scholarly journals A Genealogy of Convex Solids Via Local and Global Bifurcations of Gradient Vector Fields

2016 ◽  
Vol 26 (6) ◽  
pp. 1789-1815 ◽  
Author(s):  
Gábor Domokos ◽  
Philip Holmes ◽  
Zsolt Lángi
Keyword(s):  
Vector Fields ◽  
Global Bifurcations ◽  
Gradient Vector ◽  
Local And Global Bifurcations

ROBUST BURSTING TO THE ORIGIN: HETEROCLINIC CYCLES WITH MAXIMAL SYMMETRY EQUILIBRIA

2005 ◽  
Vol 15 (09) ◽  
pp. 2819-2832 ◽  
Author(s):  
DAVID HAWKER ◽  
PETER ASHWIN
Keyword(s):  
Limit Cycle ◽  
Vector Fields ◽  
Global Bifurcations ◽  
Heteroclinic Cycles ◽  
Maximal Symmetry ◽  
Codimension One ◽  
Local And Global Bifurcations ◽  
Resonance Bifurcation

Robust attracting heteroclinic cycles have been found in many models of dynamics with symmetries. In all previous examples, robust heteroclinic cycles appear between a number of symmetry broken equilibria. In this paper we examine the first example where there are robust attracting heteroclinic cycles that include the origin, i.e. a point with maximal symmetry. The example we study is for vector fields on ℝ3 with (ℤ2)3 symmetry. We list all possible generic (codimension one) local and global bifurcations by which this cycle can appear as an attractor; these include a resonance bifurcation from a limit cycle, direct bifurcation from a stable origin and direct bifurcation from other and more familiar robust heteroclinic cycles.

scholarly journals Gradient Vector Fields Do Not Generate Twister Dynamics

2001 ◽  
Vol 174 (1) ◽  
pp. 91-100 ◽  
Author(s):  
P. Fortuny ◽  
F. Sanz
Keyword(s):  
Vector Fields ◽  
Gradient Vector

scholarly journals Counting and Discrete Morse Theory

2019 ◽  
Vol 21 (1) ◽  
Author(s):  
Andrew Sack
Keyword(s):  
Morse Theory ◽  
Vector Fields ◽  
Discrete Morse Theory ◽  
Gradient Vector ◽  
Morse Functions

We examine enumerating discrete Morse functions on graphs up to equivalence by gradient vector fields and by restrictions on the codomain.  We give formulae for the number of discrete Morse functions on specific classes of graphs (line, cycle, and bouquet of circles).

scholarly journals Connected components of the space of proper gradient vector fields

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Maciej Starostka
Keyword(s):  
Vector Fields ◽  
Connected Components ◽  
Gradient Vector ◽  
Proper Maps

AbstractWe show that there exist two proper gradient vector fields on $$\mathbb {R}^n$$ R n which are homotopic in the category of proper maps but not homotopic in the category of proper gradient maps.

scholarly journals On Trajectories of Analytic Gradient Vector Fields

2002 ◽  
Vol 184 (1) ◽  
pp. 215-223 ◽  
Author(s):  
Aleksandra Nowel ◽  
Zbigniew Szafraniec
Keyword(s):  
Vector Fields ◽  
Gradient Vector

scholarly journals Bifurcations of Nonconstant Solutions of the Ginzburg-Landau Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-19
Author(s):  
Norimichi Hirano ◽  
Sławomir Rybicki
Keyword(s):  
Landau Equation ◽  
Conley Index ◽  
Global Bifurcations ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Local And Global Bifurcations

We study local and global bifurcations of nonconstant solutions of the Ginzburg-Landau equation from the families of constant ones. As the topological tools we use the equivariant Conley index and the degree for equivariant gradient maps.

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