Vol 68, No 3 (2022): Proceedings of the Crimean Autumn Mathematical School-Symposium

In this paper, we study spaces of vector functions that are holomorphic in the angular domain of the complex plane and with values in a separable Hilbert space. We show that, equipped with the appropriate norms, these spaces are Hilbert spaces.

We call a foliation (M,F) on a manifold M chaotic if it is topologically transitive and the union of closed leaves is dense in M. A foliated manifold M is not assumed to be compact. The chaotic foliations can be considered as multidimensional generalization of chaotic dynamical systems in the sense of Devaney. For foliations covered by fibrations we prove that a foliation is chaotic if and only if its global holonomy group is chaotic. We introduce the concept of the integrable Ehresmann connection for a foliation as a natural generalization of the integrable Ehresmann connection for smooth foliations. A description of the global structure of foliations with integrable Ehresmann connection and a criterion for the chaotic behavior of such foliations are obtained. Applying the method of suspension, a new countable family of pairwise nonisomorphic chaotic foliations of codimension two on 3-dimensional closed and nonclosed manifolds is constructed.

Gradient-like flows on surfaces have simple dynamics, which inspired many mathematicians to search for invariants of their topological equivalence. Under assumptions of different generality on the class of gradient-like flows under consideration, such classical invariants as the Leontovich- Mayer scheme, the Peixoto graph, the equipped Peixoto graph, the two-color Wang graph, the threecolor Oshemkov-Sharko graph, the Fleitas circular scheme, etc. were obtained. Thus, the problem of classifying gradient-like flows on surfaces from the point of view of topological equivalence has been solved in an exhaustive way. In recent works by Kruglov, Malyshev, and Pochinka, it was proved that for gradient-like flows the topological equivalence classes coincide with the topological conjugacy classes. The obtained result allows us to use any invariants of their equivalence for topological conjugacy of gradient-like flows. The present study is a review of the results on topological conjugacy of gradient-like flows on surfaces and efficient algorithms for its distinguishing, that is, algorithms whose running time is limited by some polynomial on the length of the input information.

We consider the system of reaction-diffusion-advection equations describing the evolution of the spatial distributions of two populations of predators and two prey populations. This model allows us to consider directed migration, the Holling functional response of the second kind, and the hyperbolic prey growth function. We obtain conditions on the parameters under which cosymmetries exist. As a result, multistability is realized, i.e., the one- and two-parameter families of stationary solutions appear. For a homogeneous environment, we analytically derive explicit formulas for equilibria. With a heterogeneous habitat, we computed distributions of species using the method of lines and the scheme of staggered grids. We present the results of violation of cosymmetry and transformation of the family in the case of invasion of a predator.