Ukrainian Mathematical Journal最新文献

We study the n-generalized Schützenberger-crossed product from the viewpoint of combinatorial group theory and introduce a new version of this product. For given monoids of this new product, we obtain a representation of the n-generalized Schützenberger-crossed product of arbitrary monoids. In addition, we give necessary and sufficient conditions for the regularity of this product.

We study locally maximal attractors of expanding dynamical systems. Our main result is a representation of these attractors with the help of topological Markov chains corresponding to the Markov partitions of these attractors, which allows us to describe the dynamics of system on them.

Ya. G. Sinai was the first who constructed and used Markov partitions for Anosov’s diffeomorphisms [Funk. Anal. Prilozh., 2, No 1, 64; No 3, 70 (1968); English translation: Funct. Anal. Appl., 2, No 1, 61; No 3, 245 (1968)]. Expanding endomorphisms regarded as the simplest representatives of endomorphisms were first studied by M. Shub [Amer. J. Math., 91, No 1, 175 (1969)]. To construct Markov partitions for expanding endomorphisms, we update Sinai’s approach in the proper way.

A more detailed historical overview can be found in the work by O. M. Sharkovsky [Ukr. Mat. Zh., 74, No. 12, 1709 (2023); English translation: Ukr. Math. J., 74, No. 12, 1950 (2023)]. In this work, Sharkovsky indicated that the methods used to prove the main results presented in [Dokl. Akad. Nauk SSSR, 170, No. 6, 1276 (1966); English translation: Sov. Math. Dokl., 7, No. 5, 1384 (1966)] were, in fact, published in the collection of papers “Dynamical systems and the problems of stability of solutions of differential equations” (1973) issued by the Institute of Mathematics of the Academy of Sciences of Ukraine. This collection is difficultly accessible and was never translated into English. Note that, in the indicated paper, these methods were applied to somewhat different objects. To the best of our knowledge, there is no information about publications of similar results. In view of the outlined history and importance of the described approach (based on Markov partitions and topological Markov chains) for the description of construction of the attractors, it seems reasonable to publish these results anew.

We consider a novel numerical approach for solving boundary-value problems for the second-order Volterra-Fredholm integrodifferential equation with layer behavior and an integral boundary condition. A finite-difference scheme is proposed on suitable Shishkin-type mesh to obtain an approximate solution of the presented problem. It is proved that the method is first-order convergent in the discrete maximum norm. Two numerical examples are included to show the efficiency of the method.

We discuss the construction of SRB measures for some families of stretched Hénon-like maps.

As usual, we denote a 2-dimensional sphere by ({mathbb{S}}^{2}). We study the periods of periodic orbits of the maps f : ({mathbb{S}}^{2}to {mathbb{S}}^{2}) that are either continuous or C¹ with all their periodic orbits being hyperbolic, or transversal, or holomorphic, or transversal holomorphic. For the first time, we summarize all known results on the periodic orbits of these distinct kinds of self-maps on ({mathbb{S}}^{2}) together. We note that every time when a map f : ({mathbb{S}}^{2}to {mathbb{S}}^{2}) increases its structure, the number of periodic orbits provided by its action on the homology increases.

Transitivity, the existence of periodic points, and positive topological entropy can be used to characterize complexity in dynamical systems. It is known that, for every graph that is not a tree and any ε > 0, there exist (complicated) totally transitive maps (then with cofinite set of periods) such that the topological entropy is smaller than ε (simplicity). To numerically measure the complexity of the set of periods, we introduce a notion of the boundary of cofiniteness. Larger boundary of cofiniteness corresponds to a simpler set of periods. We show that, for any continuous circle maps of degree one, every totally transitive (and, hence, robustly complicated) map with small topological entropy has arbitrarily large (simplicity) boundary of cofiniteness.

We consider the difference maximum principle with input data of variable sign and its application to the investigation of the monotonicity and convergence of finite-difference schemes (FDSs). Namely, we consider the Dirichlet initial-boundary-value problem for multidimensional quasilinear parabolic equations with unbounded nonlinearity. Unconditionally monotone linearized finite-difference schemes of the second-order of accuracy are constructed on uniform grids. A two-sided estimate for the grid solution, which is completely consistent with similar estimates for the exact solution, is obtained. These estimates are used to prove the convergence of FDSs in the grid L2-norm. We also present a study aimed at constructing second-order monotone difference schemes for the parabolic convection-diffusion equation with boundary conditions of the third kind and unlimited nonlinearity without using the initial differential equation on the domain boundaries. The goal is a combination of the assumption of existence and uniqueness of a smooth solution and the regularization principle. In this case, the boundary conditions are directly approximated on a two-point stencil of the second order.

We briefly describe some results that evolved from the Sharkovsky theorem.

Numerical bifurcation of a delayed diffusive hematopoiesis model with Dirichlet boundary condition is studied by using a nonstandard finite-difference scheme. We prove that a series of numerical Neimark– Sacker bifurcations appears at the positive equilibrium as the time delay increases. At the same time, the parameter conditions for the existence of numerical Neimark–Sacker bifurcations at the point of positive equilibrium are presented. Finally, we present several examples to verify the accuracy of the accumulated results.

Our main result can be formulated as follows: Consider the set of natural numbers in which the following relation is introduced: n₁ precedes n₂ (n₁ ⪯ n₂) if, for any continuous map of the real line into itself, the existence of a cycle of order n₂ follows from the existence of a cycle of order n₁. The following theorem is true:

Theorem. The introduced relation turns the set of natural numbers into an ordered set with the following ordering: