The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of...
Transcript of The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of...
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The Topologyof Chaos
RobertGilmore The Topology of Chaos
Robert Gilmore
Physics DepartmentDrexel University
Philadelphia, PA [email protected]
Colloquium, Physics DepartmentUniversity of Florida, Gainesville, FL
October 6, 2008
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The Topologyof Chaos
RobertGilmore
The Topology of Chaos
Robert Gilmore
Physics Department
Drexel University
Philadelphia, PA 19104
Colloquium, Physics DepartmentUniversity of Florida, Gainesville, FL
October 23, 2008
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The Topologyof Chaos
RobertGilmore
Table of Contents
Outline1 Overview
2 Experimental Challenge
3 Topology of Orbits
4 Topological Analysis Program
5 Basis Sets of Orbits
6 Bounding Tori
7 Covers and Images
8 Quantizing Chaos
9 Representation Theory of Strange Attractors
10 Summary
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The Topologyof Chaos
RobertGilmore
Background
J. R. Tredicce
Can you explain my data?
I dare you to explain my data!
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The Topologyof Chaos
RobertGilmore
Motivation
Where is Tredicce coming from?
Feigenbaum: α = 4.66920 16091 .....δ = −2.50290 78750 .....
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The Topologyof Chaos
RobertGilmore
Experiment
Laser with Modulated LossesExperimental Arrangement
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The Topologyof Chaos
RobertGilmore
Our Hope
Original Objectives
Construct a simple, algorithmic procedure for:
Classifying strange attractors
Extracting classification information
from experimental signals.
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The Topologyof Chaos
RobertGilmore
Our Result
Result
There is now a classification theory.
1 It is topological
2 It has a hierarchy of 4 levels
3 Each is discrete
4 There is rigidity and degrees of freedom
5 It is applicable to R3 only — for now
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The Topologyof Chaos
RobertGilmore
Topology Enters the Picture
The 4 Levels of Structure
• Basis Sets of Orbits
• Branched Manifolds
• Bounding Tori
• Extrinsic Embeddings
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The Topologyof Chaos
RobertGilmore
Topological Components
Organization
LINKS OF PERIODIC ORBITSorganize
BOUNDING TORIorganize
BRANCHED MANIFOLDSorganize
LINKS OF PERIODIC ORBITS
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The Topologyof Chaos
RobertGilmore
Experimental Schematic
Laser Experimental Arrangement
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The Topologyof Chaos
RobertGilmore
Experimental Motivation
Oscilloscope Traces
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The Topologyof Chaos
RobertGilmore
Results, Single Experiment
Bifurcation Schematics
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The Topologyof Chaos
RobertGilmore
Some Attractors
Coexisting Basins of Attraction
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The Topologyof Chaos
RobertGilmore
Many Experiments
Bifurcation Perestroikas
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The Topologyof Chaos
RobertGilmore
Real Data
Experimental Data: LSA
Lefranc - Cargese
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The Topologyof Chaos
RobertGilmore
Real Data
Experimental Data: LSA
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The Topologyof Chaos
RobertGilmore
Mechanism
Stretching & Squeezing in a Torus
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The Topologyof Chaos
RobertGilmore
Time Evolution
Rotating the Poincare Sectionaround the axis of the torus
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The Topologyof Chaos
RobertGilmore
Time Evolution
Rotating the Poincare Sectionaround the axis of the torus
Lefranc - Cargese
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The Topologyof Chaos
RobertGilmore
Another Visualization
Cutting Open a Torus
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The Topologyof Chaos
RobertGilmore
Satisfying Boundary Conditions
Global Torsion
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The Topologyof Chaos
RobertGilmore
Experimental Schematic
A Chemical Experiment
The Belousov-Zhabotinskii Reaction
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The Topologyof Chaos
RobertGilmore
Chaos
Chaos
Motion that is
•Deterministic: dxdt = f(x)
•Recurrent
•Non Periodic
• Sensitive to Initial Conditions
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The Topologyof Chaos
RobertGilmore
Strange Attractor
Strange Attractor
The Ω limit set of the flow. There areunstable periodic orbits “in” thestrange attractor. They are
• “Abundant”
•Outline the Strange Attractor
•Are the Skeleton of the StrangeAttractor
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The Topologyof Chaos
RobertGilmore
Skeletons
UPOs Outline Strange attractors
BZ reaction
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The Topologyof Chaos
RobertGilmore
Skeletons
UPOs Outline Strange attractors
Lefranc - Cargese
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The Topologyof Chaos
RobertGilmore
Dynamics and Topology
Organization of UPOs in R3:
Gauss Linking Number
LN(A,B) =1
4π
∮ ∮(rA − rB)·drA×drB
|rA − rB|3
# Interpretations of LN ' # Mathematicians in World
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The Topologyof Chaos
RobertGilmore
Linking Numbers
Linking Number of Two UPOs
Lefranc - Cargese
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The Topologyof Chaos
RobertGilmore
Evolution in Phase Space
One Stretch-&-Squeeze Mechanism
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The Topologyof Chaos
RobertGilmore
Motion of Blobs in Phase Space
Stretching — Squeezing
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The Topologyof Chaos
RobertGilmore
Collapse Along the Stable Manifold
Birman - Williams Projection
Identify x and y if
limt→∞|x(t)− y(t)| → 0
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The Topologyof Chaos
RobertGilmore
Fundamental Theorem
Birman - Williams Theorem
If:
Then:
Certain Assumptions
Specific Conclusions
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The Topologyof Chaos
RobertGilmore
Fundamental Theorem
Birman - Williams Theorem
If:
Then:
Certain Assumptions
Specific Conclusions
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The Topologyof Chaos
RobertGilmore
Fundamental Theorem
Birman - Williams Theorem
If:
Then:
Certain Assumptions
Specific Conclusions
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The Topologyof Chaos
RobertGilmore
Birman-Williams Theorem
Assumptions, B-W Theorem
A flow Φt(x)
• on Rn is dissipative, n = 3, so thatλ1 > 0, λ2 = 0, λ3 < 0.
•Generates a hyperbolic strangeattractor SA
IMPORTANT: The underlined assumptions can be relaxed.
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The Topologyof Chaos
RobertGilmore
Birman-Williams Theorem
Conclusions, B-W Theorem
• The projection maps the strangeattractor SA onto a 2-dimensionalbranched manifold BM and the flow Φt(x)on SA to a semiflow Φ(x)t on BM.•UPOs of Φt(x) on SA are in 1-1correspondence with UPOs of Φ(x)t onBM. Moreover, every link of UPOs of(Φt(x),SA) is isotopic to the correspondlink of UPOs of (Φ(x)t,BM).
Remark: “One of the few theorems useful to experimentalists.”
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The Topologyof Chaos
RobertGilmore
A Very Common Mechanism
Rossler:
Attractor Branched Manifold
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The Topologyof Chaos
RobertGilmore
A Mechanism with Symmetry
Lorenz:
Attractor Branched Manifold
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The Topologyof Chaos
RobertGilmore
Examples of Branched Manifolds
Inequivalent Branched Manifolds
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The Topologyof Chaos
RobertGilmore
Aufbau Princip for Branched Manifolds
Any branched manifold can be built upfrom stretching and squeezing units
subject to the conditions:•Outputs to Inputs•No Free Ends
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The Topologyof Chaos
RobertGilmore
Dynamics and Topology
Rossler System
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The Topologyof Chaos
RobertGilmore
Dynamics and Topology
Lorenz System
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The Topologyof Chaos
RobertGilmore
Dynamics and Topology
Poincare Smiles at Us in R3
•Determine organization of UPOs ⇒
•Determine branched manifold ⇒
•Determine equivalence class of SA
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The Topologyof Chaos
RobertGilmore
Topological Analysis Program
Topological Analysis Program
Locate Periodic Orbits
Create an Embedding
Determine Topological Invariants (LN)
Identify a Branched Manifold
Verify the Branched Manifold
—————————————————————————-
Model the Dynamics
Validate the Model
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The Topologyof Chaos
RobertGilmore
Locate UPOs
Method of Close Returns
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The Topologyof Chaos
RobertGilmore
Embeddings
Embeddings
Many Methods: Time Delay, Differential, Hilbert Transforms,SVD, Mixtures, ...
Tests for Embeddings: Geometric, Dynamic, Topological†
None Good
We Demand a 3 Dimensional Embedding
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The Topologyof Chaos
RobertGilmore
Locate UPOs
An Embedding and Periodic Orbits
Lefranc - Cargese
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The Topologyof Chaos
RobertGilmore
Determine Topological Invariants
Linking Number of Orbit Pairs
Lefranc - Cargese
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The Topologyof Chaos
RobertGilmore
Determine Topological Invariants
Compute Table of Expt’l LN
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The Topologyof Chaos
RobertGilmore
Determine Topological Invariants
Compare w. LN From Various BM
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The Topologyof Chaos
RobertGilmore
Determine Topological Invariants
Guess Branched Manifold
Lefranc - Cargese
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The Topologyof Chaos
RobertGilmore
Determine Topological Invariants
Identification & ‘Confirmation’
• BM Identified by LN of small number of orbits
• Table of LN GROSSLY overdetermined
• Predict LN of additional orbits
• Rejection criterion
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The Topologyof Chaos
RobertGilmore
Determine Topological Invariants
What Do We Learn?• BM Depends on Embedding• Some things depend on embedding, some don’t• Depends on Embedding: Global Torsion, Parity, ..• Independent of Embedding: Mechanism
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The Topologyof Chaos
RobertGilmore
Perestroikas of Strange Attractors
Evolution Under Parameter Change
Lefranc - Cargese
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The Topologyof Chaos
RobertGilmore
Perestroikas of Strange Attractors
Evolution Under Parameter Change
Lefranc - Cargese
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The Topologyof Chaos
RobertGilmore
An Unexpected Benefit
Analysis of Nonstationary Data
Lefranc - Cargese
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The Topologyof Chaos
RobertGilmore
Last Steps
Model the DynamicsA hodgepodge of methods exist: # Methods ' # Physicists
Validate the ModelNeeded: Nonlinear analog of χ2 test. OPPORTUNITY:Tests that depend on entrainment/synchronization.
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The Topologyof Chaos
RobertGilmore
Our Hope → Now a Result
Compare withOriginal Objectives
Construct a simple, algorithmic procedure for:
Classifying strange attractors
Extracting classification information
from experimental signals.
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The Topologyof Chaos
RobertGilmore
Orbits Can be “Pruned”
There Are Some Missing Orbits
Lorenz Shimizu-Morioka
![Page 61: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/61.jpg)
The Topologyof Chaos
RobertGilmore
Linking Numbers, Relative Rotation Rates, Braids
Orbit Forcing
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The Topologyof Chaos
RobertGilmore
An Ongoing Problem
Forcing Diagram - Horseshoe
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The Topologyof Chaos
RobertGilmore
An Ongoing Problem
Status of Problem
Horseshoe organization - active
More folding - barely begun
Circle forcing - even less known
Higher genus - new ideas required
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The Topologyof Chaos
RobertGilmore
Perestroikas of Branched Manifolds
Constraints on Branched Manifolds
“Inflate” a strange attractor
Union of ε ball around each point
Boundary is surface of bounded 3D manifold
Torus that bounds strange attractor
![Page 65: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/65.jpg)
The Topologyof Chaos
RobertGilmore
Torus and Genus
Torus, Longitudes, Meridians
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The Topologyof Chaos
RobertGilmore
Flows on Surfaces
Surface Singularities
Flow field: three eigenvalues: +, 0, –
Vector field “perpendicular” to surface
Eigenvalues on surface at fixed point: +, –
All singularities are regular saddles∑s.p.(−1)index = χ(S) = 2− 2g
# fixed points on surface = index = 2g - 2
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The Topologyof Chaos
RobertGilmore
Flows in Vector Fields
Flow Near a Singularity
![Page 68: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/68.jpg)
The Topologyof Chaos
RobertGilmore
Some Bounding Tori
Torus Bounding Lorenz-like Flows
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The Topologyof Chaos
RobertGilmore
Canonical Forms
Twisting the Lorenz Attractor
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The Topologyof Chaos
RobertGilmore
Constraints Provided by Bounding Tori
Two possible branched manifoldsin the torus with g=4.
![Page 71: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/71.jpg)
The Topologyof Chaos
RobertGilmore
Use in Physics
Bounding Tori contain all knownStrange Attractors
![Page 72: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/72.jpg)
The Topologyof Chaos
RobertGilmore
Labeling Bounding Tori
Labeling Bounding Tori
Poincare section is disjoint union of g-1 disks
Transition matrix sum of two g-1 × g-1 matrices
One is cyclic g-1 × g-1 matrix
Other represents union of cycles
Labeling via (permutation) group theory
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The Topologyof Chaos
RobertGilmore
Some Bounding Tori
Bounding Tori of Low Genus
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The Topologyof Chaos
RobertGilmore
Motivation
Some Genus-9 Bounding Tori
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The Topologyof Chaos
RobertGilmore
Aufbau Princip for Bounding Tori
Any bounding torus can be built upfrom equal numbers of stretching andsqueezing units
•Outputs to Inputs•No Free Ends• Colorless
![Page 76: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/76.jpg)
The Topologyof Chaos
RobertGilmore
Aufbau Princip for Bounding Tori
Application: Lorenz Dynamics, g=3
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The Topologyof Chaos
RobertGilmore
Poincare Section
Construction of Poincare Section
![Page 78: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/78.jpg)
The Topologyof Chaos
RobertGilmore
Exponential Growth
The Growth is Exponential
![Page 79: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/79.jpg)
The Topologyof Chaos
RobertGilmore
Exponential Growth
The Growth is ExponentialThe Entropy is log 3
![Page 80: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/80.jpg)
The Topologyof Chaos
RobertGilmore
Extrinsic Embedding of Bounding Tori
Extrinsic Embedding of Intrinsic Tori
Partial classification by links of homotopy group generators.Nightmare Numbers are Expected.
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The Topologyof Chaos
RobertGilmore
Modding Out a Rotation Symmetry
Modding Out a Rotation Symmetry X
YZ
→ u
vw
=
Re (X + iY )2
Im (X + iY )2
Z
![Page 82: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/82.jpg)
The Topologyof Chaos
RobertGilmore
Lorenz Attractor and Its Image
![Page 83: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/83.jpg)
The Topologyof Chaos
RobertGilmore
Lifting an Attractor: Cover-Image Relations
Creating a Cover with Symmetry X
YZ
← u
vw
=
Re (X + iY )2
Im (X + iY )2
Z
![Page 84: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/84.jpg)
The Topologyof Chaos
RobertGilmore
Cover-Image Related Branched Manifolds
Cover-Image Branched Manifolds
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The Topologyof Chaos
RobertGilmore
Covering Branched Manifolds
Two Two-fold LiftsDifferent Symmetry
Rotation InversionSymmetry Symmetry
![Page 86: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/86.jpg)
The Topologyof Chaos
RobertGilmore
Topological Indices
Topological Index: Choose Group
Choose Rotation Axis (Singular Set)
![Page 87: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/87.jpg)
The Topologyof Chaos
RobertGilmore
Locate the Singular Set wrt Image
Different Rotation Axes ProduceDifferent (Nonisotopic) Lifts
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The Topologyof Chaos
RobertGilmore
Nonisotopic Locally Diffeomorphic Lifts
![Page 89: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/89.jpg)
The Topologyof Chaos
RobertGilmore
Indices (0,1) and (1,1)
Two Two-fold CoversSame Symmetry
![Page 90: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/90.jpg)
The Topologyof Chaos
RobertGilmore
Indices (0,1) and (1,1)
Three-fold, Four-fold Covers
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The Topologyof Chaos
RobertGilmore
Two Inequivalent Lifts with V4 Symmetry
![Page 92: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/92.jpg)
The Topologyof Chaos
RobertGilmore
How to Construct Covers/Images
Algorithm
• Construct Invariant Polynomials, Syzygies, Radicals
• Construct Singular Sets
• Determine Topological Indices
• Construct Spectrum of Structurally Stable Covers
• Structurally Unstable Covers Interpolate
![Page 93: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/93.jpg)
The Topologyof Chaos
RobertGilmore
Surprising New Findings
Symmetries Due to Symmetry
Schur’s Lemmas & Equivariant Dynamics
Cauchy Riemann Symmetries
Clebsch-Gordon Symmetries
Continuations
Analytic ContinuationTopological ContinuationGroup Continuation
![Page 94: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/94.jpg)
The Topologyof Chaos
RobertGilmore
Covers of a Trefoil Torus
Granny Knot Square Knot
Trefoil Knot
![Page 95: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/95.jpg)
The Topologyof Chaos
RobertGilmore
You Can Cover a Cover = Lift a Lift
Covers of Covers of Covers
Rossler Lorenz
Ghrist
![Page 96: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/96.jpg)
The Topologyof Chaos
RobertGilmore
Universal Branched Manifold
EveryKnot Lives Here
![Page 97: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/97.jpg)
The Topologyof Chaos
RobertGilmore
Isomorphisms and Diffeomorphisms
Local Stuff
Groups:Local IsomorphismsCartan’s Theorem
Dynamical Systems:Local Diffeomorphisms??? Anything Useful ???
![Page 98: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/98.jpg)
The Topologyof Chaos
RobertGilmore
Universal Covering Group
Cartan’s Theorem for Lie Groups
![Page 99: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/99.jpg)
The Topologyof Chaos
RobertGilmore
Universal Image Dynamical System
Locally Diffeomorphic Covers of D
D: Universal Image Dynamical System
![Page 100: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/100.jpg)
The Topologyof Chaos
RobertGilmore
Useful Analogs
Local Isomorphisms & Diffeomorphisms
Lie Groups
Local Isomorphisms
Dynamical Systems
Local Diffeos
![Page 101: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/101.jpg)
The Topologyof Chaos
RobertGilmore
Useful Analogs
Local Isomorphisms & Diffeomorphisms
Lie Groups
Local Isomorphisms
Dynamical Systems
Local Diffeos
![Page 102: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/102.jpg)
The Topologyof Chaos
RobertGilmore
Useful Analogs
Local Isomorphisms & Diffeomorphisms
Lie Groups
Local Isomorphisms
Dynamical Systems
Local Diffeos
![Page 103: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/103.jpg)
The Topologyof Chaos
RobertGilmore
Useful Analogs
Local Isomorphisms & Diffeomorphisms
Lie Groups
Local Isomorphisms
Dynamical Systems
Local Diffeos
![Page 104: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/104.jpg)
The Topologyof Chaos
RobertGilmore
Useful Analogs
Local Isomorphisms & Diffeomorphisms
Lie Groups
Local Isomorphisms
Dynamical Systems
Local Diffeos
![Page 105: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/105.jpg)
The Topologyof Chaos
RobertGilmore
Useful Analogs
Local Isomorphisms & Diffeomorphisms
Lie Groups
Local Isomorphisms
Dynamical Systems
Local Diffeos
![Page 106: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/106.jpg)
The Topologyof Chaos
RobertGilmore
Useful Analogs
Local Isomorphisms & Diffeomorphisms
Lie Groups
Local Isomorphisms
Dynamical Systems
Local Diffeos
![Page 107: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/107.jpg)
The Topologyof Chaos
RobertGilmore
Creating New Attractors
Rotating the Attractor
d
dt
[XY
]=[F1(X,Y )F2(X,Y )
]+[a1 sin(ωdt+ φ1)a2 sin(ωdt+ φ2)
][u(t)v(t)
]=[
cos Ωt − sin Ωtsin Ωt cos Ωt
] [X(t)Y (t)
]d
dt
[uv
]= RF(R−1u) +Rt + Ω
[−v+u
]Ω = n ωd q Ω = p ωd
Global Diffeomorphisms Local Diffeomorphisms(p-fold covers)
![Page 108: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/108.jpg)
The Topologyof Chaos
RobertGilmore
Two Phase Spaces: R3 and D2 × S1
Rossler Attractor: Two Representations
R3 D2 × S1
![Page 109: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/109.jpg)
The Topologyof Chaos
RobertGilmore
Other Diffeomorphic Attractors
Rossler Attractor:
Two More Representations with n = ±1
![Page 110: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/110.jpg)
The Topologyof Chaos
RobertGilmore
Subharmonic, Locally Diffeomorphic Attractors
Rossler Attractor:
Two Two-Fold Covers with p/q = ±1/2
![Page 111: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/111.jpg)
The Topologyof Chaos
RobertGilmore
Subharmonic, Locally Diffeomorphic Attractors
Rossler Attractor:
Two Three-Fold Covers with p/q = −2/3,−1/3
![Page 112: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/112.jpg)
The Topologyof Chaos
RobertGilmore
Subharmonic, Locally Diffeomorphic Attractors
Rossler Attractor:
And Even More Covers (with p/q = +1/3,+2/3)
![Page 113: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/113.jpg)
The Topologyof Chaos
RobertGilmore
New Measures
Angular Momentum and Energy
L(0) = limτ→∞
1τ
∫ τ
0XdY−Y dX
L(Ω) = 〈uv − vu〉
= L(0) + Ω〈R2〉
K(0) = limτ→∞
1τ
∫ τ
0
12
(X2+Y 2)dt
K(Ω) = 〈12
(u2 + v2)〉
= K(0) + ΩL(0) +12
Ω2〈R2〉
〈R2〉 = limτ→∞
1τ
∫ τ
0(X2 + Y 2)dt = lim
τ→∞
1τ
∫ τ
0(u2 + v2)dt
![Page 114: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/114.jpg)
The Topologyof Chaos
RobertGilmore
New Measures, Diffeomorphic Attractors
Energy and Angular Momentum
Diffeomorphic, Quantum Number n
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The Topologyof Chaos
RobertGilmore
New Measures, Subharmonic Covering Attractors
Energy and Angular Momentum
Subharmonics, Quantum Numbers p/q
![Page 116: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/116.jpg)
The Topologyof Chaos
RobertGilmore
Embeddings
Embeddings
An embedding creates a diffeomorphism between an(‘invisible’) dynamics in someone’s laboratory and a (‘visible’)attractor in somebody’s computer.
Embeddings provide a representation of an attractor.
Equivalence is by Isotopy.
Irreducible is by Dimension
![Page 117: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/117.jpg)
The Topologyof Chaos
RobertGilmore
Representation Labels
Inequivalent Irreducible Representations
Irreducible Representations of 3-dimensional Genus-oneattractors are distinguished by three topological labels:
ParityGlobal TorsionKnot Type
PNKT
ΓP,N,KT (SA)
Mechanism (stretch & fold, stretch & roll) is an invariant ofembedding. It is independent of the representation labels.
![Page 118: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/118.jpg)
The Topologyof Chaos
RobertGilmore
Creating Isotopies
Equivalent Reducible Representations
Topological indices (P,N,KT) are obstructions to isotopy forembeddings of minimum dimension (irreduciblerepresentations).
Are these obstructions removed by injections into higherdimensions (reducible representations)?
Systematically?
![Page 119: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/119.jpg)
The Topologyof Chaos
RobertGilmore
Creating Isotopies
Equivalences by InjectionObstructions to Isotopy
R3
Global TorsionParityKnot Type
→ R4
Global Torsion
→ R5
There is one Universal reducible representation in RN , N ≥ 5.In RN the only topological invariant is mechanism.
![Page 120: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/120.jpg)
The Topologyof Chaos
RobertGilmore
The Road Ahead
Summary
1 Question Answered ⇒
2 Questions Raised
We must be on the right track !
![Page 121: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/121.jpg)
The Topologyof Chaos
RobertGilmore
Our Hope
Original Objectives Achieved
There is now a simple, algorithmic procedure for:
Classifying strange attractors
Extracting classification information
from experimental signals.
![Page 122: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/122.jpg)
The Topologyof Chaos
RobertGilmore
Our Result
Result
There is now a classification theory
for low-dimensional strange attractors.
1 It is topological
2 It has a hierarchy of 4 levels
3 Each is discrete
4 There is rigidity and degrees of freedom
5 It is applicable to R3 only — for now
![Page 123: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/123.jpg)
The Topologyof Chaos
RobertGilmore
Four Levels of Structure
The Classification Theory has4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
3 Bounding Tori
4 Extrinsic Embeddings
![Page 124: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/124.jpg)
The Topologyof Chaos
RobertGilmore
Four Levels of Structure
The Classification Theory has4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
3 Bounding Tori
4 Extrinsic Embeddings
![Page 125: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/125.jpg)
The Topologyof Chaos
RobertGilmore
Four Levels of Structure
The Classification Theory has4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
3 Bounding Tori
4 Extrinsic Embeddings
![Page 126: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/126.jpg)
The Topologyof Chaos
RobertGilmore
Four Levels of Structure
The Classification Theory has4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
3 Bounding Tori
4 Extrinsic Embeddings
![Page 127: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/127.jpg)
The Topologyof Chaos
RobertGilmore
Four Levels of Structure
The Classification Theory has4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
3 Bounding Tori
4 Extrinsic Embeddings
![Page 128: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/128.jpg)
The Topologyof Chaos
RobertGilmore
Four Levels of Structure
![Page 129: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/129.jpg)
The Topologyof Chaos
RobertGilmore
Topological Components
Poetic Organization
LINKS OF PERIODIC ORBITSorganize
BOUNDING TORIorganize
BRANCHED MANIFOLDSorganize
LINKS OF PERIODIC ORBITS
![Page 130: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/130.jpg)
The Topologyof Chaos
RobertGilmore
Answered Questions
Some Unexpected ResultsPerestroikas of orbits constrained by branched manifoldsRoutes to Chaos = Paths through orbit forcing diagramPerestroikas of branched manifolds constrained bybounding toriGlobal Poincare section = union of g − 1 disksSystematic methods for cover - image relationsExistence of topological indices (cover/image)Universal image dynamical systemsNLD version of Cartan’s Theorem for Lie GroupsTopological Continuation – Group ContinuuationCauchy-Riemann symmetriesQuantizing ChaosRepresentation labels for inequivalent embeddingsRepresentation Theory for Strange Attractors
![Page 131: The Topology of Chaos - Physics Departmentbob/Presentations/univ_florida.pdf · The Topology of Chaos Robert Gilmore Birman-Williams Theorem Conclusions, B-W Theorem The projection](https://reader033.fdocuments.in/reader033/viewer/2022043003/5f8473ca589c367ec06d6464/html5/thumbnails/131.jpg)
The Topologyof Chaos
RobertGilmore
Unanswered Questions
We hope to find:Robust topological invariants for RN , N > 3A Birman-Williams type theorem for higher dimensions
An algorithm for irreducible embeddings
Embeddings: better methods and tests
Analog of χ2 test for NLD
Better forcing results: Smale horseshoe, D2 → D2,n×D2 → n×D2 (e.g., Lorenz), DN → DN , N > 2Representation theory: complete
Singularity Theory: Branched manifolds, splitting points(0 dim.), branch lines (1 dim).
Singularities as obstructions to isotopy