ChaosBook.org Chapter 10 overheads: Relativity for cyclists

8/6/2019 ChaosBook.org Chapter 10 overheads: Relativity for cyclists

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relativity for cyclists symmetry reduction Summary

Chapter 10: Relativity for cyclists

Predrag Cvitanovic

10 May 2011


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Die Vorlesung

Das Problem

If a problem has a symmetry

you MUST use it!


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complex Lorenz flow example

from complex Lorenz flow 5D attractor unimodal map

complex Lorenz equations

x1x2

y1y2z

=

x1 + y1x2 + y2

(1 z)x1 2x2 y1 ey22x1 + (1 z)x2 + ey1 y2

bz + x1y1 + x2y2

1 = 28, 2 = 0, b = 8/3, = 10,e = 1/10

A typical {x1, x2, z} trajectory

superimposed: a trajectory

whose initial point is close to the

relative equilibrium Q1

attractor


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what to do?

the goal

reduce this messy strange attractor to

a 1-dimensional return map

attractor


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the goal attained

but it will cost you

after symmetry reduction; must learn

how to quotient the SO(2) symmetry

1D return map!

0 100 200 300 400 500

0

100

200

300

400

500

sn

sn1

l i i f li d i S


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Lie groups, algebras

example: SO(2) rotations for complex Lorenz equations

Let g. be an element of a Lie group, for examp., SO(2) rotationby finite angle :

g() =

cos sin 0 0 0 sin cos 0 0 0

0 0 cos sin 00 0 sin cos 0

0 0 0 0 1

l ti it f li t t d ti S


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in/equivariance

symmetries of dynamics

A flow x = v(x) is G-equivariant if

v(x) = g1 v(g x) , for all g G.



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in/equivariance

foliation by group orbits

group orbits

group orbitMx of x is the set of allgroup actions

Mx = {g x | g G}



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in/equivariance


group orbits

group orbit Mx(0) of state spacepoint x(0), and the group orbit Mx(t)reached by the trajectory x(t) time tlater.



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in/equivariance


group orbits

any point on the manifold Mx(t) isequivalent to any other.



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in/equivariance


group orbits

action of a symmetry group foliates

the state space into a union of group

orbits, each group orbit an

equivalence class



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in/equivariance


group orbits

the goal:replace each group orbit by

a unique point

in a lower-dimensional reduced state

space(or orbit space)



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y y y y y

in/equivariance

a traveling wave

TW orbit in phase space

Returns at

T = L / c ,

wavelength L

relative equilibrium

(traveling wave, rotating

wave)

xTW() MTW : thedynamical flow field points

along the group tangent

field, with constant angular

velocity c, and the trajectorystays on the group orbit



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y y y y y

in/equivariance

a traveling wave


Returns at

T = L / c ,

wavelength L

relative equilibrium

velocity and group tangent

coincide

v(x) = ct(x) , x MTW

x() = g(c) x(0) .



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in/equivariance

a traveling wave


Returns at

T = L / c ,

wavelength L

group orbit g(c) x(0)coincides with the

dynamical orbit x() MTWand is thus flow invariant



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in/equivariance

a traveling wave

Axial shifts of TW state

TW orbit

coincides with itsgroup orbit



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in/equivariance

a traveling wave

Reduction of TW orbit to point by shifts



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in/equivariance

a traveling wave

Reduce all TWs into a single slice



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in/equivariance

a traveling wave



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in/equivariance

a traveling wave

All TWs points in the slice



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in/equivariance

a relative periodic orbit

x1

pgx(T )p

x2

pg t

pg v

x(0)

x3

t

v

relative periodic orbit

xp(0) = g(p)xp(Tp)

exactly recurs at a fixed

relative period Tp, but

shifted by a fixed group

action g(p)



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in/equivariance

a relative periodic orbit

x1

pgx(T )p

x2

pg t

pg v

x(0)

x3

t

v

The group action

parameters

= (1, 2, N)are irrational: trajectory

sweeps out ergodically the

group orbit without ever

closing into a periodic orbit.



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in/equivariance

relativity for pedestrians

try a co-moving coordinate frame?

(a)

v1v2

v3

A relative periodic orbit of the Kuramoto-Sivashinsky flow,

traced for four periods Tp, projected on

(a) a stationary state space coordinate frame{

v1,

v2,

v3}

;



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in/equivariance


try a co-moving coordinate frame?

(b)

v

1

v

2

v

3

A relative periodic orbit of the Kuramoto-Sivashinsky flow,

traced for four periods Tp, projected on

(b) a co-moving{

v1,

v2,

v3}

frame



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in/equivariance


no good global co-moving frame!

(b)

v

1

v

2

v

3

this is no symmetry reduction at all;

all other relative periodic orbits require their own frames,

moving at different velocities.



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symmetry reduction

in the reduced space

all points related by symmetries are mapped to the same

point

families of solutions are mapped to a single solution

relative equilibria become equilibria

relative periodic orbits become periodic orbits



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Happy families are all alike;

every unhappy family is unhappy in its own way

everybody, her mother,

and Robert MacKay knows how to do this

except the author of



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masters of group theory



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reduction methods

1 Hilbert polynomial basis: rewrite equivariant dynamics in

invariant coordinates2 moving frames, or slices: cut group orbits by a

hypersurface (kind of Poincar section), each group orbit of

symmetry-equivalent points represented by single point



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reduction methods

1 Hilbert polynomial basis: global, but useless in

dimensions larger than 10 - will not discuss here2 moving frames, or slices: each group orbit of

symmetry-equivalent points represented by single point -

local



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state space portrait of complex Lorenz flow

drift induced by continuous symmetry

x1 x2

z

E0

Q1

01

x1 x2

z

E0

W0u

W1u

Q1

01

A generic chaotic trajectory (blue), the E0 equilibrium, a

representative of its unstable manifold (green), the Q1 relative

equilibrium (red), its unstable manifold (brown), and one repeat

of the 01 relative periodic orbit (purple).



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slice & dice

Lie groups elements, Lie algebra generators

An element of a compact Lie group:

g() = eT , T =

aTa, a= 1, 2, , N

T is a Lie algebraelement, and a are the parameters of thetransformation.


li & di


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slice & dice


SO(2) rotation by finite angle :

g() =

cos sin 0 0 0 sin cos 0 0 0

0 0 cos sin 00 0 sin cos 00 0 0 0 1


li & di


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slice & dice

Lie algebra generators

Ta generate infinitesimal transformations: a set of N linearlyindependent [dd] anti-hermitian matrices, (Ta)

= Ta, actinglinearly on the d-dimensional state space M


T =

0 1 0 0 0

1 0 0 0 00 0 0 1 0

0 0 1 0 0

0 0 0 0 0

The action of SO(2) on the complex Lorenz equations statespace decomposes into m = 0 G-invariant subspace (z-axis)and m = 1 subspace with multiplicity 2.


slice & dice


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slice & dice

group tangent fields

flow field at the state space point x induced by the action of the

group is given by the set of N tangent fields

ta(x)i = (Ta)ijxj


slice & dice


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slice & dice

slice & dice

flow reduced to a slice

Slice M through the slice-fixing point x, normal to the grouptangent t at x, intersects group orbits (dotted lines). The full

state space trajectory x() and the reduced state spacetrajectory x() are equivalent up to a group rotation g().


slice & dice


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slice & dice

flow within the slice

slice fixed by x

reduced state space

M flow u(x)

u(x) = v(x) (x) t(x) , x M

a(x) = (v(x)Tta)/(t(x)

T t) .

together with the reconstruction equation for the group phases

flow


slice & dice


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slice & dice

a traveling wave

Application to a relative periodic orbit (RPO)

Can project anytra ector into the slice relativity for cyclists symmetry reduction Summary

slice & dice


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s ce & d ce

a traveling wave

Application to a relative periodic orbit (RPO)

RPO PO within


slice & dice


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a traveling wave

TW

RPO

N dim

(N-1) dim (N)

Automatic removal of strong shift (gives c for TW)

TW point

RPO PO


slice & dice


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slice trouble 1

glitches!

group tangent of a generic trajectory orthogonal to the slicetangent at a sequence of instants k

t(k)T t = 0


slice & dice


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slice trouble 1

portrait of complex Lorenz flow in reduced state space

(a)

x2

y2

z

W

0

u

01

Q1

(b)

x2

y2

zW

0

u

01

Q1

E0

all choices of the slice fixing point x exhibit flow discontinuities

/ jumps


slice & dice


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slice trouble 2

slice cuts a relative periodic orbitmultiple times

Relative periodic orbit

intersects a hyperplaneslice in 3 closed-loop

images of the relative

periodic orbit and 3

images that appear to

connect to a closed loop.



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summary

conclusion

Symmetry reduction by slicing:

efficiently implemented, allows exploration of

high-dimensional flows with continuous symmetry.

stretching and folding of unstable manifolds in reduced

state space organizes the flow

to be done

construct Poincar sections and return maps

find all (relative) periodic orbits up to a given period.

use the information quantitatively (periodic orbit theory).



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amazing theory! amazing numerics! frustration...

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ChaosBook.org Chapter 10 overheads: Relativity for cyclists