braid dynamics of non-periodic trajectories

8/12/2019 Braid Dynamics of Non-Periodic Trajectories

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Department of Mathematics

10th AIMS Conference on Dynamical Systems, Differential Equations and ApplicationsMadrid, Spain, July 2014

Marko Budi ! i" Jean-Luc Thiffeault

BRAID DYNAMICS OF NON-PERIODIC TRAJECTORIES

Supported by NSF grantsDMS-0806821 (JLT) andCMMI-1233935 (MB, JLT)
http://www.math.wisc.edu/


2/13

Jul 7, 2014 M. Budi ! i" : Braid dynamics of non-periodic trajectories

Braids represent tangling of trajectories in space-time.

2

60 40 20

40

50

60

70

Latitude

L o n g

i t u

d e

Spaghetti plot ofoceanic oats

050 100

405060

7060

40

20

TimeLongitude

L a t i t u d e

Trajectories in space-time,physical braid

0 50 10040

50

60

70

Time

L a t i t u d e

Track the order ofstrands along an axis

Algebraic braid

k 1k

k

k + 1

k

k + 1

2 2 11 2

11


3/13


Deformation of material by the ow isrepresented by action of a braid on rubber band loops.

3

2 2 11 2

11

11

2

` Z 2 S 4

State space ! Loops

How to represent this action operationally (for numerics) ?(Dynnikovcoordinates)

Flow map ! Braid (nonlinear maps,involving maxoperation)

Takeaway : Braid dynamics are a projection ofthe full dynamics onto integer-based dynamics.

i : Z2 S 4

Z2 S 4

[Dynnikov, 2002][Hall, Yurtas, 2009]


4/13


Braids in synthesis of dynamics: optimizing stirring.

4

. . . 1 12 1 12

| {z } B

1 12

. . .. . . 1 2 1 2 | {z }

A

1 2 . . .

Material is mixed by stirring the uid using three rods.

Braids A and B are formed by rod trajectories . Both braids are periodic .

Protocol A Protocol B

Growth of loops Braiding factor

Rate of physical material growth is larger than the braiding factor .

Synthesis: To increase the ow mixing rate,

nd periodic protocols with large braiding factors.

| ` | , |A ` | , . . . , |A n ` | eh (A ) n

[Boyland, Aref, Stremler, 2000][Thiffeault, 2005, 2010]


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Braids in analysis of dynamics: detecting coherent sets.

5

Input: 40 nite-length trajectories

seeded in a moving uid.1. Form the (nite-length) braid.2. Find loops ( ! vectors) with

minimal growth.

3. Those loops contain strandsseeded inside coherentstructures.

Coherent structures found fromonly 40 time-traces,without detailed knowledgeof the velocity eld.

Squares: Initial conditions oftrajectories forming the braid

Blue: Poincar plot of dynamicsRed: Loops enclosing

coherent trajectories [Allshouse, Thiffeault, 2012]


6/13

Sep 26, 2013 M. Budi ! i" : Mesochronic Analysis 6

Synthesis : braiddenes the ow.

Rods form a

distinguished braid. ]

:

Analysis: owdenes the braid.Distinguished braid

unknown.

Goal: Study what properties connect all these braids,in order to understand the underlying ow.

Classical braid theory tells us a lot about a single braid.From a single ow , we can get many algebraic braids.

length of trajectories, number of trajectories, and positions of initial conditions.

Variables: effect of braids applied to loops

braiding factor structure of braids number and

distribution of generators ! k .

Observe/explain:


7/13Jul 7, 2014 M. Budi ! i" : Braid dynamics of non-periodic trajectories

Numerical investigation: braidlab applied toregularized Aref's Blinking Vortex ow.

7

braidlab is a MATLAB toolbox used to construct and analyze braids.

It is actively maintained and freely (GPL) available at: https://bitbucket.org/jeanluc/braidlab

Parametrized bynon-dimensionalcirculation " .Red circles regularizesingularities.

Flow blinks between two equal vortices.Vortices are fully determined by a log-typestream function.Full ow is piecewise integrable.

In the studied range of parameters " , the ow is mixing on most of the domain.

Study: Action of the braid : scaling of braiding factor as more trajectories are

added to the braid. Structure of the braid : distribution of generators in the braid.
https://bitbucket.org/jeanluc/braidlab



Braiding factor increases according to a power lawas trajectories are added to the braid.

8

100 101 102 10310

1

100

D i s t a n c e

t o l i m i t

# strands [S]

10 100 200 300 400 500 600 700 800 90010000

0.4

0.8

1.2

1.6

2

# strands [S]

B r a

i d i n g

f a c

t o r

Circulation = 4.0, # samples = 12Limit E = 1.76, rate R = 0.38

Power law E + a SR

prediction bounds

Add more concurrenttrajectories to the braid.

(uniformly initialized)

Non-periodic braid:i

|A n `

| e h (A ) n

Braiding factor

h(

B) := lim

1 ln |

B

`|



(Empirical) Power law models braiding factor acrossa range of circulation values.

9

100 101 102 10310

1

100

D i s t a n c e

t o l i m

i t

# strands [S]

10 100 200 300 400 500 600 700 800 90010000

0.40.81.21.6

22.42.83.23.64

# strands [S]

B r a

i d i n g

f a c

t o r

Circulation = 10.0, # samples = 12Limit E = 3.76, rate R = 0.30

Power law E + a SR

prediction bounds

The worstt

is stilla good t.

Can we extrapolate?

3 4 5 6 7 8 9 10

123

45

Circulation

B r a

i d i n g

f a c

t o r

LimitS = 500

3 4 5 6 7 8 9 100.6

0.5

0.40.3

0.2

0.10

Circulation

E x p o n

e n t



How does the braiding factorcompare with topological entropy?

10

Connection:

Topological entropy measuresexponential growth of material lines.Braiding factor measures

exponential growth of loops.Motivation:

Estimate topological entropy using only anite number of trajectories. 3 4 5 6 7 8 9 10

1

1.5

2

2.5

3

Circulation

D e f o r m a t

i o n r a

t e

Material deformation = 4, # = 4, dT = 0.0050

Computed by material line interpolation.

Computed braiding factor matches topological entropy.

Extrapolation overestimates topological entropy.

Causes?a) Material line computation did

not resolve all the entropy.b) Braiding factor does not truly

scale as the tted power law.3 4 5 6 7 8 9 100

123456

Circulation

E n t r o p

i e s

Braid f. S = 500MaterialLimit braid f.

E x p e c t e d

E x t r a p o l a t e d

Overlay of braid factor and material deformation



The simplest quantier of structure of a braidis the winding number of a pair of trajectories.

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What is the distribution of the winding number after a xed amount of time?

1 + 1 + (-1) = 1

1

1

1

1

Two points give a single generator.

Winding number is the length of the braid with 2 strands,after cancellations.

When dynamics are a random walk , distributions are known analytically:

Cauchy

Sech

Sech 2

Gaussian

Diffusion in plane

RW insidereecting circle

RW insideabsorbing circle

For reference

1 (1 + x 2 )

4cosh2 ( x/ 2)

12 cosh( x/ 2)

Winding number at xed time[Tumasz, PhD thesis, UW-M][Wen, Thiffeault, in progress]



Winding number of Aref Blinking Vortex is different fromrandom walk: it is a Gaussian random variable.

12

Winding around origin Winding of a pair of trajectories

Difference?

Two points : simple random walk on Z Three points:Cayley graphsfor braid group? 1

2

1

2


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Sep 26, 2013 M. Budi ! i" : Mesochronic Analysis 13

Continuousdynamics on

R 2

Physical braidof trajectoriesin space-time

Analytics:Algebraic

Braid

Numerics:Discrete

dynamics on

ZN

Goal:Move the braids from data analysis toolto dynamical analysis tool.

Objective:Understand both structure and action ofbraids arising in dynamical systems.

Two examples of this approach: Braiding factor scaling with number of trajectories analyzed:

empirical power law, but lacks predictive power . Winding number of braids do not distribute as random walk /

diffusion on the same domain.

braid dynamics of non-periodic trajectories

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