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relativity for cyclists symmetry reduction Summary
Chapter 10: Relativity for cyclists
Predrag Cvitanovic
10 May 2011
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relativity for cyclists symmetry reduction Summary
Die Vorlesung
Das Problem
If a problem has a symmetry
you MUST use it!
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relativity for cyclists symmetry reduction Summary
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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relativity for cyclists symmetry reduction Summary
complex Lorenz flow example
from complex Lorenz flow 5D attractor unimodal map
what to do?
the goal
reduce this messy strange attractor to
a 1-dimensional return map
attractor
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relativity for cyclists symmetry reduction Summary
complex Lorenz flow example
from complex Lorenz flow 5D attractor unimodal map
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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relativity for cyclists symmetry reduction Summary
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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relativity for cyclists symmetry reduction Summary
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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relativity for cyclists symmetry reduction Summary
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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relativity for cyclists symmetry reduction Summary
in/equivariance
foliation by group orbits
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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relativity for cyclists symmetry reduction Summary
in/equivariance
foliation by group orbits
group orbits
any point on the manifold Mx(t) isequivalent to any other.
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relativity for cyclists symmetry reduction Summary
in/equivariance
foliation by group orbits
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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relativity for cyclists symmetry reduction Summary
in/equivariance
foliation by group orbits
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
TW orbit in phase space
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
TW orbit in phase space
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
relativity for pedestrians
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
relativity for pedestrians
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.
relativity for cyclists symmetry reduction Summary
li & di
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slice & dice
example: SO(2) rotations for complex Lorenz equations
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
relativity for cyclists symmetry reduction Summary
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
example: SO(2) rotations for complex Lorenz equations
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.
relativity for cyclists symmetry reduction Summary
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
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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().
relativity for cyclists symmetry reduction Summary
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
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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
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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
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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
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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
relativity for cyclists symmetry reduction Summary
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...