scholarly journals The bifurcation set as a topological invariant for one-dimensional dynamics

Nonlinearity ◽  
2021 ◽  
Vol 34 (3) ◽  
pp. 1366-1388
Author(s):  
Gabriel Fuhrmann ◽  
Maik Gröger ◽  
Alejandro Passeggi
Keyword(s):  
Topological Invariant ◽  
One Dimensional ◽  
Bifurcation Set

Resonances and Torsion Numbers of Driven Dissipative Nonlinear Oscillators

1986 ◽  
Vol 41 (4) ◽  
pp. 605-614 ◽  
Author(s):  
Ulrich Parlitz ◽  
Werner Lauterborn
Keyword(s):  
Nonlinear Oscillators ◽  
Winding Number ◽  
Local Flow ◽  
One Dimensional ◽  
Bifurcation Set ◽  
Local Bifurcations ◽  
Closed Orbits

The torsion of the local flow around closed orbits and its relation to the superstructure in the bifurcation set of strictly dissipative nonlinear oscillators is investigated. The torsion number describing the twisting behaviour of the flow turns out to be a suitable invariant for the classification of local bifurcations and resonances in those systems. Furthermore, the notions of winding number and resonance are generalized to arbitrary one-dimensional dissipative oscillators.

scholarly journals Topological entropy and partially hyperbolic diffeomorphisms

2008 ◽  
Vol 28 (3) ◽  
pp. 843-862 ◽  
Author(s):  
YONGXIA HUA ◽  
RADU SAGHIN ◽  
ZHIHONG XIA
Keyword(s):  
Topological Entropy ◽  
Topological Invariant ◽  
Compact Manifolds ◽  
Two Dimensional ◽  
One Dimensional ◽  
Hyperbolic Diffeomorphisms ◽  
Lower Semi ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms

AbstractWe consider partially hyperbolic diffeomorphisms on compact manifolds. We define the notion of the unstable and stable foliations stably carrying some unique non-trivial homologies. Under this topological assumption, we prove the following two results: if the center foliation is one-dimensional, then the topological entropy is locally a constant; and if the center foliation is two-dimensional, then the topological entropy is continuous on the set of all $C^{\infty }$ diffeomorphisms. The proof uses a topological invariant we introduced, Yomdin’s theorem on upper semi-continuity, Katok’s theorem on lower semi-continuity for two-dimensional systems, and a refined Pesin–Ruelle inequality we proved for partially hyperbolic diffeomorphisms.

One-Dimensional Cell Complexes with Homeotopy Group Equal to Zero

1964 ◽  
Vol 16 ◽  
pp. 353-357 ◽  
Author(s):  
Louis V. Quintas
Keyword(s):  
Automorphism Group ◽  
Simplicial Complex ◽  
Topological Invariant ◽  
One Dimensional ◽  
Cell Complexes ◽  
Linear Graph ◽  
Group Of Homeomorphisms ◽  
Multiple Edges ◽  
Dimensional Cell ◽  
Dimensional Simplicial Complex

Let K denote a connected finite 1-dimensional cell complex (1, p. 95), G(K) its group of homeomorphisms, and D(K) the group of homeomorphisms of K which are isotopic to the identity. The group (K) = G(K)/D(K) is a topological invariant of K and is called the homeotopy group ofK (4). K may be thought of as a linear graph (connected finite 1- dimensional simplicial complex) extended to admit loops and multiple edges and (K) as the topological analogue of the automorphism group A(L), (the permutations of vertices which preserve edge incidence relations) of a linear graph L.

scholarly journals An invariant of basic sets of Smale flows

1997 ◽  
Vol 17 (6) ◽  
pp. 1437-1448 ◽  
Author(s):  
MICHAEL C. SULLIVAN
Keyword(s):  
Topological Invariant ◽  
Invariant Sets ◽  
Stable And Unstable Manifolds ◽  
One Dimensional ◽  
Smooth Flow ◽  
Unstable Manifolds ◽  
Smale Flows ◽  
Basic Sets

We consider one-dimensional flows which arise as hyperbolic invariant sets of a smooth flow on a manifold. Included in our data is the twisting in the local stable and unstable manifolds. A topological invariant sensitive to this twisting is obtained.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

Electron-Mirror Microscopic Aspects of Ferroelectric Domains of BaTiO3 And Ca2Sr (C2H5CO2)6

1970 ◽  
Vol 28 ◽  
pp. 478-479
Author(s):  
Teruo Someya ◽  
Jinzo Kobayashi
Keyword(s):  
Surface Layer ◽  
Electric Fields ◽  
Quantitative Study ◽  
Recent Progress ◽  
Ferroelectric Domain ◽  
Resolving Power ◽  
Surface Pattern ◽  
Crystal Surfaces ◽  
One Dimensional ◽  
Ferroelectric Domains

Recent progress in the electron-mirror microscopy (EMM), e.g., an improvement of its resolving power together with an increase of the magnification makes it useful for investigating the ferroelectric domain physics. English has recently observed the domain texture in the surface layer of BaTiO3. The present authors ) have developed a theory by which one can evaluate small one-dimensional electric fields and/or topographic step heights in the crystal surfaces from their EMM pictures. This theory was applied to a quantitative study of the surface pattern of BaTiO3).

Structure and function of neural networks in the retina

1988 ◽  
Vol 46 ◽  
pp. 26-27
Author(s):  
Peter Sterling
Keyword(s):  
Ganglion Cells ◽  
Three Dimensional ◽  
Structure And Function ◽  
Synaptic Connections ◽  
Visual Axis ◽  
One Dimensional ◽  
Cat Retina ◽  
Ultrathin Sections ◽  
And Function ◽  
Insight Into

The synaptic connections in cat retina that link photoreceptors to ganglion cells have been analyzed quantitatively. Our approach has been to prepare serial, ultrathin sections and photograph en montage at low magnification (˜2000X) in the electron microscope. Six series, 100-300 sections long, have been prepared over the last decade. They derive from different cats but always from the same region of retina, about one degree from the center of the visual axis. The material has been analyzed by reconstructing adjacent neurons in each array and then identifying systematically the synaptic connections between arrays. Most reconstructions were done manually by tracing the outlines of processes in successive sections onto acetate sheets aligned on a cartoonist's jig. The tracings were then digitized, stacked by computer, and printed with the hidden lines removed. The results have provided rather than the usual one-dimensional account of pathways, a three-dimensional account of circuits. From this has emerged insight into the functional architecture.

Intergrowth of modulated structures in high Tc Bi-Sr-Ca-Cu-O superconductors

1990 ◽  
Vol 48 (4) ◽  
pp. 76-77
Author(s):  
A.Q. He ◽  
G.W. Qiao ◽  
J. Zhu ◽  
H.Q. Ye
Keyword(s):  
Twin Structure ◽  
Modulated Structure ◽  
High Tc ◽  
One Dimensional ◽  
Lattice Constants ◽  
Superconducting Phases ◽  
Modulated Structures ◽  
Layer Structures

Since the first discovery of high Tc Bi-Sr-Ca-Cu-O superconductor by Maeda et al, many EM works have been done on it. The results show that the superconducting phases have a type of ordered layer structures similar to that in Y-Ba-Cu-O system formulated in Bi2Sr2Can−1CunO2n+4 (n=1,2,3) (simply called 22(n-1) phase) with lattice constants of a=0.358, b=0.382nm but the length of c being different according to the different value of n in the formulate. Unlike the twin structure observed in the Y-Ba-Cu-O system, there is an incommensurate modulated structure in the superconducting phases of Bi system superconductors. Modulated wavelengths of both 1.3 and 2.7 nm have been observed in the 2212 phase. This communication mainly presents the intergrowth of these two kinds of one-dimensional modulated structures in 2212 phase.

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