О числе классов топологической сопряженности диффеоморфизмов Пикстона
For a wide class of dynamical systems known as Pixton diffeomorphisms the topological
conjugacy class is completely defined by the Hopf knot equivalence class, i.e.
the knot whose equivalence class under homotopy of the loops is a generator of the
fundamental group π1(S2×S1).Moreover, any Hopf knot can be realized by a Pixton
diffeomorphism. Nevertheless, the number of the classes of topological conjugacy of
these diffeomorphisms is still unknown. This problem can be reduced to finding topological
invariants of Hopf knots. In the present paper we describe a first order invariant
for these knots. This result allows one to model countable families of pairwise nonequivalent
Hopf knots and, therefore, infinite set of topologically non-conjugate Pixton
We suggest a universal map capable of recovering the behavior of a wide range of dynamical systems given by ODEs. The map is built as an artificial neural network whose weights encode a modeled system. We assume that ODEs are known and prepare training datasets using the equations directly without computing numerical time series. Parameter variations are taken into account in the course of training so that the network model captures bifurcation scenarios of the modeled system. The theoretical benefit from this approach is that the universal model admits applying common mathematical methods without needing to develop a unique theory for each particular dynamical equations. From the practical point of view the developed method can be considered as an alternative numerical method for solving dynamical ODEs suitable for running on contemporary neural network specific hardware. We consider the Lorenz system, the Rцssler system and also the Hindmarch – Rose model. For these three examples the network model is created and its dynamics is compared with ordinary numerical solutions. A high similarity is observed for visual images of attractors, power spectra, bifurcation diagrams and Lyapunov exponents.
Using the tools of the Markov Decision Processes, we justify the dynamic programming approach to the optimal impulse control of deterministic dynamical systems. We prove the equivalence of the integral and diﬀerential forms of the optimality equation. The theory is illustrated by an example from mathematical epidemiology. The developed methods can be also useful for the study of piecewise deterministic Markov processes.
Building of adequate dynamical models of microblogging social networks is a topical task that is of interest from both theoretical and practical aspects. Experimental and theoretical results of studies related to choice of the adequate model are presented. The choice was made between two models: a nonlinear dynamical system and a nonlinear random dynamical system. By results of the fractal analysis of observable network time series and defining their probability density function it was established that the nonlinear random dynamical system was more adequate than the nonlinear dynamical system. The character of the observable time series was also explored. The possibility that microblogging social networks can be analyzed by means of Tsallis entropy and self-organized criticality is 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.