The Topology of Chaos - Department of Physics - Drexel University

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Birman-Williams Theorem 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 Assumptions, B-W Theorem A flow Φ t (x) • on R n is dissipative, n = 3, so that λ 1 > 0, λ 2 = 0, λ 3 < 0. • Generates a hyperbolic strange attractor SA Exp’tal-06 Exp’tal-07 Exp’tal-08 Embed-01 IMPORTANT: The underlined assumptions can be relaxed. Embed-02 Embed-03
Birman-Williams Theorem 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 Exp’tal-08 Conclusions, B-W Theorem • The projection maps the strange attractor SA onto a 2-dimensional branched manifold BM and the flow Φ t (x) on SA to a semiflow Φ(x) t on BM. • UPOs of Φ t (x) on SA are in 1-1 correspondence with UPOs of Φ(x) t on BM. Moreover, every link of UPOs of (Φ t (x), SA) is isotopic to the correspond link of UPOs of (Φ(x) t , BM). Remark: “One of the few theorems useful to experimentalists.” 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 and 14: 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: Fundamental Theorem The Topology of
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 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
Page 71 and 72: Basis Set of Orbits The Topology of
Page 73 and 74: Perestroikas of Branched Manifolds
Page 75 and 76: Torus and Genus The Topology of Cha
Page 77 and 78: Flows in Vector Fields The Topology
Page 79 and 80: Canonical Forms The Topology of Cha
Page 81 and 82: Labeling Bounding Tori The Topology
Page 83 and 84:
Exponential Growth The Topology of
Page 85 and 86:
Aufbau Princip for Bounding Tori Th
Page 87 and 88:
Poincaré Section The Topology of C
Page 89 and 90:
Represerntation Theory Redux The To
Page 91 and 92:
Embeddings The Topology of Chaos Ro
Page 93 and 94:
Extrinsic Embedding of Bounding Tor
Page 95 and 96:
The Road Ahead The Topology of Chao
Page 97 and 98:
Our Result The Topology of Chaos Ro
Page 99 and 100:
Four Levels of Structure The Topolo
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Answered Questions The Topology of
Page 107 and 108:
Unanswered Questions The Topology o
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The Topology of Chaos - Department of Physics - Drexel University

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