the topology of chaos chapter 3: topology of...

The Topologyof Chaos

Chapter 3:Topology of

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RobertGilmore

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The Topology of ChaosChapter 3: Topology of Orbits

Robert Gilmore

Physics DepartmentDrexel University

Philadelphia, PA 19104robert.gilmore@drexel.edu

Physics and Topology WorkshopDrexel University, Philadelphia, PA 19104

September 5, 2008

The Topologyof Chaos

Chapter 3:Topology of

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RobertGilmore

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Chaos

Chaos

Motion that is

•Deterministic: dxdt = f(x)

•Recurrent

•Non Periodic

• Sensitive to Initial Conditions

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Chapter 3:Topology of

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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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Skeletons

UPOs Outline Strange attractors

BZ reaction

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Skeletons

UPOs Outline Strange attractors

Lefranc - Cargese

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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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Linking Numbers

Linking Number of Two UPOs

Lefranc - Cargese

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Evolution in Phase Space

One Stretch-&-Squeeze Mechanism

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Motion of Blobs in Phase Space

Stretching — Squeezing

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Collapse Along the Stable Manifold

Birman - Williams Projection

Identify x and y if

limt→∞|x(t)− y(t)| → 0

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Fundamental Theorem

Birman - Williams Theorem

If:

Then:

Certain Assumptions

Specific Conclusions

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Fundamental Theorem

Birman - Williams Theorem

If:

Then:

Certain Assumptions

Specific Conclusions

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Chapter 3:Topology of

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Fundamental Theorem

Birman - Williams Theorem

If:

Then:

Certain Assumptions

Specific Conclusions

The Topologyof Chaos

Chapter 3:Topology of

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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.

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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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A Very Common Mechanism

Rossler:

Attractor Branched Manifold

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A Mechanism with Symmetry

Lorenz:

Attractor Branched Manifold

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Examples of Branched Manifolds

Inequivalent Branched Manifolds

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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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Dynamics and Topology

Rossler System

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Dynamics and Topology

Lorenz System

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Dynamics and Topology

Poincare Smiles at Us in R3

•Determine organization of UPOs ⇒

•Determine branched manifold ⇒

•Determine equivalence class of SA