The Topology of Chaos - Department of Physics - Drexel University

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Embeddings The Topology of Chaos Robert Gilmore Intro.-01 Varieties of Embeddings x(i) → (y 1 (i), y 2 (i), y 3 (i), · · · ) Intro.-02 Intro.-03 Exp’tal-01 Exp’tal-02 Exp’tal-03 Exp’tal-04 Exp’tal-05 Exp’tal-06 Exp’tal-07 Exp’tal-08 Embed-01 Embed-02 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 Embed-03
Embeddings The Topology of Chaos Robert Gilmore Intro.-01 Intro.-02 Intro.-03 Exp’tal-01 Exp’tal-02 Exp’tal-03 Exp’tal-04 Exp’tal-05 Exp’tal-06 Exp’tal-07 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) Exp’tal-08 Embed-01 Embed-02 Embed-03
Page 1 and 2: The Topology of Chaos Robert Gilmor
Page 3 and 4: Abstract - Key Point The Topology o
Page 5 and 6: Experimental Schematic The Topology
Page 7 and 8: Ask the Masters The Topology of Cha
Page 9 and 10: Real Data The Topology of Chaos Rob
Page 11 and 12: Time Evolution The Topology of Chao
Page 13: Embeddings The Topology of Chaos Ro
Page 17 and 18: More Knotted Embeddings The Topolog
Page 19 and 20: Strange Attractor The Topology of C
Page 21 and 22: Skeletons The Topology of Chaos Rob
Page 23 and 24: Dynamics and Topology The Topology
Page 25 and 26: Evolution in Phase Space The Topolo
Page 27 and 28: Motion of Blobs in Phase Space The
Page 29 and 30: Fundamental Theorem The Topology of
Page 31 and 32: Fundamental Theorem The Topology of
Page 33 and 34: Birman-Williams Theorem The Topolog
Page 35 and 36: A Mechanism with Symmetry The Topol
Page 37 and 38: Aufbau Princip for Branched Manifol
Page 39 and 40: Dynamics and Topology The Topology
Page 41 and 42: Topological Analysis Program The To
Page 43 and 44: Embeddings The Topology of Chaos Ro
Page 45 and 46: Determine Topological Invariants Th
Page 47 and 48: Determine Topological Invariants Th
Page 49 and 50: Perestroikas of Strange Attractors
Page 51 and 52: Perestroikas of Strange Attractors
Page 53 and 54: An Unexpected Benefit The Topology
Page 55 and 56: Representations The Topology of Cha
Page 57 and 58: Representation Labels The Topology
Page 59 and 60: Inequivalence in R 3 The Topology o
Page 61 and 62: Equivalences The Topology of Chaos
Page 63 and 64: Creating Isotopies The Topology of
Page 65 and 66:
Mendeleev’s Table of the Chemical
Page 67 and 68:
Stars The Topology of Chaos Robert
Page 69 and 70:
An Experimental Study The Topology
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Basis Set of Orbits The Topology of
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Perestroikas of Branched Manifolds
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Torus and Genus The Topology of Cha
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Flows in Vector Fields The Topology
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Canonical Forms The Topology of Cha
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Labeling Bounding Tori The Topology
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Exponential Growth The Topology of
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Aufbau Princip for Bounding Tori Th
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Poincaré Section The Topology of C
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Represerntation Theory Redux The To
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Embeddings The Topology of Chaos Ro
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Extrinsic Embedding of Bounding Tor
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The Road Ahead The Topology of Chao
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Our Result The Topology of Chaos Ro
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Four Levels of Structure The Topolo
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Page 105 and 106:
Answered Questions The Topology of
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Unanswered Questions The Topology o
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The Topology of Chaos - Department of Physics - Drexel University

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