Topological characterization of deterministic chaos: enforcing orientation preservation

2007 ◽  
Vol 366 (1865) ◽  
pp. 559-567 ◽  
Author(s):  
Marc Lefranc ◽  
Pierre-Emmanuel Morant ◽  
Michel Nizette
Keyword(s):  
Periodic Orbits ◽  
Topological Analysis ◽  
Three Dimensional ◽  
Deterministic Chaos ◽  
Three Dimensions ◽  
Unstable Periodic Orbits ◽  
Topological Characterization ◽  
Triangulated Surfaces ◽  
Theoretic Characterization

The determinism principle, which states that dynamical state completely determines future time evolution, is a keystone of nonlinear dynamics and chaos theory. Since it precludes that two state space trajectories intersect, it is a core ingredient of a topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits embedded in a strange attractor. However, knot theory can be applied only to three-dimensional systems. Still, determinism applies in any dimension. We propose an alternative framework in which this principle is enforced by constructing an orientation-preserving dynamics on triangulated surfaces and find that in three dimensions our approach numerically predicts the correct topological entropies for periodic orbits of the horseshoe map.

DIMENSION REDUCTION FOR ANALYSIS OF UNSTABLE PERIODIC ORBITS USING LOCALLY LINEAR EMBEDDING

2012 ◽  
Vol 22 (01) ◽  
pp. 1230001
Author(s):  
BENJAMIN COY
Keyword(s):  
Learning Community ◽  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Topological Analysis ◽  
Three Dimensions ◽  
Locally Linear Embedding ◽  
Unstable Periodic Orbits ◽  
Linking Numbers ◽  
Linear Embedding ◽  
Locally Linear

An autonomous four-dimensional dynamical system is investigated through a topological analysis. This system generates a chaotic attractor for the range of control parameters studied and we determine the organization of the unstable periodic orbits (UPOs) associated with the chaotic attractor. Surrogate UPOs were found in the four-dimensional phase space and pairs of these orbits were embedded in three-dimensions using Locally Linear Embedding. This is a dimensionality reduction technique recently developed in the machine learning community. Embedding pairs of orbits allows the computation of their linking numbers, a topological invariant. A table of linking numbers was computed for a range of control parameter values which shows that the organization of the UPOs is consistent with that of a Lorenz-type branched manifold with rotation symmetry.

convergent-beam diffraction from surfaces

1987 ◽  
Vol 45 ◽  
pp. 30-33
Author(s):  
J. A. Eades ◽  
A. E. Smith ◽  
D. F. Lynch
Keyword(s):  
Reciprocal Lattice ◽  
Three Dimensional ◽  
Grazing Incidence ◽  
Three Dimensions ◽  
Plane Figure ◽  
Transmission Electron ◽  
Convergent Beam ◽  
Beam Patterns ◽  
Beam Diffraction

It is quite simple (in the transmission electron microscope) to obtain convergent-beam patterns from the surface of a bulk crystal. The beam is focussed onto the surface at near grazing incidence (figure 1) and if the surface is flat the appropriate pattern is obtained in the diffraction plane (figure 2). Such patterns are potentially valuable for the characterization of surfaces just as normal convergent-beam patterns are valuable for the characterization of crystals.There are, however, several important ways in which reflection diffraction from surfaces differs from the more familiar electron diffraction in transmission.GeometryIn reflection diffraction, because of the surface, it is not possible to describe the specimen as periodic in three dimensions, nor is it possible to associate diffraction with a conventional three-dimensional reciprocal lattice.

Local bond order parameters for accurate determination of crystal structures in two and three dimensions

2018 ◽  
Vol 20 (42) ◽  
pp. 27059-27068 ◽  
Author(s):  
Hossein Eslami ◽  
Parvin Sedaghat ◽  
Florian Müller-Plathe
Keyword(s):  
Crystal Structures ◽  
Bond Order ◽  
Accurate Determination ◽  
Three Dimensional ◽  
Order Parameters ◽  
Three Dimensions ◽  
Local Order ◽  
Crystalline Structures

Local order parameters for the characterization of liquid and different two- and three-dimensional crystalline structures are presented.

Characterization of blowout bifurcation by unstable periodic orbits

1997 ◽  
Vol 55 (2) ◽  
pp. R1251-R1254 ◽  
Author(s):  
Yoshihiko Nagai ◽  
Ying-Cheng Lai
Keyword(s):  
Periodic Orbits ◽  
Unstable Periodic Orbits

Characterization of the Lorentz attractor by unstable periodic orbits

Nonlinearity ◽  
1993 ◽  
Vol 6 (2) ◽  
pp. 251-258 ◽  
Author(s):  
V Franceschini ◽  
C Giberti ◽  
Zhiming Zheng
Keyword(s):  
Periodic Orbits ◽  
Unstable Periodic Orbits

scholarly journals Control of Chaos in the Rössler System

1997 ◽  
Vol 50 (2) ◽  
pp. 263 ◽  
Author(s):  
Stuart Corney
Keyword(s):  
Periodic Orbits ◽  
Time Series Data ◽  
Control Method ◽  
Three Dimensional ◽  
Dynamical Variable ◽  
Series Data ◽  
External Parameter ◽  
Unstable Periodic Orbits ◽  
System A ◽  
Linear Differential

The control method of Ott, Grebogi and Yorke (1990) as applied to the Rössler system, a set of three-dimensional non-linear differential equations, is examined. Using numerical time series data for a single dynamical variable the method was successfully employed to control several of the unstable periodic orbits in a three-dimensional embedding of the data. The method also failed for a number of unstable periodic orbits due to difficulties in linearising about the orbit or the tangential coincidence of the stable manifold and the motion of the orbit with external parameter.

A Topological Characterization of Conjugate Nets

1977 ◽  
Vol 29 (4) ◽  
pp. 707-721
Author(s):  
Paul A. Vincent
Keyword(s):  
Level Sets ◽  
Harmonic Functions ◽  
Function Theory ◽  
Topological Analysis ◽  
Topological Characterization ◽  
Families Of Curves ◽  
Open Maps

One aspect of topological analysis that authors, such as G. T. Whyburn and Marston Morse, have pointed to ([16; 6] for instance) as being fundamental in the development of function theory is the topological study of the level sets of analytic and harmonic functions or of their topological analogues, light open maps and pseudo-harmonic functions. The first step in this direction seems to have been made by H. Whitney [14] when he studied families of curves, given abstractly using a condition of regularity.

scholarly journals Progress on numerical simulation of nanofluids: impact of an isothermal spherical partition on the mixed convection of nanofluids within cubic enclosures

2020 ◽  
Vol 307 ◽  
pp. 01016
Author(s):  
A. BOUTRA ◽  
K. RAGUI ◽  
N. LABSI ◽  
Y.K. BENKAHLA ◽  
R BENNACER
Keyword(s):  
Lattice Boltzmann ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Numerical Code ◽  
Three Dimensional Flow ◽  
Boltzmann Method ◽  
Do So

The main objective of our work is to light out the three-dimensional flow of an Ag-water nanofluid within a lid-driven cubical space which equipped with a spherical heater into its center. Due to its crucial role in the characterization of the main transfer within such configurations, impact of some parameters is widely inspected. It consists the Richardson value (0,05 to 50), the solid volume fraction (0% to 10%), as well as the heater geometry (10% ≤ d ≤ 25%). To do so, a numerical code based on the Lattice-Boltzmann method, coupled with a finite difference one, is used. The latter has been validated after comparison between the present results and those of the literature. It is to note that the three dimensions D3Q19 model is adopted based on a cubic Lattice, where each pattern of the latter is characterized by nineteen discrete speeds.

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