Envelope singularities of families of planes in control theory

Generic singularities of the local transitivity set for control systems with nonconvex indicatrix on three-dimensional manifolds are classified. A simple recognition criterion for generic singularities of envelopes of bi-tangents to space curves is described. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 262, pp. 73–86.     

Convex hulls of surfaces with boundaries and corners and singularities of transitivity zone in ℝ<Superscript>3</Superscript>

Generic singularities of the boundary of the local transitivity set of a control system on two- and three-dimensional manifolds are classified. The indicatrices of the system are assumed to be given by generic equations and inequalities.     

Convex hulls of indicatrices and singularities of the transitivity zone in {{\mathbb{R}}^3}

We consider a control system on two- and three-dimensional manifolds, the indicatrices of which are the images of smooth mappings, and present a survey on classification of the generic singularities of the boundary of the local transitivity set.     

Structural stability of simplest dynamical inequalities

The structural stability of families of orbits is proved for the simplest generic smooth dynamical inequality in the plane with bounded complement of the domain of complete controllability. Typical singularities of the boundaries of nonlocal transitivity zones for such inequalities are found. The stability of these singularities under small perturbations of the generic inequality is proved.     

Generic singularities of implicit systems of first order differential equations on the plane

For the implicit systems of first order ordinary differential equations on the plane there is presented the complete local classification of generic singularities of family of its phase curves up to smooth orbital equivalence. Besides the well-known singularities of generic vector fields on the plane and the singularities described by a generic first order implicit differential equations, there exists only one generic singularity described by the implicit first order equation supplied by Whitney umbrella surface generically embedded to the space of directions on the plane.     

Local controllability of generic dynamic inequalities

Sufficient conditions for small time local transitivity and for the absence of local controllability of a generic dynamic inequality with locally bounded derivatives are presented. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 21, Geometric Problems in Control Theory, 2004.     

Perestroikas of Convexified Apparent Contours and Global Phase Diagrams in Thermodynamics

We obtain the classification of singularities occurring in families of convex hulls of apparent contours up to codimension 3. The results for codimension 2 singularities allow us to supplement Varchenko's classification of local singularities of thermodynamic phase diagrams of binary mixtures. Singularities of three-parameter families specify so-called global phase diagrams in three-dimensional parameter spaces and define all local perestroikas of phase diagrams in generic one-parameter families of binary mixtures.     

Singularités de solutions d'équations aux dérivées partielles

We develop the general mathematical setting necessary to study the singularities of local solutions of the quasi-linear first-order systems of PDEs with a free initial condition. In the good cases, it is possible to describe these singularities as a function of the free initial conditions satisfied by the solutions. Using the transversality theorems, it is then possible to describe the singularities of generic solutions, and of generic families of solutions under deformation of the initial conditions. We apply this study by giving classifications of an important classe of hyperbolic quasi-linear first-order systems in the plane, the reducible systems, and of an almost general class of hyperbolic quasi-linear second-order equations in the plane.     

Polynomial normal forms of constrained differential equations with three parameters

We study generic constrained differential equations (CDEs) with three parameters, thereby extending Takens's classification of singularities of such equations. In this approach, the singularities analyzed are the Swallowtail, the Hyperbolic, and the Elliptic Umbilics. We provide polynomial local normal forms of CDEs under topological equivalence. Generic CDEs are important in the study of slow–fast (SF) systems. Many properties and the characteristic behavior of the solutions of SF systems can be inferred from the corresponding CDE. Therefore, the results of this paper show a first approximation of the flow of generic SF systems with three slow variables.     

Generic Profit Singularities of One-Parameter Cyclic Processes with Discount

On the circle for a smooth one-parameter family of controllable cyclic processes with discount, we classify generic singularities of time-average profit as a function of the parameter. The stability of these singularities with respect to small perturbations of generic families is proved.     

Resolution of Corank 1 Singularities of a Generic Front

We construct a resolution of singularities for wave fronts having only stable singularities of corank 1. It is based on a transformation that takes a given front to a new front with singularities of the same type in a space of smaller dimension. This transformation is defined by the class A_µ of Legendre singularities. The front and the ambient space obtained by the A_µ-transformation inherit topological information on the closure of the manifold of singularities A_µ of the original front. The resolution of every (reducible) singularity of a front is determined by a suitable iteration of A_µ-transformations. As a corollary, we obtain new conditions for the coexistence of singularities of generic fronts.     

Generic bifurcations of low codimension of planar Filippov Systems

In this article some qualitative and geometric aspects of non-smooth dynamical systems theory are discussed. The main aim of this article is to develop a systematic method for studying local (and global) bifurcations in non-smooth dynamical systems. Our results deal with the classification and characterization of generic codimension-2 singularities of planar Filippov Systems as well as the presentation of the bifurcation diagrams and some dynamical consequences.     

Timelike Constant Mean Curvature Surfaces with Singularities

We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz–Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behavior of the surfaces at the big cell boundary, generalize the definition of CMC surfaces to include those with finite, generic singularities, and show how to construct surfaces with prescribed singularities by solving a singular geometric Cauchy problem. The solution shows that the generic singularities of the generalized surfaces are cuspidal edges, swallowtails, and cuspidal cross caps.     

Oriented bordism of codimension one immersions of 7-manifolds and relative Thom polynomials

We give a formula to detect the oriented bordism class of a codimension one immersion of an oriented 7-manifold in terms of singularities of its singular Seifert surface, that is, a generic map from a compact 8-manifold which extends the given immersion. Our argument involves a study of a relative version of the Thom polynomials for certain singularities of generic maps from manifolds with non-empty boundaries.     

n维Anti-de Sitter空间中类空超曲面的类时Gauss像的奇点

本文利用奇点理论研究了n维Anti-deSitter空间中的类空超曲面,介绍了类空超曲面的局部微分几何,定义了类时Anti-deSitterGauss像及Anti-deSitter高度函数,并进一步的利用Anti-deSitter高度函数族和流形间的切触理论研究了类时Anti-deSitterGauss像的几何意义及类空超曲面的通有性.最后研究了类空超曲面的AdS-Monge型.     

On the Monodromy at Infinity of a Polynomial Map, II

To a polynomial map f : Cn + 1 C one can attach a monodromy transformation on the complex cohomology of its generic fiber that reflects its asymptotic behaviour. In this paper this transformation is determined for a class of generic polynomials in terms of data attached to projective hypersurfaces with isolated singularities.     

Singularities of lightcone Gauss images of spacelike hypersurfaces in de Sitter space

We define the notions of lightcone Gauss images of spacelike hypersurfaces in de Sitter space. We investigate the relationships between singularities of these maps and geometric properties of spacelike hypersurfaces as an application of the theory of Legendrian singularities. We classify the singularities and give some examples in the generic case in de Sitter 3-space.     

The Quenching Set of a MEMS Capacitor in Two-Dimensional Geometries

The formation of finite time singularities in a nonlinear parabolic fourth order partial differential equation (PDE) is investigated for a variety of two-dimensional geometries. The PDE is a variant of a canonical model for Micro–Electro Mechanical systems (MEMS). The singularities are observed to form at specific points in the domain and correspond to solutions whose values remain finite but whose derivatives diverge as the finite time singularity is approached. This phenomenon is known as quenching. An asymptotic analysis reveals that the quenching set can be predicted by simple geometric considerations suggesting that the phenomenon described is generic to higher order parabolic equations which exhibit finite time singularity.     

Hereditarily hypercyclic operators and mixing

We show that hereditary transitivity (respectively strongly hereditary transitivity) is equivalent to weak mixing (respectively strong mixing) in a discrete dynamical system with Polish phase space. We also study the connection between local orbit structure and hypercyclicity, and obtain a “local hypercyclicity criterion.”