Braiding and Mixing. Jean-Luc Thiffeault and Matthew Finn. Department of Mathematics Imperial College London.

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1 Braiding and Mixing Jean-Luc Thiffeault and Matthew Finn jeanluc Department of Mathematics Imperial College London Braiding and Mixing p.1/23

2 Experiment of Boyland et al. [P. L. Boyland, H. Aref, and M. A. Stremler, J. Fluid Mech. 403, 277 (2000)] (movie by Matthew Finn) Braiding and Mixing p.2/23

3 Four Basic Operations ements PSfrag replacements σ 1 1 σ 2 σ 1 2 σ 1 σ 1 σ 1 1 σ 1 2 σ 2 ements σ 1 σ 2 σ 1 2 PSfrag replacements σ 1 1 σ 1 σ 1 1 σ 2 σ 1 2 σ 1 and σ 2 are referred to as the generators of the 3-braid group. Braiding and Mixing p.3/23

4 Two Stirring Protocols σ 1 σ 2 protocol σ 1 1 σ 2 protocol [P. L. Boyland, H. Aref, and M. A. Stremler, J. Fluid Mech. 403, 277 (2000)] Braiding and Mixing p.4/23

5 Braiding σ 1 σ 2 protocol σ 1 1 σ 2 protocol [P. L. Boyland, H. Aref, and M. A. Stremler, J. Fluid Mech. 403, 277 (2000)] Braiding and Mixing p.5/23

6 Matrix Representation of σ 2 I I II II Let I and II denote the lengths of the two segments. After a σ 2 operation, we have ( ) ( ) ( ) ( ) ( ) I I + II 1 1 I I II = = = σ 2. II 0 1 II II Hence, the matrix representation for σ 2 is ( ) 1 1 σ 2 =. 0 1 Braiding and Mixing p.6/23

7 Matrix Representation of σ 1 1 II I II I Similarly, after a σ1 1 operation we have ( ) ( ) ( ) ( ) I I 1 0 I II = = I + II 1 1 II = σ 1 1 ( ) I II. Hence, the matrix representation for σ1 1 is ( ) σ =. 1 1 Braiding and Mixing p.7/23

8 Matrix Representation of the Braid Group We now invoke the faithfulness of the representation to complete the set, ( ) ( ) σ 1 = ; σ 2 = ; σ 1 1 = ( ) ; σ 1 2 = ( ) Our two protocols have representation ( ) 1 1 σ 1 σ 2 = ; σ σ 2 = ( ) Braiding and Mixing p.8/23

9 The Difference between the Protocols The matrix associated with each generator has unit eigenvalues. Braiding and Mixing p.9/23

10 The Difference between the Protocols The matrix associated with each generator has unit eigenvalues. The first stirring protocol has eigenvalues on the unit circle Braiding and Mixing p.9/23

11 The Difference between the Protocols The matrix associated with each generator has unit eigenvalues. The first stirring protocol has eigenvalues on the unit circle The second has eigenvalues (3 ± 5)/2 = for the larger eigenvalue. Braiding and Mixing p.9/23

12 The Difference between the Protocols The matrix associated with each generator has unit eigenvalues. The first stirring protocol has eigenvalues on the unit circle The second has eigenvalues (3 ± 5)/2 = for the larger eigenvalue. So for the second protocol the length of the lines I and II grows exponentially! Braiding and Mixing p.9/23

13 The Difference between the Protocols The matrix associated with each generator has unit eigenvalues. The first stirring protocol has eigenvalues on the unit circle The second has eigenvalues (3 ± 5)/2 = for the larger eigenvalue. So for the second protocol the length of the lines I and II grows exponentially! The larger eigenvalue is a lower bound on the growth factor of the length of material lines. Braiding and Mixing p.9/23

14 The Difference between the Protocols The matrix associated with each generator has unit eigenvalues. The first stirring protocol has eigenvalues on the unit circle The second has eigenvalues (3 ± 5)/2 = for the larger eigenvalue. So for the second protocol the length of the lines I and II grows exponentially! The larger eigenvalue is a lower bound on the growth factor of the length of material lines. That is, material lines have to stretch by at least a factor of each time we execute the protocol σ 1 1 σ 2. Braiding and Mixing p.9/23

15 The Difference between the Protocols The matrix associated with each generator has unit eigenvalues. The first stirring protocol has eigenvalues on the unit circle The second has eigenvalues (3 ± 5)/2 = for the larger eigenvalue. So for the second protocol the length of the lines I and II grows exponentially! The larger eigenvalue is a lower bound on the growth factor of the length of material lines. That is, material lines have to stretch by at least a factor of each time we execute the protocol σ 1 1 σ 2. This is guaranteed to hold in some neighbourhood of the rods (Thurston Nielsen theorem). Braiding and Mixing p.9/23

16 Freely-moving Rods in a Cavity Flow [A. Vikhansky, Physics of Fluids 15, 1830 (2003)] Braiding and Mixing p.10/23

17 Particle Orbits are Topological Obstacles Choose any fluid particle orbit (green dot). Material lines must bend around the orbit: it acts just like a rod! The idea: pick any three fluid particles and follow them. How do they braid around each other? Braiding and Mixing p.11/23

18 Braiding and Mixing p.12/23

19 Detecting Braiding Events σ 2 σ 1 2 σ 1 2 ements PSfrag replacements σ 2 In the second case there is no net braid: the two elements cancel each other. Braiding and Mixing p.13/23

20 Random Sequence of Braids We end up with a sequence of braids, with matrix representation Σ (N) = σ (N) σ (2) σ (1) where σ (µ) {σ 1, σ 2, σ 1 2 events detected after a time t. 1, σ 1 } and N is the number of braiding Braiding and Mixing p.14/23

21 Random Sequence of Braids We end up with a sequence of braids, with matrix representation where σ (µ) {σ 1, σ 2, σ 1 2 events detected after a time t. Σ (N) = σ (N) σ (2) σ (1) 1, σ 1 } and N is the number of braiding The largest eigenvalue of Σ (N) is a measure of the complexity of the braiding motion, called the braiding factor. Random matrix theory says that the braiding factor can grow exponentially! We call the rate of exponential growth the braiding Lyapunov exponent or just braiding exponent. Braiding and Mixing p.14/23

22 Non-braiding Motion ents First consider the motion of of three points in concentric circles with irrationally-related frequencies. y x Eigenvalue of Σ (N) The braiding factor grows linearly, which means that the braiding exponent is zero. Notice that the eigenvalue often returns to unity. t Braiding and Mixing p.15/23

23 Blinking-vortex Flow To demonstrate good braiding, we need a chaotic flow on a bounded domain (a spatially-periodic flow won t do). Aref s blinking-vortex flow is ideal. Active Vortex Inactive Vortex First half of period Second half of period The only parameter is the circulation Γ of the vortices. Braiding and Mixing p.16/23

24 Blinking Vortex: Non-braiding Motion ents For Γ = 0.5, the blinking vortex has only small chaotic regions. y x Eigenvalue of Σ (N) One of the orbits is chaotic, the other two are closed. t Braiding and Mixing p.17/23

25 Blinking Vortex: Braiding Motion ents For Γ = 13, the blinking vortex is globally chaotic. y x Eigenvalue of Σ (N) The braiding factor now grows exponentially. In the same time interval as for Γ = 0.5, the final value is now of order rather than 80! t Braiding and Mixing p.18/23

26 Averaging over many Triplets 400 Log of eigenvalue of Σ (N) replacements slope = Γ = Averaged over 100 random triplets. t Braiding and Mixing p.19/23

27 Comparison with Lyapunov Exponents Lyapunov exponent g replacements Line stretching exponent Lyapunov exponent Braiding Lyapunov exponent Γ varies from 8 to 20. Braiding and Mixing p.20/23

28 Beyond Three Particles even odd Braiding exponent eplacements Γ = n Braiding and Mixing p.21/23

29 But does it Saturate? Braiding exponent eplacements even odd Γ = n Braiding and Mixing p.22/23

30 Conclusions Topological chaos involves moving obstacles in a 2D flow, which create nontrivial braids. The complexity of a braid can be represented by the largest eigenvalue of a product of matrices the braiding factor. Any collection of n particles can potentially braid. The complexity of the braid is a good measure of chaos. No need for infinitesimal separation of trajectories or derivatives of the velocity field. For instance, can use all the floats in a data set (J. La Casce). Test in 2D turbulent simulations (F. Paparella). Many issues to investigate: faithfulness of representation, lower-bound for topological entropy... Braiding and Mixing p.23/23