CONDITIONS FOR TOPOLOGICAL EQUIVALENCE OFLOCAL QUALITATIVE SINGULARITIES OF DYNAMICALSYSTEMS WITH IMPACT INTERACTIONS

For the introduced class of dynamical systems with impact interactions, local singularities (six types) are determined. Properties that allow us to prove the topological equivalence of these singularities are described for them. A number of unsolved problems are formulated, which are adjacent to the problems considered in the article.

Bifurcations of the Riccati Quadratic Polynomial Differential Systems

In this paper, we characterize the global phase portrait of the Riccati quadratic polynomial differential system [Formula: see text] with [Formula: see text], [Formula: see text] nonzero (otherwise the system is a Bernoulli differential system), [Formula: see text] (otherwise the system is a Liénard differential system), [Formula: see text] a polynomial of degree at most [Formula: see text], [Formula: see text] and [Formula: see text] polynomials of degree at most 2, and the maximum of the degrees of [Formula: see text] and [Formula: see text] is 2. We give the complete description of the phase portraits in the Poincaré disk (i.e. in the compactification of [Formula: see text] adding the circle [Formula: see text] of the infinity) modulo topological equivalence.

Topological Equivalence Classification of Balanced Linearly Separable Boolean Functions on n-Dimensional Hypercube

The concept of conformal transformation is proposed through the study of the spatial structure of [Formula: see text]-dimensional hypercubes. Based on conformal transformation, a novel algorithm, called topological equivalence classification algorithm, is proposed for classifying balanced linearly separable Boolean functions. By the proposed algorithm, the topological equivalence classes of all balanced linearly separable Boolean functions and the number of Boolean functions in each of the topological equivalence classes are obtained. In addition, the properties of conformal transformation also show an application prospect for decomposing nonlinearly separable Boolean functions.

Expansive flows on uniform spaces

<p style='text-indent:20px;'>In this paper we study several dynamical properties on uniform spaces. We define expansive flows on uniform spaces and provide some equivalent ways of defining expansivity. We also define the concept of expansive measures for flows on uniform spaces. We prove for flows on compact uniform spaces that every expansive measure vanishes along the orbits and has no singularities in the support. We also prove that every expansive measure for flows on uniform spaces is aperiodic and is expansive with respect to time-<inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula> map. Furthermore we show that every expansive measure for flows on compact uniform spaces maintains expansive under topological equivalence.</p>

Bimeron clusters in chiral antiferromagnets

A magnetic bimeron is an in-plane topological counterpart of a magnetic skyrmion. Despite the topological equivalence, their statics and dynamics could be distinct, making them attractive from the perspectives of both physics and spintronic applications. In this work, we demonstrate the stabilization of bimeron solitons and clusters in the antiferromagnetic (AFM) thin film with interfacial Dzyaloshinskii–Moriya interaction (DMI). Bimerons demonstrate high current-driven mobility as generic AFM solitons, while featuring anisotropic and relativistic dynamics excited by currents with in-plane and out-of-plane polarizations, respectively. Moreover, these spin textures can absorb other bimeron solitons or clusters along the translational direction to acquire a wide range of Néel topological numbers. The clustering involves the rearrangement of topological structures, and gives rise to remarkable changes in static and dynamical properties. The merits of AFM bimeron clusters reveal a potential path to unify multibit data creation, transmission, storage, and even topology-based computation within the same material system, and may stimulate spintronic devices enabling innovative paradigms of data manipulations.

Complete transition diagrams of generic Hamiltonian flows with a few heteroclinic orbits

We study the transition graph of generic Hamiltonian surface flows, whose vertices are the topological equivalence classes of generic Hamiltonian surface flows and whose edges are the generic transitions. Using the transition graph, we can describe time evaluations of generic Hamiltonian surface flows (e.g., fluid phenomena) as walks on the graph. We propose a method for constructing the complete transition graph of all generic Hamiltonian flows. In fact, we construct two complete transition graphs of Hamiltonian surface flows having three and four genus elements. Moreover, we demonstrate that a lower bound on the transition distance between two Hamiltonian surface flows with any number of genus elements can be calculated by solving an integer programming problem using vector representations of Hamiltonian surface flows.