math matters, ima march 7, 2007 · math matters, ima march 7, 2007 martin golubitsky department of...

Patterns Patterns Everywhere

Math Matters, IMAMarch 7, 2007

Martin GolubitskyDepartment of Mathematics

University of Houston

– p. 1/64

Why Study Patterns I

Patterns are surprising and pretty

Thanks to Stephen Morris and Michael Gorman

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Mud Plains

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Leopard Spots

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Convection Cells

Bodenschatz, de Bruyn, Cannell, and Ahlers, 1991– p. 5/64

Sand Dunes in Namibian Desert

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Zebra Stripes

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Convection Cells

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Porous Plug Burner Flames (Gorman)

burner

Air & fuel Inert gas

Flame front

Cooling coils

Dynamic patterns

A film in three parts

stationary patternstime-periodic patternsmore complicatedspatio-temporal patterns

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Why Study Patterns II1) Patterns are surprising and pretty

2) Emergent phenomena; self organization

Patterns often provide functionScience behind patterns

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Patterns We Use

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Patterns We Use

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Columnar Joints on Staffa near Mull

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Columns along Snake River

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Experiment on Corn Starch

Goehring and Morris, 2005

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Why Study Patterns III1) Patterns are surprising and pretty

2) Emergent phenomena; self organizationPatterns provide functionScience behind patterns

3) Change in patterns provide tests for models

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A Brief History of Navier-StokesNavier-Stokes equations for an incompressible fluid

ut = ν∇2u − (u · ∇)u −1

ρ∇p

0 = ∇ · u

u = velocity vector ρ = mass densityp = pressure ν = kinematic viscosity

Navier (1821); Stokes (1856); Taylor (1923)

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The Couette Taylor Experiment

Ω Ωo i

Ωi = speed of innercylinderΩo = speed of outercylinder

Andereck, Liu, and Swinney (1986)

Couette Taylor Spiraltime independent time periodic

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G.I. Taylor: Theory & Experiment (1923)

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Why Study Patterns IV1) Patterns are surprising and pretty2) Emergent phenomena; self organization

Patterns provide functionScience behind patterns

3) Change in patterns provide tests for models

4) Model independenceMathematics provides menu of patternsTwo Examples

Stripes and Spots in planar systemsOscillations with circle symmetry

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Planar Symmetry-BreakingEuclidean symmetry: translations, rotations, reflections

Symmetry-breaking from translation invariant state inplanar systems with Euclidean symmetry leads to

stripes or spots

Stripes: invariant under translation in one directionSand dunes, zebra, convection cells

Spots: states centered at lattice pointsmud plains, leopard, convection cells

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Circle Symmetry-Breaking Oscillation

There exist two types of time-periodic solutionsnear a circularly symmetric equilibrium

Rotating waves:Time evolution is the same as spatial rotationStanding waves:Fixed line of symmetry for all time

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Burner with Circular Symmetry

Porous plug burner (Gorman)

Parameters:flow rate and stochiometry

burner

Air & fuel Inert gas

Flame front

Cooling coils

Rotating wave time evolution = spatial rotation

Standing wave line of symmetry

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Taylor-Couette ExperimentThe states: Andereck, Liu, and Swinney (1986)

vortexTaylor

flow

Wavyoutflow

Wavy inflow+ twists

Ripple

Corkscrewwavelets

Twists

Couette flow

Wavy vortices

Wavy inflow

Modulated waves

Turbulent Taylor vortices

Spiral turbulence

-4000 -3000 -2000 -1000 0 1000

0

2000

1000

Featureless turbulenceUnexplored

Spirals

Wavy spirals

Interpenetrating spirals

Couette flow

Intermittency Wavyvortexflow

Ro

Ri

Modulated

wav

es

1

Prediction of Ribbons: Chossat, Iooss, Demay (1987)

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Taylor-Couette ExperimentThe states: Andereck, Liu, and Swinney (1986)

Prediction of Ribbons: Chossat, Iooss, Demay (1987)

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Summary on Pattern Formation

The mathematics of pattern formation leads to amenu of patterns

This menu is model independent

Physics and biology choose from the menu of pattens

This choice is model dependent

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Planar Dynamics; Symmetric ChaosLet f : R2

→ R2 be a function

Choose a point z0 in the plane

Let z1 = f(z0), z2 = f(z1), . . .

The trajectory z0, z1, z2, . . . is chaotic if it hassensitive dependence on initial conditions

To be explained by pictures — please wait

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Let z1 = f(z0), z2 = f(z1), . . .

f(x, y) = (y − x3,−x − xy − y3)

z0 = (.25, .26) z1 = (.2444,−.3326) z2 = (.3472,−.1263)



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Let z1 = f(z0), z2 = f(z1), . . .

−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0

1

2

3

4



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Let z1 = f(z0), z2 = f(z1), . . .

−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0

1

2

3

4

5

6

7

8

9

10

11

1213

14



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Let z1 = f(z0), z2 = f(z1), . . .

−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5



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Let z1 = f(z0), z2 = f(z1), . . .



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Finite Symmetries on the PlaneD3 = symmetries of an equilateral triangleD4 = symmetries of a squareD5 = symmetries of a regular pentagon

Dm consists of rotations and reflections

A planar map has symmetries Dm if

f(gz) = gf(z)

for every symmetry g in Dm

Example of map with Dm-symmetry is

f(z) = (` + azz + bRe(zm))z + czm−1

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Four Rotations Four Reflections



f(gz) = gf(z)




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f(gz) = gf(z)




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Symmetric Chaos

m=3 m=5

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IMA Attractor (Mike Field)

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Emperor’s Cloak

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Wild Chaos

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Sensitive Dependence: Golden Flintstone

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Sensitive Dependence (1 iterate)

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Sensitive Dependence (3 iterates)

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St. John’s Wort Attractor

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Mercedes Attractor

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Pentagon Attractor

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HexNuts Attractor

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Mosaic Attractor

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Consequence of Attractor SymmetryLet u(x, t) be a time series

Ergodic Theorem: Time Average = Space Average

Time average has same symmetries as attractor

Let U(x) = average of u(x, t) over t

Then U(σx) = U(x) for every symmetry σ

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Faraday Surface Wave ExperimentVibrate a fluid layer at fixed frequency and amplitude

At small amplitude — surface is flat

At large amplitudes — surface deforms

Take picture at each period of forcing

Light is transmitted through the fluidDark areas: surface is concave upBright areas: surface is concave down

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The Faraday Experiment (2)

Gollub, Gluckman, Marcq, & Bridger (1993)

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Patterns in Neuroscience

Animal GaitsStewart, Buono, Collins

Geometric Visual HallucinationsErmentrout, Cowan, Bressloff, Thomas, Wiener

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Quadruped GaitsBound of the Siberian Souslik

Amble of the Elephant

Pace of the Horse

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Standard Gait Phases

0

0 1/2

1/20

1/4 3/4

0

0

1/2

1/2

0 0

1/2 1/2

0

1/2

0.1

0.6

0

1/2

0 0

0 0

WALK PACE TROT BOUND

TRANSVERSEGALLOP

ROTARYGALLOP

PRONK0.1

0.6

1/2

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The Pronk

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Gait SymmetriesGait Spatio-temporal symmetriesTrot (Left/Right, 1

2) and (Front/Back, 1

2)

Pace (Left/Right, 1

2) and (Front/Back, 0)

Walk (Figure Eight, 1

4)

0 1/2 1

PACE:

0 1

TROT:

1/2

WALK: 1/4 1/2 3/4 1 0

Three gaits are different Collins and Stewart (1993)

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Central Pattern Generators (CPG)Assumption: There is a network in the nervous system thatproduces the characteristic rhythms of each gait

Design simplest network to produce walk, trot, and pace

Guess at simplest network

One unit ‘signals’ each leg1 2

43

No four-unit network can produce independent gaits ofwalk, trot and pace

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Simplest Coupled Unit Gait ModelUse gait symmetries to construct coupled network1) walk =⇒ four-cycle ω in symmetry group2) pace or trot =⇒ transposition κ in symmetry group3) Simplest network

LF

LH RH

RF

LH

LF RF

RH

1 2

3 4

5 6

7 8

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Primary Gaits: Oscillations from Stand

Phase Diagram Gait0

@

0 0

0 0

1

A pronk0

@

01

2

01

2

1

A pace0

@

1

20

01

2

1

A trot0

@

0 0

1

2

1

2

1

A bound0

@

±1

4±

3

4

01

2

1

A walk±

0

@

0 0

±1

4±

1

4

1

A jump±

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The Jump

Average Right Rear to Right Front = 31.2 frames

Average Right Front to Right Rear = 11.4 frames

31.211.4

= 2.74

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Biped Network

LF

LH RH

RF

LH

LF RF

RH

1 2

3 4

5 6

7 8

3 4

1 2

left right

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Biped Prediction

0 0.5 0.5 0

left right

0 0.5

left right

0 0.5

(b)(a)

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Biped Gaits: Walk and Run

0 0.5 0.5 0

left right

0 0.5

left right

0 0.5

(b)(a)

Units control timing of muscle groups

Electromyographic signals from ankle muscles

During walking: gastrocnemius (GA) andtibialis anterior (TA) are activated out-of-phase

During running: (GA) and (TA) are co-activated

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Geometric Visual Hallucinations

Patterns fall into four form constants (Klüver, 1928)

tunnels and funnels

spirals

lattices includes honeycombs and phosphenes

cobwebs

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Klüver Form Constants

funnel spiral

honeycomb cobweb– p. 60/64

Cowan-Ermentrout-Bressloff TheoryDrug uniformly forces activation of cortical cells

Leads to spontaneous pattern formation on cortex

Map from retina to primary visual cortex;translates pattern on cortex to visual image

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Planforms in Visual Field

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ConclusionsPatterns that appear in physical and biological systemsoften have their genesis in symmetryanimal stripes, convection cells, geometric visualhallucinations, etc.

Using symmetry patterns in systems with differentphysics can have the same mathematical descriptionhose swinging in a circle, rotating flame front,spiral vortices in Taylor-Couette

Complicated patterns can be built in stages by relativelysimple mechanisms—once the mathematical structureof the system is identifiedsymmetric chaos, patterns in animal gaits

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Thank You

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math matters, ima march 7, 2007 · math matters, ima march 7, 2007 martin golubitsky department of...

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