scholarly journals A robustness measure for singular point and index estimation in discretized orientation and vector fields

PAMM ◽  
2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Karl B. Hoffmann ◽  
Ivo F. Sbalzarini
Keyword(s):  
Singular Point ◽  
Vector Fields ◽  
Robustness Measure ◽  
Index Estimation

Quantization of Bending Deformations of Polygons In , Hypergeometric Integrals and the Gassner Representation

2001 ◽  
Vol 44 (1) ◽  
pp. 36-60 ◽  
Author(s):  
Michael Kapovich ◽  
John J. Millson
Keyword(s):  
Singular Point ◽  
Lie Algebra ◽  
Moduli Space ◽  
Braid Group ◽  
Vector Fields ◽  
Flat Connection ◽  
Hamiltonian Action ◽  
Pure Braid Group ◽  
Hypergeometric Integrals

AbstractThe Hamiltonian potentials of the bending deformations of n-gons instudied in [KM] and [Kly] give rise to a Hamiltonian action of the Malcev Lie algebra𝓟nof the pure braid groupPnon the moduli spaceMrofn-gon linkages with the side-lengthsr = (r1, … , rn)in. Ife∈Mris a singular point wemay linearize the vector fields in𝓟nate. This linearization yields a flat connection ∇ on the spaceof n distinct points on. We show that the monodromy of ∇ is the dual of a quotient of a specialized reduced Gassner representation.

scholarly journals Local Bifurcations of Reversible Piecewise Smooth Planar Dynamical Systems

2020 ◽  
pp. 1-15
Author(s):  
V. Sh. Roitenberg
Keyword(s):  
Dynamical System ◽  
Vector Field ◽  
Dynamical Systems ◽  
Singular Point ◽  
Vector Fields ◽  
Singular Points ◽  
Piecewise Smooth ◽  
Smooth Vector ◽  
Local Bifurcations ◽  
Piecewise Smooth Vector Fields

There are quite a few works, which consider local bifurcations of piecewise-smooth vector fields on the plane. A number of papers also studied  the local bifurcations of smooth vector fields on the plane that are reversible with respect to involution. In the paper, we introduce reversible dynamical systems defined by piecewise-smooth vector fields on the coordinate plane (x, y) for which the discontinuity line y = 0 coincides with the set of fixed points of the system involution. We consider the generic one-parameter perturbations of such a vector field. The bifurcations of the singular point O lying on this line are described in two cases. In the first case, the point O is a rough saddle of the smooth vector fields that coincide with a piecewise smooth vector field in the half-planes y > 0 and y < 0. The parameter can be chosen so that for parameter values less than or equal to zero, the dynamical system has a unique singular point with four hyperbolic sectors in a vicinity of the point O. For positive values of the parameter in the vicinity of the point O, there are three singular points, a quasi-centre and two saddles, the separatrixes of which form a simple closed contour that bounds the cell from closed trajectories. In the second case, O is a rough node of the corresponding vector fields. The parameter can be chosen so that for values of the parameter less than or equal to zero, the dynamical system has a unique singular point in a vicinity of the point O, and all other trajectories are closed. For positive values of the parameter in the vicinity of the point O, there are three singular points, two nodes and a quasi-saddle, whose two separatrixes go to the nodes.

ON THE CENTER PROBLEM FOR DEGENERATE SINGULAR POINTS OF PLANAR VECTOR FIELDS

2002 ◽  
Vol 12 (04) ◽  
pp. 687-707 ◽  
Author(s):  
VÍCTOR MAÑOSA
Keyword(s):  
Singular Point ◽  
Stability Problem ◽  
Vector Fields ◽  
Blow Up ◽  
Singular Points ◽  
Center Problem ◽  
Focal Value ◽  
The Status ◽  
Planar Vector ◽  
The Stability

The center problem for degenerate singular points of planar systems (the degenerate-center problem) is a poorly-understood problem in the qualitative theory of ordinary differential equations. It may be broken down into two problems: the monodromy problem, to decide if the singular point is of focus-center type, and the stability problem, to decide whether it is a focus or a center. We present an outline on the status of the center problem for degenerate singular points, explaining the main techniques and obstructions arising in the study of the problem. We also present some new results. Our new results are the characterization of a family of vector fields having a degenerate monodromic singular point at the origin, and the computation of the generalized first focal value for this family V1. This gives the solution of the stability problem in the monodromic case, except when V1 = 1. Our approach relies on the use of the blow-up technique and the study of the blow-up geometry of singular points. The knowledge of the blow-up geometry is used to generate a bifurcation of a limit cycle.

Parallel Computation of Complex Antennas around the Coated Object Using Iterative Vector Fields Technique

2014 ◽  
Vol E97.C (7) ◽  
pp. 661-669
Author(s):  
Ying YAN ◽  
Xunwang ZHAO ◽  
Yu ZHANG ◽  
Changhong LIANG ◽  
Zhewang MA
Keyword(s):  
Parallel Computation ◽  
Vector Fields
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