The Topology of Chaos - Drexel Universityeinstein.drexel.edu/~bob/Presentations/dinconX.pdf · The...

The Topologyof Chaos

RobertGilmore

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The Topology of Chaos

Robert Gilmore

Physics DepartmentDrexel University

Philadelphia, PA 19104robert.gilmore@drexel.edu

2011 Brazilian Conference on Dynamics, Control, and ApplicationsAguas de Lindoia, Sao Paulo, Brazil

August 29 — September 2, 2011

August 16, 2011

RobertGilmore

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Abstract

Data generated by a low-dimensional dynamical systemoperating in a chaotic regime can be analyzed using topologicalmethods. The process is (almost) straightforward. On a scalartime 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 isdetermined;

4 Some orbits are used to identify an underlying branchedmanifold;

5 The branched manifold is used as a tool to predict theremaining topological invariants.

This algorithm has its own built in rejection criterion.

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Abstract - Key Point

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 ofwhat you learn is about the embedding and howmuch is about the underlying dynamics?

This question has been answered by creating a representationtheory of low dimensional strange attractors. It is now possibleto totally disentangle the mechanism generating the underlyingdynamics from topological structure induced by the embeddingstep.

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Table of Contents

Outline1 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

Laser Experimental Arrangement

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Real Data

Experimental Data: LSA

Lefranc - Cargese

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Ask the Masters

Periodic Orbits are the Key

Joseph Fourier Henri PoincareLinear Systems Nonlinear Systems

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Periodic Orbit Surrogates

Searching for Periodic Orbits

Θ(i, i+ p) = |x(i)− x(i+ p)|

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Real Data

“Periodic Orbits” in Real Data

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Mechanism

Stretching & Squeezing in a Torus

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Time Evolution

Rotating the Poincare Section

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Time Evolution

Rotating the Poincare Section

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Lefranc - Cargese

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Embeddings

Creating Something from “Nothing”

scalar time series −→ vector time series

Embedding is an Art.

Perhaps more like Black Magic.

There are many ways to conjure an embedding.

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Embeddings

Varieties of Embeddings

x(i)→ (y1(i), y2(i), y3(i), · · · )

Delay (x(i), x(i− τ1), x(i− τ2), x(i− τ3), · · · )Delay yj(i) = x(i− [j − 1] τ) τ,NDifferential y1 = x, y2 = dx/dt, y3 = d2x/dt2, · · ·Int.−Diff . y1 =

∫ x−∞ dx, y2 = x, y3 = dx/dt, · · ·

SVD EoMHilbert−Tsf .“Circular′′

KnottedOther

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Embeddings

Circular and Knotted Embeddings

If there is a “hole in the middle” then parameterize the scalarobservable 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) + y1n(θ) + y2b(θ)

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Tori-20

Summary-01

Summary-02

Summary-03

Summary-04

Summary-05

Summary-06

Summary-07

Summary-08

Summary-09

Summary-10

Some Knotted Embeddings

Simple “Unknot” Trefoil Knot

RobertGilmore

Intro.-01

Intro.-02

Intro.-03

Exp’tal-01

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Exp’tal-03

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Tori-20

Summary-01

Summary-02

Summary-03

Summary-04

Summary-05

Summary-06

Summary-07

Summary-08

Summary-09

Summary-10

More Knotted Embeddings

Granny Knot Square Knot

RobertGilmore

Intro.-01

Intro.-02

Intro.-03

Exp’tal-01

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Summary-01

Summary-02

Summary-03

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Summary-05

Summary-06

Summary-07

Summary-08

Summary-09

Summary-10

Motion that is

•Deterministic: dxdt = f(x)

•Recurrent

•Non Periodic

• Sensitive to Initial Conditions

RobertGilmore

Intro.-01

Intro.-02

Intro.-03

Exp’tal-01

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Summary-01

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Summary-10

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

RobertGilmore

Intro.-01

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Exp’tal-01

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Tori-20

Summary-01

Summary-02

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Summary-08

Summary-09

Summary-10

Skeletons

UPOs Outline Strange Attractors

BZ reaction

RobertGilmore

Intro.-01

Intro.-02

Intro.-03

Exp’tal-01

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Tori-20

Summary-01

Summary-02

Summary-03

Summary-04

Summary-05

Summary-06

Summary-07

Summary-08

Summary-09

Summary-10

Skeletons

UPOs Outline Strange attractors

Laser w. Modulated Losses

RobertGilmore

Intro.-01

Intro.-02

Intro.-03

Exp’tal-01

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Tori-20

Summary-01

Summary-02

Summary-03

Summary-04

Summary-05

Summary-06

Summary-07

Summary-08

Summary-09

Summary-10

Organization

How Are Orbits Organized

Ask the Master:

Carl Friedrich Gauss

RobertGilmore

Intro.-01

Intro.-02

Intro.-03

Exp’tal-01

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Tori-20

Summary-01

Summary-02

Summary-03

Summary-04

Summary-05

Summary-06

Summary-07

Summary-08

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Summary-10

Dynamics and Topology

Organization of UPOs in R3:

Gauss Linking Number

LN(A,B) =1

∮ ∮(rA − rB)·drA×drB

|rA − rB|3

# Interpretations of LN ' # Mathematicians in World

RobertGilmore

Intro.-01

Intro.-02

Intro.-03

Exp’tal-01

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Summary-01

Summary-02

Summary-03

Summary-04

Summary-05

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Summary-07

Summary-08

Summary-09

Summary-10

Linking Numbers

Linking Number of Two UPOs

Lefranc - Cargese

RobertGilmore

Intro.-01

Intro.-02

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Exp’tal-01

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Summary-01

Summary-02

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Summary-04

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Summary-10

Evolution in Phase Space

One Stretch-&-Squeeze Mechanism

“Rossler Mechanism”

RobertGilmore

Intro.-01

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Exp’tal-01

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Summary-01

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Motion of Blobs in Phase Space

Another Stretch-&-Squeeze Mechanism

“Lorenz Mechanism”

RobertGilmore

Intro.-01

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Summary-01

Summary-02

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Summary-10

Motion of Blobs in Phase Space

Stretching — Folding

RobertGilmore

Intro.-01

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Exp’tal-01

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Summary-01

Summary-02

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Summary-09

Summary-10

Collapse Along the Stable Manifold

Birman - Williams Projection

Identify x and y if

limt→∞|x(t)− y(t)| → 0

RobertGilmore

Intro.-01

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Summary-01

Summary-02

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Summary-08

Summary-09

Summary-10

Fundamental Theorem

Birman - Williams Theorem

Certain Assumptions

Specific Conclusions

RobertGilmore

Intro.-01

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Summary-01

Summary-02

Summary-03

Summary-04

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Summary-07

Summary-08

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Summary-10

Fundamental Theorem

Certain Assumptions

RobertGilmore

Intro.-01

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Summary-01

Summary-02

Summary-03

Summary-04

Summary-05

Summary-06

Summary-07

Summary-08

Summary-09

Summary-10

Fundamental Theorem

Certain Assumptions

RobertGilmore

Intro.-01

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Summary-01

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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.

RobertGilmore

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Summary-01

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Birman-Williams Theorem

Conclusions, B-W Theorem

• The projection maps the strange attractor SA onto a2-dimensional branched 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 1-1 correspondence withUPOs of Φ(x)t on BM. Moreover, every link of UPOs of(Φt(x),SA) is isotopic to the correspond link of UPOs of(Φ(x)t,BM).

Remark: “One of the few theorems useful to experimentalists.”

RobertGilmore

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Summary-01

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A Very Common Mechanism

Rossler:

Attractor Branched Manifold

RobertGilmore

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A Mechanism with Symmetry

Lorenz:

Attractor Branched Manifold

RobertGilmore

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Examples of Branched Manifolds

Inequivalent Branched Manifolds

RobertGilmore

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Aufbau Princip for Branched Manifolds

Any branched manifold can be built up from stretchingand squeezing units

subject to the conditions:

•Outputs to Inputs

•No Free Ends

RobertGilmore

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Rossler System

RobertGilmore

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Lorenz System

RobertGilmore

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Poincare Smiles at Us in R3

•Determine organization of UPOs ⇒

•Determine branched manifold ⇒

•Determine equivalence class of SA

RobertGilmore

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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

RobertGilmore

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Locate UPOs

Method of Close Returns

RobertGilmore

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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

RobertGilmore

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Determine Topological Invariants

Compute Table of Expt’l LN

RobertGilmore

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Compare w. LN From Various BM

RobertGilmore

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Guess Branched Manifold

Lefranc - Cargese

RobertGilmore

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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)

RobertGilmore

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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

RobertGilmore

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Perestroikas of Strange Attractors

Evolution Under Parameter Change

Lefranc - Cargese

RobertGilmore

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Lefranc - Cargese

RobertGilmore

Intro.-01

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Used and Martin — Zaragosa

RobertGilmore

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Used and Martin — Zaragosa

RobertGilmore

Intro.-01

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Summary-01

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An Unexpected Benefit

Analysis of Nonstationary Data

Lefranc - Cargese

RobertGilmore

Intro.-01

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Summary-01

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Our Hope → Now a Result

Compare with

Original Objectives

Construct a simple, algorithmic procedure for:

Classifying strange attractors

Extracting classification information

from experimental signals.

RobertGilmore

Intro.-01

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Summary-01

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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

RobertGilmore

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Summary-01

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Representations

We know about representations from studies of groups andalgebras.

We use this knowledge as a guiding light.

RobertGilmore

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Representation Labels

Inequivalent Irreducible Representations

Irreducible Representations of 3-dimensional Genus-oneattractors are distinguished by three topological labels:

ParityGlobal TorsionKnot Type

ΓP,N,KT (SA)

Mechanism (stretch & fold, stretch & roll) is an invariant ofembedding. It is independent of the representation labels.

RobertGilmore

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Representation Labels

Global Torsion & Parity

RobertGilmore

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Summary-01

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Inequivalence in R3

RobertGilmore

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Summary-01

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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?

RobertGilmore

Intro.-01

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Equivalences

Crossing Exchange in R4

Parity reversal is also possible in R4 by isotopy.

RobertGilmore

Intro.-01

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Isotopies

2 Twists = 1 Writhe = Identity

Z −→ Z2

Global Torsion −→ Binary Op

RobertGilmore

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Summary-01

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Creating Isotopies

Equivalences by Injection

Obstructions to Isotopy

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.

RobertGilmore

Intro.-01

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Summary-01

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Classifications

Can We Classify

Strange Attractors?

Chemists have their classification.

Nuclear Physicists have their classification.

Particle Physicists have their classification.

Astronomers have their classification.

RobertGilmore

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Mendeleev’s Table of the Chemical Elements

RobertGilmore

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Atomic Nuclei

Table of Atomic Nuclei

RobertGilmore

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Hertzsprung-Russell Diagram

RobertGilmore

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Challenge

How Does it Work Out?

Chemical Elements 1 Integer: NP

Atomic Nuclei 2 Integers: NP , NN

Stars 1 Continuous variable M + exceptions

Strange Attractors ? ? ? ?

RobertGilmore

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An Experimental Study

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 operatingconditions.

There was a total of 25 different data sets.

We wanted to “prove experimentally” that changing theabsorber/operating condition served merely to push to flowaround on the same branched manifold.

RobertGilmore

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An Experimental Observation

What We Found

When certain orbits (UPOs) were present they were invariablyaccompanied 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 completecollection of UPOs present in an attractor.

RobertGilmore

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Basis Set of Orbits

Forcing Diagram - Horseshoe

RobertGilmore

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An Ongoing Problem

Status of Problem

Horseshoe organization - active

More folding - barely begun

Circle forcing - even less known

Higher genus - new ideas required

Higher dimension - ???

RobertGilmore

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Perestroikas of Branched Manifolds

Constraints

Branched manifolds largely constrain the ‘perestroikas” thatforcing diagrams can undergo.

Is there some mechanism/structure that constrains the types ofperestroikas that branched manifolds can undergo?

RobertGilmore

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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

RobertGilmore

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Torus and Genus

Torus, Longitudes, Meridians

Tori are identified by genus g and dressed with a surface flowinduced from that creating the strange attractor.

RobertGilmore

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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

Singularities organize the surface flow dressing the torus

RobertGilmore

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Flows in Vector Fields

Flow Near a Singularity

RobertGilmore

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Some Bounding Tori

Torus Bounding Lorenz-like Flows

RobertGilmore

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Canonical Forms

Twisting the Lorenz Attractor

RobertGilmore

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Constraints Provided by Bounding Tori

Two possible branched manifolds

in the torus with g=4.

RobertGilmore

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Labeling Bounding Tori

Poincare 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 Poincare sections to P’sections.

Bounding tori labeled by (permutation) group theory.

RobertGilmore

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Some Bounding Tori

Bounding Tori of Low Genus

RobertGilmore

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Exponential Growth

The Growth is Exponential

RobertGilmore

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Motivation

Some Genus-9 Bounding Tori

RobertGilmore

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Aufbau Princip for Bounding Tori

These units (”pants, trinions”) surround the stretching andsqueezing units of branched manifolds.

RobertGilmore

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Any bounding torus can be built upfrom equal numbers of stretching andsqueezing units

•Outputs to Inputs•No Free Ends• Colorless

RobertGilmore

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Poincare Section

Construction of Poincare Section

RobertGilmore

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Application: Lorenz Dynamics, g=3

RobertGilmore

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Represerntation Theory Redux

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-genusattractors?

Yes. The results are similar.

Begin as follows:

RobertGilmore

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Application: Lorenz Dynamics, g=3

RobertGilmore

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Embeddings

RobertGilmore

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Embeddings

Preparations for Embedding tori into R3

Equivalent to embedding a specific class of directed networksinto R3

RobertGilmore

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Extrinsic Embedding of Bounding Tori

Extrinsic Embedding of Intrinsic Tori

A specific simple example.Partial classification by links of homotopy group generators.Nightmare Numbers are Expected.

RobertGilmore

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Creating Isotopies

Equivalences by Injection

Obstructions to Isotopy

Index R3 R4 R5

Global Torsion Z⊗3(g−1) Z⊗2(g−1)2 -

Parity Z2 - -Knot Type Gen. KT. - -

In R5 all representations (embeddings) of a genus-g strangeattractor become equivalent under isotopy.

RobertGilmore

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The Road Ahead

Summary

1 Question Answered ⇒

2 Questions Raised

We must be on the right track !

RobertGilmore

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Our Hope

Original Objectives Achieved

There is now a simple, algorithmic procedure for:

Classifying strange attractors

Extracting classification information

from experimental signals.

RobertGilmore

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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

RobertGilmore

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Four Levels of Structure

The Classification Theory has

4 Levels of Structure

2 Branched Manifolds

3 Bounding Tori

4 Extrinsic Embeddings

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3 Bounding Tori

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3 Bounding Tori

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3 Bounding Tori

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3 Bounding Tori

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RobertGilmore

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Topological Components

Poetic Organization

LINKS OF PERIODIC ORBITS

organize

BOUNDING TORI

organize

BRANCHED MANIFOLDS

organize

LINKS OF PERIODIC ORBITS

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Answered Questions

There is a Representation Theory for Strange Attractors

There is a complete set of rerpesentation labels for strangeattractors of any genus g.

The labels are complete and discrete.

Representations can become equivalent when immersed inhigher dimension.

All representations (embeddings) of a 3-dimensional strangeattractor become isotopic (equivalent) in R5.

The Universal Representation of an attractor in R5 identifiesmechanism. No embedding artifacts are left.

The topological index in R5 that identifies mechanism remainsto be discovered.

RobertGilmore

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Answered Questions

Some Unexpected ResultsPerestroikas of orbits constrained by branched manifolds

Routes to Chaos = Paths through orbit forcing diagram

Perestroikas of branched manifolds constrained bybounding tori

Global Poincare 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

RobertGilmore

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Unanswered Questions

We hope to find:Robust topological invariants for RN , N > 3

A 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 > 2

Representation theory: complete

Singularity Theory: Branched manifolds, splitting points(0 dim.), branch lines (1 dim).

Singularities as obstructions to isotopy

RobertGilmore

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Thanks

To my colleagues and friends:

Jorge Tredicce Elia EschenaziHernan G. Solari Gabriel B. MindlinNick Tufillaro Mario NatielloFrancesco Papoff Ricardo Lopez-RuizMarc Lefranc Christophe LetellierTsvetelin D. Tsankov Jacob KatrielDaniel Cross Tim Jones

Thanks also to:NSF PHY 8843235NSF PHY 9987468NSF PHY 0754081

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