A relation between the curvature ellipse and the curvature parabola

2019 ◽  
Vol 19 (3) ◽  
pp. 389-399
Author(s):  
P. Benedini Riul ◽  
R. Oset Sinha
Keyword(s):  
Singular Point ◽  
Singular Surface ◽  
Second Order ◽  
Regular Surface ◽  
Normal Plane ◽  
Tangent Direction ◽  
Curvature Ellipse

Abstract At each point in an immersed surface in ℝ4 there is a curvature ellipse in the normal plane which codifies all the local second order geometry of the surface. Recently, at the singular point of a corank 1 singular surface in ℝ3, a curvature parabola in the normal plane which codifies all the local second order geometry has been defined. When projecting a regular surface in ℝ4 to ℝ3 in a tangent direction, corank 1 singularities appear generically. The projection has a cross-cap singularity unless the direction of projection is asymptotic, where more degenerate singularities can appear. In this paper we relate the geometry of an immersed surface in ℝ4 at a certain point to the geometry of the projection of the surface to ℝ3 at the singular point. In particular we relate the curvature ellipse of the surface to the curvature parabola of its singular projection.

scholarly journals Stability on the basis of Orthogonal Trajectories

1965 ◽  
Vol 8 (5) ◽  
pp. 647-658
Author(s):  
T. A. Burton
Keyword(s):  
Singular Point ◽  
Differential Equations ◽  
Second Order ◽  
System Of Differential Equations ◽  
Partial Derivatives ◽  
Connected Set ◽  
Image Position ◽  
Simply Connected

We consider a system of differential equations of second order given by1(' = d/dt) where P and Q have continuous first partial derivatives with respect to x and y in some open and simply connected set R containing O = (0, 0) which we assume to be the only singular point in R. In fact, let R be the whole plane; for if not then the following discussion can be modified.

Решения гамильтоновой системы с двумерным управлением в окрестности особой экстремали второго порядка

2021 ◽  
Vol 76 (5(461)) ◽  
pp. 201-202
Author(s):  
Мария Игоревна Ронжина ◽  
Mariya Igorevna Ronzhina ◽  
Лариса Анатольевна Манита ◽  
Larisa Anatol'evna Manita ◽  
Лев Вячеславович Локуциевский ◽  
...  
Keyword(s):  
Singular Point ◽  
Hamiltonian System ◽  
Infinite Number ◽  
Finite Time ◽  
Second Order ◽  
Two Dimensional ◽  
Optimal Solutions ◽  
Countable Number ◽  
Bounded Control ◽  
Singular Extremal

We consider a Hamiltonian system that is affine in two-dimensional bounded control that takes values in an ellipse. In the neighborhood of a singular extremal of the second order, we find two families of optimal solutions: chattering trajectories that attain the singular point in a finite time with a countable number of control switchings, and logarithmic-like spirals that reach the singular point in a finite time and undergo an infinite number of rotations.

scholarly journals A connection between blowing-up and gluings in one-dimensional rings

1984 ◽  
Vol 94 ◽  
pp. 75-87 ◽  
Author(s):  
Grazia Tamone
Keyword(s):  
Singular Point ◽  
Singular Surface ◽  
One Dimensional ◽  
Closed Point ◽  
Blowing Up

Let C be an affine curve, contained on a non-singular surface X as a closed 1-dimensional subscheme. If P is a closed point on C, the blowing-up C′ of C with center P (induced by the blowing-up of X with center P) is an affine curve. It is known that there is a sequence:where C is the normalization of C, and each Ci + 1 is the blowing-up of Ci with center a singular point Pt on Ci (i = 0, …, k – 1).

ON THE ASYMPTOTIC EXPANSIONS OF NUMERICAL SOLUTIONS TO SECOND ORDER DIFFERENTIAL EQUATIONS WITH A REGULAR SINGULAR POINT

2001 ◽  
Vol 11 (01) ◽  
pp. 163-177
Author(s):  
RICHARD WEISS ◽  
FRANK R. de HOOG ◽  
ROBERT S. ANDERSSEN
Keyword(s):  
Differential Equation ◽  
Singular Point ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Approximation ◽  
Numerical Solutions ◽  
Second Order ◽  
Regular Singular Point ◽  
Second Order Differential Equations ◽  
Uniform Asymptotic Expansion

When difference schemes with uniformly spaced gridpoints are applied to second order ordinary differential equations with a regular singular point, it is often the case that the resulting numerical approximation does not have a uniform asymptotic expansion. As a consequence, postprocessing, such as h2-extrapolation is not an option. This paper examines the cause of this phenomenon and finds that the existence of such expansions requires the discretization of the boundary conditions at the singular point to be compatible with the discretization of the differential equation. In addition, it is shown how an understanding of the need for compatible discretization can assist in the construction of schemes for several classes of equations that arise when symmetry is used to reduce partial differential equations to ordinary differential equations with a regular singular point.

Stability and Response of Systems Satisfying a Second-Order Linear Differential Equation with Time-Dependent Coefficients

1957 ◽  
Vol 8 (1) ◽  
pp. 78-86
Author(s):  
A. W. Babister
Keyword(s):  
Differential Equation ◽  
Singular Point ◽  
Linear Differential Equation ◽  
Second Order ◽  
Time Dependent ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
Oscillatory Nature ◽  
Linear Differential ◽  
Image Position

SummaryThe differential equation considered iswhere all the a’s and b’s are real constants.The nature of the solution is investigated in the neighbourhood of the singular point and the conditions are found for logarithmic terms to be absent.The conditions for stability for large values of τ are determined; the system is stable ifare all positive for large values of τ.The form of the response is considered and its oscillatory (or non-oscillatory) nature investigated. The Sonin-Polya theorem is used to determine simple inequalities which must hold between the coefficients of the differential equation in any interval for the relative maxima of | x | to form an increasing or decreasing sequence in that interval.

New Double Bifurcation of Nilpotent Focus

2021 ◽  
Vol 31 (04) ◽  
pp. 2150053
Author(s):  
Feng Li ◽  
Hongwei Li ◽  
Yuanyuan Liu
Keyword(s):  
Hopf Bifurcation ◽  
Singular Point ◽  
Limit Cycles ◽  
Second Order ◽  
Singular Points ◽  
Weak Focus ◽  
Bifurcation Phenomenon ◽  
Planar Systems ◽  
Nilpotent Singular Point

In this paper, a new bifurcation phenomenon of nilpotent singular point is analyzed. A nilpotent focus or center of the planar systems with 3-multiplicity can be broken into two complex singular points and a second order elementary weak focus. Then, two more limit cycles enclosing the second order elementary weak focus can bifurcate through the multiple Hopf bifurcation.

scholarly journals Of one over determined system differential equation at private derivative second order with one singular point and one singular line

2021 ◽  
pp. 14-18
Author(s):  
B. M. Shoimkulov ◽  
Keyword(s):  
Partial Differential Equations ◽  
Singular Point ◽  
Differential Equations ◽  
Compatibility Condition ◽  
Second Order ◽  
Integral Representations ◽  
Singular Line ◽  
Cauchy Type ◽  
National Academy Of Sciences ◽  
Partial Differential

In this paper, a over determined system of second-order partial differential equations with one singular point and one singular line is investigated. A compatibility condition is found for over determined systems of second-order partial differential equations with one singular point and one singular line. If the compatibility condition is met, integral representations of the variety of solutions are found explicitly in terms of three arbitrary constants, when the singular line is in the boundaries of the domain for which initial data problems (Cauchy-type Problems) can be set. In this paper considers a redefined system of second-order partial differential equations, when the coefficients and right parts have one singular point and one singular line. Obtaining a variety of solutions and studying boundary value problems for linear differential equations of the hyperbolic type of the second order, some linear redefined systems of the first and second order with one and two supersingular lines and supersingular points is devoted to the monograph of academician of the National Academy of Sciences of the Republic of Tatarstan Rajabov N. - 1992 "Introduction to the theory of partial differential equations with supersingular coefficients" [6, p.126]. Using the obtained results of The monograph of Rajabov N., a variety of solutions of redefined systems of partial differential equations of the second order with one singular point and one singular line in an explicit form, through three arbitrary constants, was found.

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