The discriminant of quasi m-boundary singularities

Bifurcation diagrams and caustics of simple quasi border singularities

We numerically study some of the three-dimensional dynamical systems which exhibit complete synchronization as well as generalized synchronization to show that these systems can be conveniently partitioned into equivalent classes facilitating the study of bifurcation diagrams within each class. We demonstrate how bifurcation diagrams may be helpful in predicting the nature of the driven system by knowing the bifurcation diagram of driving system and vice versa. The study is extended to include the possible generalized synchronization between elements of two different equivalent classes by taking the Rossler-driven-Lorenz-system as an example.

<p>We describe the generators of the vector fields tangent to the bifurcation diagrams and caustics of simple quasi boundary singularities. As an application, submersions on the pair $ (G, B) $, which consists of a cuspidal edge $ G $ in $ \mathbb{R}^3 $ that contains a distinguishing regular curve $ B $, are classified. This classification was used as a means to investigate the contact that a general cuspidal edge $ G $ equipped with a regular curve $ B\subset G $ has with planes. The singularities of the height functions on $ (G, B) $ are discussed and they are related to the curvatures and torsions of the distinguished curves on the cuspidal edge. In addition to this, the discriminants of the versal deformations of the submersions that were accomplished are described and they are related to the duality of the cuspidal edge.</p>

The crease flow, replacing the Hamiltonian system used for the evolution of crease sets on black hole horizons, is introduced and its bifurcation properties for null hypersurfaces are discussed. We state the conditions of nondegeneracy and typicality for the crease submanifolds, and find their normal forms and versal unfoldings (codimension 3). The allowed boundary singularities are thus prescribed by the Arnold–Kazaryan–Shcherbak theorem for 3-parameter versal families, and hence identified as swallowtails and Whitney umbrellas of particular kinds. We further present the bifurcation diagrams describing crease evolution at the crossings of the bifurcation sets and elsewhere, and a typical example is studied. Some remarks on the connection of these results to the crease evolution on black hole horizons are also given.

In this paper an expository account on singularities of reversible vector fields on manifolds and boundary singularities is presented. Also we present the bifurcation diagram of a boundary cusp of codimension three, i.e, a Bogdanov-Takens singular point in the boundary of the semi plane {(x, y)∈R2:x>-0} whose topological unfolding is given by the quadratic three parameter family\(y\frac{\partial }{{\partial x}} + (x^2 + ax + c + \alpha x(x + b))\frac{\partial }{{\partial y}}\),\(\alpha = \pm 1\). This study can be applied to the analysis of the behavior of singularity of the germ of vector fieldX0(x, y)=(y, 2x(x4+x2y)) in the class of reversible vector fields.

In the absence of an active dynamo, purely poloidal magnetic field configurations are unstable to large-scale dynamical perturbations, and decay via reconnection on an Alfvenic time-scale. Nevertheless, a number of classes of dynamo-free stars do exhibit significant, long-lived, surface magnetic fields. Numerical simulations suggest that the large-scale poloidal field in these systems is stabilized by a toroidal component of the field in the stellar interior. Using the principle of conservation of total helicity, we develop a variational principle for computing the structure of the magnetic field inside a conducting sphere surrounded by an insulating vacuum. We show that, for a fixed total helicity, the minimum energy state corresponds to a force-free configuration. We find a simple class of axisymmetric solutions, parametrized by angular and radial quantum numbers. However, these solutions have a discontinuity in the toroidal magnetic field at the stellar surface which will exert a toroidal stress on the surface of the star. We then describe two other classes of solutions, the standard spheromak solutions and ones with fixed surface magnetic fields, the latter being relevant for neutron stars with rigid crusts. We discuss the implications of our results for the structure of neutron star magnetic fields, the decay of fields, and the origin of variability and outbursts in magnetars.

The Gauss–Codazzi equations imposed on the elements of the first and the second quadratic forms of a surface embedded in \(\mathbb{R}^{3} \) are integrable by the dressing method. This method allows constructing classes of Combescure-equivalent surfaces with the same “rotation coefficients.” Each equivalence class is defined by a function of two variables (“master function of a surface”). Each class of Combescure-equivalent surfaces includes the sphere. Different classes of surfaces define different systems of orthogonal coordinates of the sphere. The simplest class (with the master function zero) corresponds to the standard spherical coordinates.

Kinnersley's SU(2, 1) theory is investigatecl to extract all generating transformations by introducing an equivalence relation into the solutions of the Einstein-l'vfaxwell equations. It is shown that all equivalence classes of generated solutions are obtained by using two simple SU(2, 1) matrices Uo, U,. vVhen Kinnersley's functions u, v, ware linearly dependent, we fmd that solutions of Kinnersley's equations are divided into three classes and that within one class all solutions are generated, starting with a given solution, by means of the matrices Uo, U,. The Bonner-MPST solution is generalized to a five-parameter solution with recourse to the matrix Uo. The form of the solution is simpler than that of Esposito and 'Witten's solution. A special solution is obtained which describes a system with mass 2rn, electric: charge 2e, n1agnetic dipole 2rndm, and angular n1ornentun1 2edm. § I. Introduction Recently, Kinncrsley1J found that the Einstein-Maxwell equations possess SU(2, 1) symmetry under the condition that the metric admits at least one nonnull Killing vector ( vvhich vvas chosen to be time-like in his paper). Owing to this symmetry, every SU(2, 1) matrix can be used to generate new solutions from old ones. Kinnersley ga\'e five simple classes (I)~ (V) o£ SU(2, 1) transforma­ tions. Then he sho·wecl that first three classes (I)~ (III) generate the solutions which are essentially the same as old ones, and that only last t\VO classes (IV) and (V) generate new solutions. Using the transformations (IV) and (V) Esposito and vVitten') derived a five-parameter solution from the Bonner-.IVIPST'),,) solution.

The stability and Hopf bifurcation of a class of flywheel governor systems subject to stochastic excitation under time-delay feedback are studied. The model is simplified using center manifold reduction to obtain a paradigm for a two-dimensional central subsystem. The system converges asymptotically to an Itô stochastic differential equation based on polar transformations and stochastic averaging method. The stochastic stability of the system is analyzed by applying the maximum Lyapunov exponent and the singular boundary theory, and its probability density function is calculated. In addition, it is found through Monte Carlo numerical simulations that the effects of noise intensity and time-delay on the stochastic P-bifurcation of the system are opposite, and proper noise intensity makes the flywheel governor system converge to the equilibrium position, while a large time-delay leads to stochastic bifurcation and makes the flywheel governor system unstable. Finally, the two-parameter bifurcation diagram is used to further analyze the impact of time-delay and noise intensity on the bifurcation of the system under variable load torque. It is found that the influence of noise intensity and time-delay on the periodic oscillation state is the same, and the stability of the flywheel governor system will decrease with both of their increases. This also further demonstrates the duality of the impact of noise intensity on the same system under different conditions. The above research provides better theoretical references for the design and safe operation of speed control systems.

A chord is a straight line joining two points of a pair of hypersurfaces in an affine space such that the tangent hyperplanes at these points are parallel. We classify the singularities of envelopes of the families of chords determined by generic pairs of plane curves and surfaces in three-space. The list contains all bifurcation diagrams of simple boundary singularities (of the corresponding multiplicity).

We seek discrete approximations to solutions $u:\Omega\to\mathbb{R}$ of semilinear elliptic PDE of the form $\Delta u+f_s(u)=0$, where $f_s$ is a one-parameter family of nonlinear functions and $\Omega$ is a domain in $\mathbb{R}^d$. The main achievement of this paper is the approximation of solutions to the PDE on the cube $\Omega=(0,\pi)^3\subseteq\mathbb{R}^3$. There are 323 possible isotropy subgroups of functions on the cube, which fall into 99 conjugacy classes. The bifurcations with symmetry in this problem are quite interesting, including many with 3-dimensional critical eigenspaces. Our automated symmetry analysis is necessary with so many isotropy subgroups and bifurcations among them, and it allows our code to follow one branch in each equivalence class that is created at a bifurcation point. Our most complicated result is the complete analysis of a degenerate bifurcation with a 6-dimensional critical eigenspace. This article extends our work in [Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), pp. 2531--2556], wherein we combined symmetry analysis with modified implementations of the gradient Newton--Galerkin algorithm (GNGA [J. M. Neuberger and J. W. Swift, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), pp. 801--820]) to automatically generate bifurcation diagrams and solution graphics for small, discrete problems with large symmetry groups. The code described in the current paper is efficiently implemented in parallel, allowing us to investigate a relatively fine-mesh discretization of the cube. We use the methodology and corresponding library presented in our previous paper [Int. J. Parallel Program., 40 (2012), pp. 443--464].

We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse X ↘ Y of finite spaces induces a simplicial collapse K ( X ) ↘ K ( Y ) of their associated simplicial complexes. Moreover, a simplicial collapse K ↘ L induces a collapse X ( K ) ↘ X ( L ) of the associated finite spaces. This establishes a one-to-one correspondence between simple homotopy types of finite simplicial complexes and simple equivalence classes of finite spaces. We also prove a similar result for maps: We give a complete characterization of the class of maps between finite spaces which induce simple homotopy equivalences between the associated polyhedra. This class describes all maps coming from simple homotopy equivalences at the level of complexes. The advantage of this theory is that the elementary move of finite spaces is much simpler than the elementary move of simplicial complexes: It consists of removing (or adding) just a single point of the space.

Numerous studies have explored the special mathematical properties of the diatonic set. However, much less attention has been paid to the sets associated with the other scales that play an important role in Western tonal music, such as the harmonic minor scale and ascending melodic minor scale. This paper focuses on the special properties of the class, τ, of sets associated with the major and minor scales (including the harmonic major scale). It is observed that τ is the set of pitch class sets associated with the shortest simple pitch class cycles in which every interval between consecutive pitch classes is either a major or a minor third, and at least one of each type of third appears in the cycle. Employing Rothenberg's definition of stability and propriety, τ is also the union of the three most stable inversional equivalence classes of proper 7-note sets. Following Clough and Douthett's concept of maximal evenness, a method of measuring the evenness of a set is proposed and it is shown that τ is also the union of the three most even 7-note inversional equivalence classes.

Matrices permutation equivalent to primitive matrices

We consider the classification of germs of functions up to a nonstandard equivalence relation similar to the quasi boundary equivalence and quasi equivalence of projections recently introduced by the second author. In fact, it is more rough than the classification of functions with respect to the group of diffeomorphisms preserving a corner (that is, a union of a pair of transversal hypersurfaces). We present the list of all simple classes and discuss its relation to the singularities of Lagrangian projections with corners. Also, we describe the bifurcation diagrams and caustics of simple quasi corner singularities.