the topology of chaos chapter 3: topology of...
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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 [email protected]
Physics and Topology WorkshopDrexel University, Philadelphia, PA 19104
September 5, 2008
The Topologyof Chaos
Chapter 3:Topology of
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Chaos
Chaos
Motion that is
•Deterministic: dxdt = f(x)
•Recurrent
•Non Periodic
• Sensitive to Initial Conditions
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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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Fundamental Theorem
Birman - Williams Theorem
If:
Then:
Certain Assumptions
Specific Conclusions
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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