неподвижной точке. Эти методы могут быть применены к большому классу физических моделей, например, модели фазового поля, моделей горения, модели сильно затухающих волн с памятью. In this paper we give an overview on some methods that are useful to solve a class of integrodi erential inverse problems. Precisely, we present some methods to solve integrodi erential inverse problems of parabolic type that are based on the theory of analytic semigroups, optimal regularity results and xed point arguments. A large class of physical models can be treated with this procedure, for example phase- eld models, combustion models and the strongly damped wave equation with memory to mention some of them. Фабрицио Коломбо, профессор, кафедра математики, Миланский политехнический университет (г. Милан, Италия), fabrizio.colombo@polimi.it. F. Colombo, Politecnico di Milano, Milan, Italy, fabrizio.colombo@polimi.it