The Topology of Chaos - Department of Physics - Drexel University

The Topology
of Chaos
Robert
Gilmore
The Topology of Chaos
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Robert Gilmore
Physics Department
Drexel University
Philadelphia, PA 19104
robert.gilmore@drexel.edu
2011 Brazilian Conference on Dynamics, Control, and Applications
Águas de Lindóia, Sao Paulo, Brazil
August 29 — September 2, 2011
August 16, 2011
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Abstract
The Topology
of Chaos
Robert
Gilmore
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Data generated by a low-dimensional dynamical system
operating in a chaotic regime can be analyzed using topological
methods. The process is (almost) straightforward. On a scalar
time series, the following steps are taken:
1 Unstable periodic orbits are identified;
2 An embedding is constructed; ⋆ ⋆
3 The topological organization of these periodic orbits is
determined;
4 Some orbits are used to identify an underlying branched
manifold;
5 The branched manifold is used as a tool to predict the
remaining topological invariants.
This algorithm has its own built in rejection criterion.

Abstract - Key Point
The Topology
of Chaos
Robert
Gilmore
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One soft spot in this analysis program is the embedding step.
Different embeddings can yield different topological results.
This makes the following question exciting:
When you analyze embedded data: How much of
what you learn is about the embedding and how
much is about the underlying dynamics?
This question has been answered by creating a representation
theory of low dimensional strange attractors. It is now possible
to totally disentangle the mechanism generating the underlying
dynamics from topological structure induced by the embedding
step.
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Table of Contents
The Topology
of Chaos
Robert
Gilmore
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Outline
1 Overview
2 Experimental Challenge
3 Embedding Problems
4 Topological Analysis Program
5 Representation Theory of Strange Attractors
6 Classification of Strange Attractors
7 Basis Sets of Orbits
8 Bounding Tori
9 Summary
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Experimental Schematic
The Topology
of Chaos
Robert
Gilmore
Laser Experimental Arrangement
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Real Data
The Topology
of Chaos
Robert
Gilmore
Experimental Data: LSA
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Lefranc - Cargese
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Ask the Masters
The Topology
of Chaos
Robert
Gilmore
Periodic Orbits are the Key
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Joseph Fourier
Linear Systems
Henri Poincare
Nonlinear Systems

Periodic Orbit Surrogates
The Topology
of Chaos
Robert
Gilmore
Intro.-01
Searching for Periodic Orbits
Θ(i, i + p) = |x(i) − x(i + p)|
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Real Data
The Topology
of Chaos
Robert
Gilmore
“Periodic Orbits” in Real Data
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Mechanism
The Topology
of Chaos
Robert
Gilmore
Stretching & Squeezing in a Torus
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Time Evolution
The Topology
of Chaos
Robert
Gilmore
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Rotating the Poincaré Section
around the axis of the torus
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Time Evolution
The Topology
of Chaos
Robert
Gilmore
Intro.-01
Rotating the Poincaré Section
around the axis of the torus
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Lefranc - Cargese

Embeddings
The Topology
of Chaos
Robert
Gilmore
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Creating Something from “Nothing”
scalar time series −→ vector time series
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Embedding is an Art.
Perhaps more like Black Magic.
There are many ways to conjure an embedding.
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Embeddings
The Topology
of Chaos
Robert
Gilmore
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Varieties of Embeddings
x(i) → (y 1 (i), y 2 (i), y 3 (i), · · · )
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Delay (x(i), x(i − τ 1 ), x(i − τ 2 ), x(i − τ 3 ), · · · )
Delay y j (i) = x(i − [j − 1] τ) τ, N
Differential y 1 = x, y 2 = dx/dt, y 3 = d 2 x/dt 2 , · · ·
Int. − Diff. y 1 = ∫ x
−∞ dx, y 2 = x, y 3 = dx/dt, · · ·
SVD
EoM
Hilbert − Tsf.
“Circular ′′
Knotted
Other
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Embeddings
The Topology
of Chaos
Robert
Gilmore
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Circular and Knotted Embeddings
If there is a “hole in the middle” then parameterize the scalar
observable by an angle θ: x(t) → x(θ)
Introduce Knot coordinates (“harmonic knots”)
K(θ) = (ξ(θ), η(θ), ζ(θ)) = K(θ + 2π)
Repere Mobile: {t(θ), n(θ), b(θ)}
x(t) → x(θ) → K(θ) + y 1 n(θ) + y 2 b(θ)
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Some Knotted Embeddings
The Topology
of Chaos
Robert
Gilmore
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Simple “Unknot”
Trefoil Knot

More Knotted Embeddings
The Topology
of Chaos
Robert
Gilmore
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Granny Knot
Square Knot

Chaos
The Topology
of Chaos
Robert
Gilmore
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Motion that is
• Deterministic:
• Recurrent
• Non Periodic
Chaos
dx
dt = f(x)
• Sensitive to Initial Conditions
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Strange Attractor
The Topology
of Chaos
Robert
Gilmore
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Strange Attractor
The Ω limit set of the flow. There are
unstable periodic orbits “in” the
strange attractor. They are
• “Abundant”
• Outline the Strange Attractor
• Are the Skeleton of the Strange
Attractor
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Skeletons
The Topology
of Chaos
Robert
Gilmore
UPOs Outline Strange Attractors
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BZ reaction

Skeletons
The Topology
of Chaos
Robert
Gilmore
UPOs Outline Strange attractors
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Laser w. Modulated Losses

Organization
The Topology
of Chaos
Robert
Gilmore
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How Are Orbits Organized
Ask the Master:
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Carl Friedrich Gauss

Dynamics and Topology
The Topology
of Chaos
Robert
Gilmore
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Organization of UPOs in R 3 :
Gauss Linking Number
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LN(A, B) = 1 ∮ ∮ (rA − r B )·dr A ×dr B
4π |r A − r B | 3
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# Interpretations of LN ≃ # Mathematicians in World
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Linking Numbers
The Topology
of Chaos
Robert
Gilmore
Linking Number of Two UPOs
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Lefranc - Cargese

Evolution in Phase Space
The Topology
of Chaos
Robert
Gilmore
One Stretch-&-Squeeze Mechanism
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Motion of Blobs in Phase Space
The Topology
of Chaos
Robert
Gilmore
Another Stretch-&-Squeeze Mechanism
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“Lorenz Mechanism”
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Motion of Blobs in Phase Space
The Topology
of Chaos
Robert
Gilmore
Stretching — Folding
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Collapse Along the Stable Manifold
The Topology
of Chaos
Robert
Gilmore
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Birman - Williams Projection
Identify x and y if
lim |x(t) − y(t)| → 0
t→∞
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Fundamental Theorem
The Topology
of Chaos
Robert
Gilmore
Birman - Williams Theorem
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If:
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Then:
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Fundamental Theorem
The Topology
of Chaos
Robert
Gilmore
Birman - Williams Theorem
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If:
Certain Assumptions
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Then:
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Fundamental Theorem
The Topology
of Chaos
Robert
Gilmore
Birman - Williams Theorem
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If:
Certain Assumptions
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Then:
Specific Conclusions

Birman-Williams Theorem
The Topology
of Chaos
Robert
Gilmore
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Assumptions, B-W Theorem
A flow Φ t (x)
• on R n
is dissipative, n = 3, so that
λ 1 > 0, λ 2 = 0, λ 3 < 0.
• Generates a hyperbolic strange
attractor SA
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IMPORTANT: The underlined assumptions can be relaxed.
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Birman-Williams Theorem
The Topology
of Chaos
Robert
Gilmore
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Conclusions, B-W Theorem
• The projection maps the strange attractor SA onto a
2-dimensional branched manifold BM and the flow Φ t (x)
on SA to a semiflow Φ(x) t on BM.
• UPOs of Φ t (x) on SA are in 1-1 correspondence with
UPOs of Φ(x) t on BM. Moreover, every link of UPOs of
(Φ t (x), SA) is isotopic to the correspond link of UPOs of
(Φ(x) t , BM).
Remark: “One of the few theorems useful to experimentalists.”
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A Very Common Mechanism
The Topology
of Chaos
Robert
Gilmore
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Attractor
Rössler:
Branched Manifold

A Mechanism with Symmetry
The Topology
of Chaos
Robert
Gilmore
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Attractor
Lorenz:
Branched Manifold

Examples of Branched Manifolds
The Topology
of Chaos
Robert
Gilmore
Inequivalent Branched Manifolds
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Aufbau Princip for Branched Manifolds
The Topology
of Chaos
Robert
Gilmore
Any branched manifold can be built up from stretching
and squeezing units
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subject to the conditions:
• Outputs to Inputs
• No Free Ends
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Dynamics and Topology
The Topology
of Chaos
Robert
Gilmore
Rossler System
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Dynamics and Topology
The Topology
of Chaos
Robert
Gilmore
Lorenz System
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Dynamics and Topology
The Topology
of Chaos
Robert
Gilmore
Poincaré Smiles at Us in R 3
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• Determine organization of UPOs ⇒
• Determine branched manifold ⇒
• Determine equivalence class of SA
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Topological Analysis Program
The Topology
of Chaos
Robert
Gilmore
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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

Locate UPOs
The Topology
of Chaos
Robert
Gilmore
Method of Close Returns
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Embeddings
The Topology
of Chaos
Robert
Gilmore
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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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Determine Topological Invariants
The Topology
of Chaos
Robert
Gilmore
Compute Table of Expt’l LN
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Determine Topological Invariants
The Topology
of Chaos
Robert
Gilmore
Compare w. LN From Various BM
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Determine Topological Invariants
The Topology
of Chaos
Robert
Gilmore
Guess Branched Manifold
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Lefranc - Cargese

Determine Topological Invariants
The Topology
of Chaos
Robert
Gilmore
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Identification & ‘Confirmation’
• BM Identified by LN of small number of orbits
• Table of LN GROSSLY overdetermined
• Predict LN of additional orbits
• Rejection criterion (Reject or Fail to Reject)
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Determine Topological Invariants
The Topology
of Chaos
Robert
Gilmore
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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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Perestroikas of Strange Attractors
The Topology
of Chaos
Robert
Gilmore
Evolution Under Parameter Change
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Lefranc - Cargese

Perestroikas of Strange Attractors
The Topology
of Chaos
Robert
Gilmore
Evolution Under Parameter Change
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Lefranc - Cargese

Perestroikas of Strange Attractors
The Topology
of Chaos
Robert
Gilmore
Evolution Under Parameter Change
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Used and Martin — Zaragosa
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Perestroikas of Strange Attractors
The Topology
of Chaos
Robert
Gilmore
Evolution Under Parameter Change
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Used and Martin — Zaragosa

An Unexpected Benefit
The Topology
of Chaos
Robert
Gilmore
Analysis of Nonstationary Data
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Lefranc - Cargese
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Our Hope → Now a Result
The Topology
of Chaos
Robert
Gilmore
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Compare with
Original Objectives
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Construct a simple, algorithmic procedure for:
Classifying strange attractors
Extracting classification information
from experimental signals.
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Representations
The Topology
of Chaos
Robert
Gilmore
Intro.-01
Intro.-02
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Representations
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
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Representations
The Topology
of Chaos
Robert
Gilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
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Representations
We know about representations from studies of groups and
algebras.
We use this knowledge as a guiding light.
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Representation Labels
The Topology
of Chaos
Robert
Gilmore
Intro.-01
Intro.-02
Intro.-03
Inequivalent Irreducible Representations
Irreducible Representations of 3-dimensional Genus-one
attractors are distinguished by three topological labels:
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Parity
Global Torsion
Knot Type
P
N
KT
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Γ P,N,KT (SA)
Mechanism (stretch & fold, stretch & roll) is an invariant of
embedding. It is independent of the representation labels.

Representation Labels
The Topology
of Chaos
Robert
Gilmore
Global Torsion & Parity
Intro.-01
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Inequivalence in R 3
The Topology
of Chaos
Robert
Gilmore
Inequivalence in R 3
Intro.-01
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Creating Isotopies
The Topology
of Chaos
Robert
Gilmore
Intro.-01
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Equivalent Reducible Representations
Topological indices (P,N,KT) are obstructions to isotopy for
embeddings of minimum dimension (irreducible
representations).
Are these obstructions removed by injections into higher
dimensions (reducible representations)?
Systematically?
Exp’tal-07
Exp’tal-08
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Equivalences
The Topology
of Chaos
Robert
Gilmore
Crossing Exchange in R 4
Intro.-01
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Parity reversal is also possible in R 4 by isotopy.
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Isotopies
The Topology
of Chaos
Robert
Gilmore
2 Twists = 1 Writhe = Identity
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Z −→ Z 2
Global Torsion −→ Binary Op
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Creating Isotopies
The Topology
of Chaos
Robert
Gilmore
Intro.-01
Equivalences by Injection
Obstructions to Isotopy
Intro.-02
Intro.-03
Exp’tal-01
Exp’tal-02
Exp’tal-03
Exp’tal-04
R 3
Global Torsion
Parity
Knot Type
→ R 4
Global Torsion
→ R 5
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There is one Universal reducible representation in R N , N ≥ 5.
In R N the only topological invariant is mechanism.
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Classifications
The Topology
of Chaos
Robert
Gilmore
Intro.-01
Can We Classify
Strange Attractors?
Intro.-02
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Chemists have their classification.
Nuclear Physicists have their classification.
Particle Physicists have their classification.
Astronomers have their classification.
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Mendeleev’s Table of the Chemical Elements
The Topology
of Chaos
Robert
Gilmore
Intro.-01
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Atomic Nuclei
The Topology
of Chaos
Robert
Gilmore
Table of Atomic Nuclei
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Stars
The Topology
of Chaos
Robert
Gilmore
Hertzsprung-Russell Diagram
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Challenge
The Topology
of Chaos
Robert
Gilmore
How Does it Work Out?
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Chemical Elements
Atomic Nuclei
Stars
1 Integer: N P
2 Integers: N P , N N
1 Continuous variable M + exceptions
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Strange Attractors ? ? ? ?
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An Experimental Study
The Topology
of Chaos
Robert
Gilmore
Intro.-01
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What We Did
We analyzed data from a Laser with Saturable Absorber (LSA).
3 Different absorbers were used.
For each absorber data were taken under 6 - 10 operating
conditions.
There was a total of 25 different data sets.
We wanted to “prove experimentally” that changing the
absorber/operating condition served merely to push to flow
around on the same branched manifold.
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An Experimental Observation
The Topology
of Chaos
Robert
Gilmore
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What We Found
When certain orbits (UPOs) were present they were invariably
accompanied by a specific set of other orbits.
This led us to propose that certain orbits ‘force’ other orbits.
Forcing is topological.
A discrete set of “basis orbits” serves to identify the complete
collection of UPOs present in an attractor.
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Basis Set of Orbits
The Topology
of Chaos
Robert
Gilmore
Forcing Diagram - Horseshoe
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An Ongoing Problem
The Topology
of Chaos
Robert
Gilmore
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Status of Problem
Horseshoe organization - active
More folding - barely begun
Circle forcing - even less known
Higher genus - new ideas required
Higher dimension - ???
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Perestroikas of Branched Manifolds
The Topology
of Chaos
Robert
Gilmore
Intro.-01
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Constraints
Branched manifolds largely constrain the ‘perestroikas” that
forcing diagrams can undergo.
Is there some mechanism/structure that constrains the types of
perestroikas that branched manifolds can undergo?
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Perestroikas of Branched Manifolds
The Topology
of Chaos
Robert
Gilmore
Intro.-01
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Constraints on Branched Manifolds
“Inflate” a strange attractor
Union of ɛ ball around each point
Boundary is surface of bounded 3D manifold
Torus that bounds strange attractor
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Torus and Genus
The Topology
of Chaos
Robert
Gilmore
Torus, Longitudes, Meridians
Intro.-01
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Tori are identified by genus g and dressed with a surface flow
induced from that creating the strange attractor.
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Flows on Surfaces
The Topology
of Chaos
Robert
Gilmore
Intro.-01
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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
Singularities organize the surface flow dressing the torus
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Flows in Vector Fields
The Topology
of Chaos
Robert
Gilmore
Flow Near a Singularity
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Some Bounding Tori
The Topology
of Chaos
Robert
Gilmore
Torus Bounding Lorenz-like Flows
Intro.-01
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Canonical Forms
The Topology
of Chaos
Robert
Gilmore
Twisting the Lorenz Attractor
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Constraints Provided by Bounding Tori
The Topology
of Chaos
Robert
Gilmore
Intro.-01
Two possible branched manifolds
in the torus with g=4.
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Labeling Bounding Tori
The Topology
of Chaos
Robert
Gilmore
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Labeling Bounding Tori
Poincaré section is disjoint union of g-1 disks.
Transition matrix sum of two g-1 × g-1 matrices.
Both are g-1 × g-1 permutation matrices.
They identify mappings of Poincaré sections to P’sections.
Bounding tori labeled by (permutation) group theory.
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Some Bounding Tori
The Topology
of Chaos
Robert
Gilmore
Bounding Tori of Low Genus
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Exponential Growth
The Topology
of Chaos
Robert
Gilmore
The Growth is Exponential
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Motivation
The Topology
of Chaos
Robert
Gilmore
Some Genus-9 Bounding Tori
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Aufbau Princip for Bounding Tori
The Topology
of Chaos
Robert
Gilmore
Aufbau Princip for Bounding Tori
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These units (”pants, trinions”) surround the stretching and
squeezing units of branched manifolds.
Exp’tal-08
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Aufbau Princip for Bounding Tori
The Topology
of Chaos
Robert
Gilmore
Intro.-01
Any bounding torus can be built up
from equal numbers of stretching and
squeezing units
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• Outputs to Inputs
• No Free Ends
• Colorless
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Poincaré Section
The Topology
of Chaos
Robert
Gilmore
Construction of Poincaré Section
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Aufbau Princip for Bounding Tori
The Topology
of Chaos
Robert
Gilmore
Application: Lorenz Dynamics, g=3
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Represerntation Theory Redux
The Topology
of Chaos
Robert
Gilmore
Intro.-01
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Representation Theory for g > 1
Can we extend the representation theory for strange attractors
“with a hole in the middle” (i.e., genus = 1) to higher-genus
attractors?
Yes. The results are similar.
Begin as follows:
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Aufbau Princip for Bounding Tori
The Topology
of Chaos
Robert
Gilmore
Application: Lorenz Dynamics, g=3
Intro.-01
Intro.-02
Intro.-03
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Embeddings
The Topology
of Chaos
Robert
Gilmore
Embeddings
Intro.-01
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Embeddings
The Topology
of Chaos
Robert
Gilmore
Preparations for Embedding tori into R
Intro.-01
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Equivalent to embedding a specific class of directed networks
into R 3
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Extrinsic Embedding of Bounding Tori
The Topology
of Chaos
Robert
Gilmore
Extrinsic Embedding of Intrinsic Tori
Intro.-01
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A specific simple example.
Partial classification by links of homotopy group generators.
Nightmare Numbers are Expected.

Creating Isotopies
The Topology
of Chaos
Robert
Gilmore
Intro.-01
Equivalences by Injection
Obstructions to Isotopy
Intro.-02
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Index R 3 R 4 R 5
Global Torsion Z ⊗3(g−1) Z ⊗2(g−1)
2 -
Parity Z 2 - -
Knot Type Gen. KT. - -
In R 5 all representations (embeddings) of a genus-g strange
attractor become equivalent under isotopy.
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The Road Ahead
The Topology
of Chaos
Robert
Gilmore
Intro.-01
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Summary
1 Question Answered ⇒
2 Questions Raised
We must be on the right track !
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Our Hope
The Topology
of Chaos
Robert
Gilmore
Original Objectives Achieved
Intro.-01
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There is now a simple, algorithmic procedure for:
Classifying strange attractors
Extracting classification information
from experimental signals.
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Our Result
The Topology
of Chaos
Robert
Gilmore
Result
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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 R 3 only — for now
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Four Levels of Structure
The Topology
of Chaos
Robert
Gilmore
Intro.-01
Intro.-02
The Classification Theory has
4 Levels of Structure
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Four Levels of Structure
The Topology
of Chaos
Robert
Gilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
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The Classification Theory has
4 Levels of Structure
1 Basis Sets of Orbits
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Four Levels of Structure
The Topology
of Chaos
Robert
Gilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
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The Classification Theory has
4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
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Exp’tal-08
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Four Levels of Structure
The Topology
of Chaos
Robert
Gilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
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The Classification Theory has
4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
3 Bounding Tori
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Exp’tal-08
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Four Levels of Structure
The Topology
of Chaos
Robert
Gilmore
Intro.-01
Intro.-02
Intro.-03
Exp’tal-01
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The Classification Theory has
4 Levels of Structure
1 Basis Sets of Orbits
2 Branched Manifolds
3 Bounding Tori
4 Extrinsic Embeddings
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Exp’tal-08
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Four Levels of Structure
The Topology
of Chaos
Robert
Gilmore
Intro.-01
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Topological Components
The Topology
of Chaos
Robert
Gilmore
Poetic Organization
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LINKS OF PERIODIC ORBITS
organize
BOUNDING TORI
organize
BRANCHED MANIFOLDS
organize
LINKS OF PERIODIC ORBITS
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Answered Questions
The Topology
of Chaos
Robert
Gilmore
Intro.-01
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There is a Representation Theory for Strange Attractors
There is a complete set of rerpesentation labels for strange
attractors of any genus g.
The labels are complete and discrete.
Representations can become equivalent when immersed in
higher dimension.
All representations (embeddings) of a 3-dimensional strange
attractor become isotopic (equivalent) in R 5 .
The Universal Representation of an attractor in R 5 identifies
mechanism. No embedding artifacts are left.
The topological index in R 5 that identifies mechanism remains
to be discovered.

Answered Questions
The Topology
of Chaos
Robert
Gilmore
Intro.-01
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Some Unexpected Results
Perestroikas of orbits constrained by branched manifolds
Routes to Chaos = Paths through orbit forcing diagram
Perestroikas of branched manifolds constrained by
bounding tori
Global Poincaré section = union of g − 1 disks
Systematic methods for cover - image relations
Existence of topological indices (cover/image)
Universal image dynamical systems
NLD version of Cartan’s Theorem for Lie Groups
Topological Continuation – Group Continuuation
Cauchy-Riemann symmetries
Quantizing Chaos

Unanswered Questions
The Topology
of Chaos
Robert
Gilmore
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We hope to find:
Robust topological invariants for R N , N > 3
A Birman-Williams type theorem for higher dimensions
An algorithm for irreducible embeddings
Embeddings: better methods and tests
Analog of χ 2 test for NLD
Better forcing results: Smale horseshoe, D 2 → D 2 ,
n × D 2 → n × D 2 (e.g., Lorenz), D N → D N , N > 2
Representation theory: complete
Singularity Theory: Branched manifolds, splitting points
(0 dim.), branch lines (1 dim).
Singularities as obstructions to isotopy

Thanks
The Topology
of Chaos
Robert
Gilmore
Intro.-01
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To my colleagues and friends:
Jorge Tredicce Elia Eschenazi
Hernan G. Solari Gabriel B. Mindlin
Nick Tufillaro Mario Natiello
Francesco Papoff Ricardo Lopez-Ruiz
Marc Lefranc Christophe Letellier
Tsvetelin D. Tsankov Jacob Katriel
Daniel Cross Tim Jones
Exp’tal-05
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Thanks also to:
NSF PHY 8843235
NSF PHY 9987468
NSF PHY 0754081