We find necessary and sufficient conditions for a foliation of codimension $q$ on $n$-dimensional manifold with transverse linear connection to admit a transverse invariant pseudo-Riemannian metric of a given signature which is parallel with the respect to the indicated connection. In particular, we obtain a criterion for a foliation with transverse linear connection to be Riemannian foliation.
In the paper we construct some example of smooth dieomorphism of closed manifold. This dieomorphism has one-dimensional (in topological sense) basic set with stable invariant manifold of arbitrary nonzero dimension and the unstable invariant manifold of arbitrary dimension not less than two. The basic set has a saddle type, i.e. is neither attractor nor repeller. In addition, it follows from the construction that the dieomorphism has a positive entropy and is conservative (i.e. its jacobian equals one) in some neighborhood of the one-dimensional solenoidal basic set. The construction represented in this paper allows to construct a dieomorphism with the properties stated above on the manifold that is dieomorphic to the prime product of the circle and the sphere of codimension one
In the paper a new method of constructing of three-dimensional flow systems with different chaotic attractors is presented. Using this method, an example of three-dimensional system possessing an asymmetric Lorenz attractor is obtained. Unlike the classical Lorenz attractor, the observed attractor does not have symmetry. However, the discovered asymmetric attractor, as well as classical one, belongs to a class of <<true>> chaotic, or, more precise, pseudohyperbolic attractors; the theory of such attractors was developed by D.~Turaev and L.P.~Shilnikov. Any trajectory of a pseudohyperbolic attractor has a positive Lyapunov exponent and this property holds for attractors of close systems. In this case, in contrast to hyperbolic attractors, pseudohyperbolic ones admit homoclinic tangencies, but bifurcations of such tangencies do not lead to generation of stable periodic orbits. In order to find the non-symmetric Lorenz attractor we applied the method of <<saddle chart>>. Using diagrams of maximal Lyapunov exponent, we show that there are no stability windows in the neighborhood of the observed attractor. In addition, we verify the pseudohyperbolicity for the non-symmetric Lorenz attractor using the LMP-method developed quite recently by Gonchenko, Kazakov and Turaev.
We study a structure of four-dimensional phase space decomposition on trajectories of Morse-Smale flows admitting heteroclinical intersections. We consider a class $G(S^4)$ of Morse-Smale flows on the sphere $S^4$ such that for any flow $f\in G(S^4)$ its non-wandering set consists of exactly four equilibria: source, sink and two saddles. Wandering set of such flows contains finite number of heteroclinical curves generating the intersection of invariant manifolds of saddle equilibria. We describe a topology of embedding of invariant manifolds of saddle equilibria that is the first step in a solution of topological classification problem.
In this paper we consider a class Phi of diffeomorphisms of a closed n -dimensional manifold that are bifurcation points of simple arcs in the space of diffeomorphisms. The authors have studied the asymptotic properties and the embedding structure of invariant manifolds of non-wandering points of such diffeomorphisms.
For three-dimensional dynamical systems with continuous time (flows), a classification of strange homoclinic attractors, which goes back to the papers S.V. Gonchenko, D.V. Turaev A.L. Shilnikov and L.P. Shilnikov, is proposed. By homoclinic we mean strange attractors containing a specific saddle equilibrium together with its unstable manifold. Moreover, the type of such an attractor is determined by the eigenvalues of the equilibrium. The classification of homoclinic attractors is based on a bifurcation analysis of systems of the form $\dot x=y+g_1(x,y,z), \dot y=z+g_2(x,y,z), \dot z=Ax+By+Cz+g_3(x,y,z), \;\; g_i(0,0,0) = (g_i)^\prime_x(0,0,0) = (g_i)^\prime_y(0,0,0) = (g_i)^\prime_z(0,0,0) = 0, \; i = 1, 2, 3$, whose linearization matrix is represented in the Frobenius form, and the eigenvalues are determined by the coefficients $A, B$ and $C$. In the parameters space $A, B$ and $C$, an extended bifurcation diagram is constructed, where 7 regions corresponding to attractors of various types are distinguished. It is noted that a wide class of three-dimensional flows can be reduced to the class of systems under consideration. The paper also discusses problems related to the pseudohyperbolicity of homoclinic attractors of three-dimensional flows (stability of chaotic dynamics to changes of system parameters). It is proved that in three-dimensional flows only two types of homoclinic attractors can be pseudo-hyperbolic: Lorenz-like attractors containing a saddle equilibrium state with a two-dimensional stable manifold whose saddle value is positive; as well as Shilnikov saddle attractors containing a saddle equilibrium with a two-dimensional unstable manifold.
In this paper 2n-gons and surfaces obtained through identification of 2n-gon's sides in pairs (i.e. through sewing) are considered. As well-known, one can get surface of any genus and orientability through sewing but it's very uneasy to calculate by only the polygon and the way of sewing, because to do this one need to calculate the number of vertices appearing after identification; even for small n the problem is almost impossible if one want to do this directly. There are different ways to solve the task. The canonical variant of 4q-gon sewing (2q-gon sewing) giving an orientable (unorientable) surface of genus q is well-known, as the Harer-Zagier's numbers, that are the numbers of variants of sewing a 2n-gon to an orientable surface of gunes q. In this paper we offer a new way of Euler characteristic's of obtained surface calculation (and, hence, its genus) undepending on its orientability by means of three-colour graph and information about closed surfaces topological classification.
This paper is the first step in stydying structure of decomposition of phase space with dimension n≥4" role="presentation" style="position: relative;">n≥4n≥4n\geq 4 on the trajectories of Morse-Smale flows (structurally stable flows with non-wandering set consisting of finite number of equilibria and closed trajectories) allowing heteroclinic intersections. More precisely, special class of Morse-Smale flows on the sphere Sn" role="presentation" style="position: relative;">SnSnSn is studied. The non-wandering set of the flow of interest consists of two nodal and two saddle equilibrium states. It is proved that for every flow from the class under consideration the intersection of invariant manifolds of two different saddle equilibrium states is nonempty and consists of a finite number of connectivity components. Heteroclinic intersections are mathematical models for magnetic field separators. Study of their structure, as well as the question of their existence, is one of the principal problems of magnetic hydrodynamics.
The subject of this article is a review of the results on foliations with transversal linear connection obtained by the author together with N.I. Zhukova, and their comparison with the results of other authors. The work consists of three parts. The first part focuses on to automorphism groups of foliations with a transversal linear connection in the category of foliations. In the second part, the question of the equivalence of the concept of completeness for the class of foliations under investigation is studied. The third part we present theorems on pseudo-Riemannian foliations that form an important class of foliations with a transversal linear connection.In particular, we present results on graphs of pseudo-Riemannian foliations that contain all information about foliations.
The vertex 3-colourability problem is to determine for a given graph whether one can divide its vertex set into three subsets of pairwise non-adjacent vertices. This problem is NP-complete in the class of planar graphs, but it becomes polynomial-time solvable for planar triangulations, i.e. planar graphs, all facets of which (including external) are triangles. Additionally, the problem is NP-complete for planar graphs whose vertices have degrees at most 4, but it becomes linear-time solvable for graphs whose vertices have maximal degree at most 3. So it is an interesting question to nd a threshold for lengths of facets and maximum vertex degree, for which the complexity of the vertex 3-colourability changes from polynomial-time solvability to NP-completeness. In this paper we answer this question and prove NP-completeness of the vertex 3-colourability problem in the class of planar graphs of the maximum vertex degree at most 5, whose facets are triangles and quadrangles only.