Physics Department Drexel University Philadelphia, PA 19104

Transcription

1 Overview- Overview- Physics Department Drexel University Philadelphia, PA Physics & Topology Workshop Drexel University, Philadelphia, PA Sept. 9, 2008

2 Table of Contents Overview- Overview- Outline 1 Overview 2 Experimental Challenge 3 Topology of Orbits 4 Topological Analysis Program 5 Basis Sets of Orbits 6 Bounding Tori 7 Covers and Images 8 Quantizing Chaos 9 Representation Theory of Strange Attractors 10 Summary

3 Background J. R. Tredicce Overview- Overview- Can you explain my data? I dare you to explain my data!

4 Motivation Where is Tredicce coming from? Overview- Overview- Feigenbaum: α = δ =

5 Experiment LSA Experimental Arrangement Overview- Overview-

6 Our Hope Original Objectives Overview- Overview- Construct a simple, algorithmic procedure for: Classifying strange attractors Extracting classification information from experimental signals.

7 Our Result Result Overview- Overview- There is now a classification theory. 1 It is topological 2 It has a hierarchy of 4 levels 3 Each is discrete 4 There is rigidity and degrees of freedom 5 It is applicable to R 3 only for now

8 Topology Enters the Picture Overview- Overview- The 4 Levels of Structure Basis Sets of Orbits Branched Manifolds Bounding Tori Extrinsic Embeddings

9 Topological Components Organization Overview- Overview- LINKS OF PERIODIC ORBITS organize BOUNDING TORI organize BRANCHED MANIFOLDS organize LINKS OF PERIODIC ORBITS

10 Experimental Schematic Laser Experimental Arrangement Overview- Overview-

11 Experimental Motivation Oscilloscope Traces Overview- Overview-

12 Results, Single Experiment Bifurcation Schematics Overview- Overview-

13 Some Attractors Coexisting Basins of Attraction Overview- Overview-

14 Many Experiments Bifurcation Perestroikas Overview- Overview-

15 Real Data Experimental Data: LSA Overview- Overview- Lefranc - Cargese

16 Real Data Experimental Data: LSA Overview- Overview-

17 Mechanism Stretching & Squeezing in a Torus Overview- Overview-

18 Time Evolution Rotating the Poincaré Section around the axis of the torus Overview- Overview-

19 Time Evolution Rotating the Poincaré Section around the axis of the torus Overview- Overview- Lefranc - Cargese

20 Another Visualization Cutting Open a Torus Overview- Overview-

21 Satisfying Boundary Conditions Global Torsion Overview- Overview-

22 Experimental Schematic Overview- Overview- A Chemical Experiment The Belousov-Zhabotinskii Reaction

23 Chaos Overview- Overview- Motion that is Deterministic: Recurrent Non Periodic Chaos dx dt = f(x) Sensitive to Initial Conditions

24 Strange Attractor Overview- Overview- Strange Attractor The Ω limit set of the flow. There are unstable periodic orbits in the strange attractor. They are Abundant Outline the Strange Attractor Are the Skeleton of the Strange Attractor

25 Skeletons UPOs Outline Strange attractors Overview- Overview- BZ reaction

26 Skeletons UPOs Outline Strange attractors Overview- Overview- Lefranc - Cargese

27 Dynamics and Topology Organization of UPOs in R 3 : Gauss Linking Number Overview- Overview- LN(A, B) = 1 (ra r B ) dr A dr B 4π r A r B 3 # Interpretations of LN # Mathematicians in World

28 Linking Numbers Linking Number of Two UPOs Overview- Overview- Lefranc - Cargese

29 Evolution in Phase Space One Stretch-&-Squeeze Mechanism Overview- Overview-

30 Motion of Blobs in Phase Space Stretching Squeezing Overview- Overview-

31 Collapse Along the Stable Manifold Overview- Birman - Williams Projection Identify x and y if lim x(t) y(t) 0 t Overview-

32 Fundamental Theorem Birman - Williams Theorem Overview- If: Overview- Then:

33 Fundamental Theorem Birman - Williams Theorem Overview- Overview- If: Certain Assumptions Then:

34 Fundamental Theorem Birman - Williams Theorem Overview- Overview- If: Certain Assumptions Then: Specific Conclusions

35 Birman-Williams Theorem Overview- Overview- 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 IMPORTANT: The underlined assumptions can be relaxed.

36 Birman-Williams Theorem Overview- Overview- 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.

37 A Very Common Mechanism Overview- Overview- Attractor Rössler: Branched Manifold

38 A Mechanism with Symmetry Overview- Overview- Attractor Lorenz: Branched Manifold

39 Examples of Branched Manifolds Inequivalent Branched Manifolds Overview- Overview-

40 Aufbau Princip for Branched Manifolds Any branched manifold can be built up from stretching and squeezing units Overview- Overview- subject to the conditions: Outputs to Inputs No Free Ends

41 Dynamics and Topology Rossler System Overview- Overview-

42 Dynamics and Topology Lorenz System Overview- Overview-

43 Dynamics and Topology Poincaré Smiles at Us in R 3 Overview- Overview- Determine organization of UPOs Determine branched manifold Determine equivalence class of SA

44 Topological Analysis Program Overview- Overview- Topological Analysis Program Locate Periodic Orbits Create an Embedding Determine Topological Invariants (LN) Identify a Branched Manifold Verify the Branched Manifold - Model the Dynamics Validate the Model

45 Locate UPOs Method of Close Returns Overview- Overview-

46 Embeddings Overview- Overview- Embeddings Many Methods: Time Delay, Differential, Hilbert Transforms, SVD, Mixtures,... Tests for Embeddings: Geometric, Dynamic, Topological None Good We Demand a 3 Dimensional Embedding

47 Locate UPOs An Embedding and Periodic Orbits Overview- Overview- Lefranc - Cargese

48 Determine Topological Invariants Linking Number of Orbit Pairs Overview- Overview- Lefranc - Cargese

49 Determine Topological Invariants Compute Table of Expt l LN Overview- Overview-

50 Determine Topological Invariants Compare w. LN From Various BM Overview- Overview-

51 Determine Topological Invariants Guess Branched Manifold Overview- Overview- Lefranc - Cargese

52 Determine Topological Invariants Overview- Overview- Identification & Confirmation BM Identified by LN of small number of orbits Table of LN GROSSLY overdetermined Predict LN of additional orbits Rejection criterion

53 Determine Topological Invariants What Do We Learn? BM Depends on Embedding Some things depend on embedding, some don t Depends on Embedding: Global Torsion, Parity,.. Independent of Embedding: Mechanism Overview- Overview-

54 Perestroikas of Strange Attractors Evolution Under Parameter Change Overview- Overview- Lefranc - Cargese

55 Perestroikas of Strange Attractors Evolution Under Parameter Change Overview- Overview- Lefranc - Cargese

56 An Unexpected Benefit Analysis of Nonstationary Data Overview- Overview- Lefranc - Cargese

57 Last Steps Overview- Overview- Model the Dynamics A hodgepodge of methods exist: # Methods # Physicists Validate the Model Needed: Nonlinear analog of χ 2 test. OPPORTUNITY: Tests that depend on entrainment/synchronization.

58 Our Hope Now a Result Overview- Overview- Compare with Original Objectives Construct a simple, algorithmic procedure for: Classifying strange attractors Extracting classification information from experimental signals.

59 Orbits Can be Pruned There Are Some Missing Orbits Overview- Overview- Lorenz Shimizu-Morioka

60 Linking Numbers, Relative Rotation Rates, Braids Orbit Forcing Overview- Overview-

61 An Ongoing Problem Forcing Diagram - Horseshoe Overview- Overview-

62 An Ongoing Problem Overview- Overview- Status of Problem Horseshoe organization - active More folding - barely begun Circle forcing - even less known Higher genus - new ideas required

63 Perestroikas of Branched Manifolds Overview- Overview- Constraints on Branched Manifolds Inflate a strange attractor Union of ɛ ball around each point Boundary is surface of bounded 3D manifold Torus that bounds strange attractor

64 Torus and Genus Torus, Longitudes, Meridians Overview- Overview-

65 Flows on Surfaces Overview- Overview- Surface Singularities Flow field: three eigenvalues: +, 0, Vector field perpendicular to surface Eigenvalues on surface at fixed point: +, All singularities are regular saddles s.p. ( 1)index = χ(s) = 2 2g # fixed points on surface = index = 2g - 2

66 Flows in Vector Fields Flow Near a Singularity Overview- Overview-

67 Some Bounding Tori Torus Bounding Lorenz-like Flows Overview- Overview-

68 Canonical Forms Twisting the Lorenz Attractor Overview- Overview-

69 Constraints Provided by Bounding Tori Two possible branched manifolds in the torus with g=4. Overview- Overview-

70 Use in Physics Bounding Tori contain all known Strange Attractors Overview- Overview-

71 Labeling Bounding Tori Overview- Overview- Labeling Bounding Tori Poincaré section is disjoint union of g-1 disks Transition matrix sum of two g-1 g-1 matrices One is cyclic g-1 g-1 matrix Other represents union of cycles Labeling via (permutation) group theory

72 Some Bounding Tori Bounding Tori of Low Genus Overview- Overview-

73 Motivation Some Genus-9 Bounding Tori Overview- Overview-

74 Aufbau Princip for Bounding Tori Any bounding torus can be built up from equal numbers of stretching and squeezing units Overview- Overview- Outputs to Inputs No Free Ends Colorless

75 Aufbau Princip for Bounding Tori Application: Lorenz Dynamics, g=3 Overview- Overview-

76 Poincaré Section Construction of Poincaré Section Overview- Overview-

77 Exponential Growth The Growth is Exponential Overview- Overview-

78 Exponential Growth The Growth is Exponential The Entropy is log 3 Overview- Overview-

79 Extrinsic Embedding of Bounding Tori Extrinsic Embedding of Intrinsic Tori Overview- Overview- Partial classification by links of homotopy group generators. Nightmare Numbers are Expected.

80 Modding Out a Rotation Symmetry Modding Out a Rotation Symmetry X Y Z u v w = Re (X + iy ) 2 Im (X + iy ) 2 Z Overview- Overview-

81 Lorenz Attractor and Its Image Overview- Overview-

82 Lifting an Attractor: Cover-Image Relations Creating a Cover with Symmetry X Y Z u v w = Re (X + iy ) 2 Im (X + iy ) 2 Z Overview- Overview-

83 Cover-Image Related Branched Manifolds Cover-Image Branched Manifolds Overview- Overview-

84 Covering Branched Manifolds Two Two-fold Lifts Different Symmetry Overview- Overview- Rotation Symmetry Inversion Symmetry

85 Topological Indices Overview- Overview- Topological Index: Choose Group Choose Rotation Axis (Singular Set)

86 Locate the Singular Set wrt Image Different Rotation Axes Produce Different (Nonisotopic) Lifts Overview- Overview-

87 Nonisotopic Locally Diffeomorphic Lifts Overview- Overview-

88 Indices (0,1) and (1,1) Two Two-fold Covers Same Symmetry Overview- Overview-

89 Indices (0,1) and (1,1) Three-fold, Four-fold Covers Overview- Overview-

90 Two Inequivalent Lifts with V 4 Symmetry Overview- Overview-

91 How to Construct Covers/Images Overview- Overview- Algorithm Construct Invariant Polynomials, Syzygies, Radicals Construct Singular Sets Determine Topological Indices Construct Spectrum of Structurally Stable Covers Structurally Unstable Covers Interpolate

92 Surprising New Findings Overview- Overview- Symmetries Due to Symmetry Schur s Lemmas & Equivariant Dynamics Cauchy Riemann Symmetries Clebsch-Gordon Symmetries Continuations Analytic Continuation Topological Continuation Group Continuation

93 Covers of a Trefoil Torus Overview- Overview- Granny Knot Square Knot Trefoil Knot

94 You Can Cover a Cover = Lift a Lift Covers of Covers of Covers Overview- Overview- Rossler Lorenz Ghrist

95 Universal Branched Manifold EveryKnot Lives Here Overview- Overview-

96 Isomorphisms and Diffeomorphisms Overview- Overview- Local Stuff Groups: Local Isomorphisms Cartan s Theorem Dynamical Systems: Local Diffeomorphisms??? Anything Useful???

97 Universal Covering Group Cartan s Theorem for Lie Groups Overview- Overview-

98 Universal Image Dynamical System Locally Diffeomorphic Covers of D Overview- Overview- D: Universal Image Dynamical System

99 Useful Analogs Local Isomorphisms & Diffeomorphisms Overview- Overview-

100 Useful Analogs Local Isomorphisms & Diffeomorphisms Lie Groups Overview- Overview-

101 Useful Analogs Overview- Local Isomorphisms & Diffeomorphisms Lie Groups Local Isomorphisms Overview-

102 Useful Analogs Overview- Local Isomorphisms & Diffeomorphisms Lie Groups Local Isomorphisms Overview-

103 Useful Analogs Local Isomorphisms & Diffeomorphisms Overview- Overview- Lie Groups Local Isomorphisms Dynamical Systems

104 Useful Analogs Local Isomorphisms & Diffeomorphisms Overview- Overview- Lie Groups Local Isomorphisms Dynamical Systems Local Diffeos

105 Useful Analogs Local Isomorphisms & Diffeomorphisms Overview- Overview- Lie Groups Local Isomorphisms Dynamical Systems Local Diffeos

106 Creating New Attractors Overview- Overview- Rotating the Attractor [ d X dt Y ] = [ u(t) v(t) [ d u dt v [ F1 (X, Y ) F 2 (X, Y ) ] [ cos Ωt sin Ωt = sin Ωt cos Ωt ] ] [ a1 sin(ω + d t + φ 1 ) a 2 sin(ω d t + φ 2 ) ] [ X(t) Y (t) [ = RF(R 1 v u) + Rt + Ω +u ] ] ] Ω = n ω d Global Diffeomorphisms q Ω = p ω d Local Diffeomorphisms (p-fold covers)

107 Two Phase Spaces: R 3 and D 2 S 1 Rossler Attractor: Two Representations R 3 D 2 S 1 Overview- Overview-

108 Other Diffeomorphic Attractors Rossler Attractor: Two More Representations with n = ±1 Overview- Overview-

109 Subharmonic, Locally Diffeomorphic Attractors Rossler Attractor: Two Two-Fold Covers with p/q = ±1/2 Overview- Overview-

110 Subharmonic, Locally Diffeomorphic Attractors Rossler Attractor: Two Three-Fold Covers with p/q = 2/3, 1/3 Overview- Overview-

111 Subharmonic, Locally Diffeomorphic Attractors Overview- Overview- Rossler Attractor: And Even More Covers (with p/q = +1/3, +2/3)

112 New Measures Overview- Overview- Angular Momentum and Energy 1 τ L(0) = lim XdY Y dx τ τ 0 L(Ω) = u v v u 1 τ 1 K(0) = lim τ τ 0 2 (Ẋ2 +Ẏ 2 )dt K(Ω) = 1 2 ( u2 + v 2 ) = L(0) + Ω R 2 = K(0) + ΩL(0) Ω2 R 2 R 2 1 = lim τ τ τ 0 (X 2 + Y 2 1 )dt = lim τ τ τ 0 (u 2 + v 2 )dt

113 New Measures, Diffeomorphic Attractors Overview- Overview- Energy and Angular Momentum Diffeomorphic, Quantum Number n

114 New Measures, Subharmonic Covering Attractors Energy and Angular Momentum Subharmonics, Quantum Numbers p/q Overview- Overview-

115 Embeddings Overview- Overview- Embeddings An embedding creates a diffeomorphism between an ( invisible ) dynamics in someone s laboratory and a ( visible ) attractor in somebody s computer. Embeddings provide a representation of an attractor. Equivalence is by Isotopy. Irreducible is by Dimension

116 Representation Labels Inequivalent Irreducible Representations Irreducible Representations of 3-dimensional Genus-one attractors are distinguished by three topological labels: Overview- Overview- Parity Global Torsion Knot Type P N KT Γ P,N,KT (SA) Mechanism (stretch & fold, stretch & roll) is an invariant of embedding. It is independent of the representation labels.

117 Creating Isotopies Overview- Overview- Equivalent Reducible Representations Topological indices (P,N,KT) are obstructions to isotopy for embeddings of minimum dimension (irreducible representations). Are these obstructions removed by injections into higher dimensions (reducible representations)? Systematically?

118 Creating Isotopies Equivalences by Injection Obstructions to Isotopy Overview- Overview- R 3 Global Torsion Parity Knot Type R 4 Global Torsion R 5 There is one Universal reducible representation in R N, N 5. In R N the only topological invariant is mechanism.

119 The Road Ahead Summary Overview- Overview- 1 Question Answered 2 Questions Raised We must be on the right track!

120 Our Hope Original Objectives Achieved Overview- Overview- There is now a simple, algorithmic procedure for: Classifying strange attractors Extracting classification information from experimental signals.

121 Our Result Result Overview- Overview- There is now a classification theory for low-dimensional strange attractors. 1 It is topological 2 It has a hierarchy of 4 levels 3 Each is discrete 4 There is rigidity and degrees of freedom 5 It is applicable to R 3 only for now

122 Four Levels of Structure The Classification Theory has 4 Levels of Structure Overview- Overview-

123 Four Levels of Structure Overview- The Classification Theory has 4 Levels of Structure 1 Basis Sets of Orbits Overview-

124 Four Levels of Structure Overview- Overview- The Classification Theory has 4 Levels of Structure 1 Basis Sets of Orbits 2 Branched Manifolds

125 Four Levels of Structure Overview- Overview- The Classification Theory has 4 Levels of Structure 1 Basis Sets of Orbits 2 Branched Manifolds 3 Bounding Tori

126 Four Levels of Structure Overview- Overview- The Classification Theory has 4 Levels of Structure 1 Basis Sets of Orbits 2 Branched Manifolds 3 Bounding Tori 4 Extrinsic Embeddings

127 Four Levels of Structure Overview- Overview-

128 Topological Components Poetic Organization Overview- Overview- LINKS OF PERIODIC ORBITS organize BOUNDING TORI organize BRANCHED MANIFOLDS organize LINKS OF PERIODIC ORBITS

129 Answered Questions Overview- Overview- Some Unexpected Results Perestroikas of orbits constrained by branched manifolds Routes to Chaos = Paths through orbit forcing diagram Perestroikas of branched manifolds constrained by bounding tori Global Poincaré section = union of g 1 disks Systematic methods for cover - image relations Existence of topological indices (cover/image) Universal image dynamical systems NLD version of Cartan s Theorem for Lie Groups Topological Continuation Group Continuuation Cauchy-Riemann symmetries Quantizing Chaos Representation labels for inequivalent embeddings Representation Theory for Strange Attractors

130 Unanswered Questions Overview- Overview- We hope to find: Robust topological invariants for R N, N > 3 A Birman-Williams type theorem for higher dimensions An algorithm for irreducible embeddings Embeddings: better methods and tests Analog of χ 2 test for NLD Better forcing results: Smale horseshoe, D 2 D 2, n D 2 n D 2 (e.g., Lorenz), D N D N, N > 2 Representation theory: complete Singularity Theory: Branched manifolds, splitting points (0 dim.), branch lines (1 dim). Singularities as obstructions to isotopy