The Topology of Chaos Chapter 3: Topology of Orbits Robert Gilmore Topology of Orbits-01 Topology of Orbits-02 Topology of Orbits-03a Topology of Orbits-03b Topology of Orbits-04a Topology of Orbits-04b Topology of Orbits-05 Topology of Orbits-06 The Topology of Chaos Chapter 3: Topology of Orbits Robert Gilmore Physics Department Drexel University Philadelphia, PA 19104 [email protected]Physics and Topology Workshop Drexel University, Philadelphia, PA 19104 September 5, 2008
22
Embed
The Topology of Chaos Chapter 3: Topology of Orbitseinstein.drexel.edu/~bob/Presentations/chapter_orbits.pdfChapter 3: Topology of Orbits Robert Gilmore Topology of Orbits-01 Topology
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
The Topology of ChaosChapter 3: Topology of Orbits
Physics and Topology WorkshopDrexel University, Philadelphia, PA 19104
September 5, 2008
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Chaos
Chaos
Motion that is
•Deterministic: dxdt = f(x)
•Recurrent
•Non Periodic
• Sensitive to Initial Conditions
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
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
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Skeletons
UPOs Outline Strange attractors
BZ reaction
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Skeletons
UPOs Outline Strange attractors
Lefranc - Cargese
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
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
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Linking Numbers
Linking Number of Two UPOs
Lefranc - Cargese
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Evolution in Phase Space
One Stretch-&-Squeeze Mechanism
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Motion of Blobs in Phase Space
Stretching — Squeezing
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Collapse Along the Stable Manifold
Birman - Williams Projection
Identify x and y if
limt→∞|x(t)− y(t)| → 0
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Fundamental Theorem
Birman - Williams Theorem
If:
Then:
Certain Assumptions
Specific Conclusions
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Fundamental Theorem
Birman - Williams Theorem
If:
Then:
Certain Assumptions
Specific Conclusions
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Fundamental Theorem
Birman - Williams Theorem
If:
Then:
Certain Assumptions
Specific Conclusions
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
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.
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
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.”
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
A Very Common Mechanism
Rossler:
Attractor Branched Manifold
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
A Mechanism with Symmetry
Lorenz:
Attractor Branched Manifold
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
Examples of Branched Manifolds
Inequivalent Branched Manifolds
The Topologyof Chaos
Chapter 3:Topology of
Orbits
RobertGilmore
Topology ofOrbits-01
Topology ofOrbits-02
Topology ofOrbits-03a
Topology ofOrbits-03b
Topology ofOrbits-04a
Topology ofOrbits-04b
Topology ofOrbits-05
Topology ofOrbits-06
Topology ofOrbits-07
Topology ofOrbits-08
Topology ofOrbits-09
Topology ofOrbits-10
Topology ofOrbits-11
Topology ofOrbits-12
Topology ofOrbits-13
Topology ofOrbits-14
Topology ofOrbits-15
Topology ofOrbits-16
Topology ofOrbits-17
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