Topological equivalence of germs of functions

1984 ◽  
Vol 35 (4) ◽  
pp. 306-310 ◽  
Author(s):  
V. I. Bakhtin
Keyword(s):  
Topological Equivalence

INTERTWINED BASINS OF ATTRACTION

2012 ◽  
Vol 22 (06) ◽  
pp. 1250130
Author(s):  
CHANGMING DING
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Basins Of Attraction ◽  
General Definition ◽  
Sufficient Condition ◽  
Topological Equivalence ◽  
Definition Of

This paper deals with intertwined basins of attraction for dynamical systems in a metric space. After giving a general definition of intertwining property, which is preserved by a topological equivalence between dynamical systems, we present a sufficient condition to guarantee the existence of intertwined basins for dynamical systems in ℝn.

scholarly journals On number of moduli for gradient surface height function flows

2018 ◽  
Vol 20 (4) ◽  
pp. 419-428
Author(s):  
V. E. Kruglov
Keyword(s):  
Finite Number ◽  
Sufficient Conditions ◽  
Height Function ◽  
Orientable Surface ◽  
Topological Conjugacy ◽  
Topological Equivalence ◽  
Gradient Flows ◽  
A Chain ◽  
Gradient Surface ◽  
Equivalence Invariant

In 1978 J. Palis invented continuum topologically non-conjugate systems in a neighbourhood of a system with a heteroclinic contact; in other words, he invented so-called moduli. W. de Melo and С. van Strien in 1987 described a diffeomorphism class with a finite number of moduli. They discovered that a chain of saddles taking part in the heteroclinic contact of such diffeomorphism includes not more than three saddles. Surprisingly, such effect does not happen in flows. Here we consider gradient flows of the height function for an orientable surface of genus g>0. Such flows have a chain of 2g saddles. We found that the number of moduli for such flows is 2g−1 which is the straight consequence of the sufficient topological conjugacy conditions for such systems given in our paper. A complete topological equivalence invariant for such systems is four-colour graph carrying the information about its cells relative position. Equipping the graph's edges with the analytical parameters --- moduli, connected with the saddle connections, gives the sufficient conditions of the flows topological conjugacy.

scholarly journals TOPOLOGICAL EQUIVALENCE FOR DISCONTINUOUS RANDOM DYNAMICAL SYSTEMS AND APPLICATIONS

2013 ◽  
Vol 14 (01) ◽  
pp. 1350007 ◽  
Author(s):  
HUIJIE QIAO ◽  
JINQIAO DUAN
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Lévy Processes ◽  
Levy Processes ◽  
Topological Equivalence ◽  
Random Differential Equation ◽  
Jump Discontinuities ◽  
Non Gaussian

After defining non-Gaussian Lévy processes for two-sided time, stochastic differential equations with such Lévy processes are considered. Solution paths for these stochastic differential equations have countable jump discontinuities in time. Topological equivalence (or conjugacy) for such an Itô stochastic differential equation and its transformed random differential equation is established. Consequently, a stochastic Hartman–Grobman theorem is proved for the linearization of the Itô stochastic differential equation. Furthermore, for Marcus stochastic differential equations, this topological equivalence is used to prove the existence of global random attractors.

Topological equivalence of vector fields after blow-up

2010 ◽  
Vol 348 (11-12) ◽  
pp. 673-676 ◽  
Author(s):  
Clementa Alonso-González
Keyword(s):  
Vector Fields ◽  
Blow Up ◽  
Topological Equivalence
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