Symmetry Breaking Bifurcations
of the Information Distortion
Dissertation Defense
April 8, 2003
Albert E. Parker III
Complex Biological Systems
Department of Mathematical Sciences
Center for Computational Biology
Montana State University
Goal: Solve the
Information Distortion Problem
The goal of my thesis is to solve the Information Distortion problem, an
optimization problem of the form
maxqG(q) constrained by D(q)D0
where
•  is a subset of Rn.
• G and D are sufficiently smooth in .
• G and D have symmetry: they are invariant to some group action.
Problems of this form arise in the study of clustering problems or optimal
source coding systems.
Goal: Another Formulation
Using the method Lagrange multipliers, the goal of finding solutions of
the optimization problem can be rephrased as finding stationary points
of the problem
maxqF(q,) = maxq(G(q)+D(q))

where
•  [0,).
•  is a subset of RNK.
• G and D are sufficiently smooth in .
• G and D have symmetry: they are invariant to some group action.
How: Determine the
Bifurcation Structure
We have described the bifurcation structure of stationary points to any
problem of the form
maxqF(q,) = maxq(G(q)+D(q))

where
•  [0,).
•  is a linear subset of RNK.
• G and D are sufficiently smooth in .
• G and D have symmetry: they are invariant to some group action.
Thesis Topics
 The Data Clustering Problem
 The Neural Coding Problem
 Information Theory / Probability Theory
 Optimization Theory
 Dynamical Systems
 Bifurcation Theory with Symmetries
 Group Theory
 Continuation Techniques
Outline of this talk
The Data Clustering Problem
A Class of Optimization Problems
Bifurcation with Symmetries
Numerical Results
The Data Clustering Problem
q(YN|Y) : a clustering
Y
K objects {yi}
YN
N objects {yNi}
• Data Classification: identifying all of the books printed in 2002
which address the martial art Kempo
• Data Compression: converting a bitmap file to a jpeg file
A Symmetry: invariance to
relabelling of the clusters of YN
class 1
class 2
q(YN|Y) : a clustering
Y
K objects {yi}
YN
N objects {yNi}
A Symmetry: invariance to
relabelling of the clusters of YN
class 2
class 1
q(YN|Y) : a clustering
Y
K objects {yi}
YN
N objects {yNi}
Requirements of a
Clustering Method
• The original data is represented reasonably
well by the clusters
– Choosing a cost function, D(Y,YN) , called a distortion function, rigorously defines
what we mean by the “data is represented reasonably well”.
• Fast implementation
Examples
optimizing at a distortion level D(Y,YN)  D0
• Deterministic Annealing (Rose 1998)
max H(YN|Y)
C, q  
A Fast Clustering Algorithm
constrained by D(Y,YN)  D0
• Rate Distortion Theory (Shannon ~1950)
Minimum Informative Compression
min I(Y,YN) constrained by D(Y,YN)  D0
q


 : q( y N | y ) |  q( y N | y )  1, y  Y    NK
y N YN


Inputs and Outputs and Clustered Outputs
Inputs
X
L objects {xi}
Outputs
p(X,Y)
Y
Clusters
q(YN |Y)
K objects {yi}
YN
N objects {yNi}
• The Information Distortion method clusters the outputs Y into
clusters YN so that the information that one can learn about X
by observing YN , I(X;YN), is as close as possible to the
mutual information I(X;Y)
• The corresponding information distortion function is
DI(Y;YN)=I(X;Y) - I(X;YN )
Two optimization problems which use the
information distortion function
• Information Distortion Method (Dimitrov and Miller 2001)
max H(YN|Y) constrained by DI(Y,YN)  D0
q
max H(YN|Y) +  I(X;YN)
q
• Information Bottleneck Method (Tishby, Pereira, Bialek 1999)
min I(Y,YN) constrained by DI(Y,YN)  D0
q
max –I(Y,YN) +  I(X;YN)
q
An annealing algorithm
to solve
maxqF(q,) = maxq(G(q)+D(q))

Let q0 be the maximizer of maxq G(q), and let 0 =0. For k  0, let (qk , k ) be
a solution to maxq G(q) +  D(q ). Iterate the following steps until
K =  max for some K.
1. Perform  -step: Let  k+1 = k + dk where dk>0
2. The initial guess for qk+1 at  k+1 is qk+1(0) = qk +  for some small
perturbation .
3. Optimization: solve maxq (G(q) +  k+1 D(q)) to get the maximizer qk+1 ,
using initial guess qk+1(0) .
Application of the annealing method to the Information Distortion problem
maxq (H(YN|Y) +  I(X;YN))
when p(X,Y) is defined by four gaussian blobs
Y
q(YN |Y)
YN
p(X,Y)
X
52 objects
Inputs
Outputs
Y
52 objects
52 objects
N objects
I(X;YN)=D(q(YN|Y))
Observed Bifurcations
for the Four Blob problem:
We just saw the optimal clusterings q* at some  *=  max . What do the clusterings look like for <  max ??
Conceptual Bifurcation Structure
q*
q* 
1
N

Observed Bifurcations for the 4 Blob Problem
??????
Observed Bifurcations for the 4 Blob Problem
Conceptual Bifurcation Structure
q*

Why are there only 3 bifurcations observed? In general, are there only N-1 bifurcations?
What kinds of bifurcations do we expect: pitchfork-like, transcritical, saddle-node, or some
other type?
How many bifurcating solutions are there?
What do the bifurcating branches look like? Are they subcritical or supercritical ?
What is the stability of the bifurcating branches? Is there always a bifurcating branch
which contains solutions of the optimization problem?
Are there bifurcations after all of the classes have resolved ?
Bifurcations with symmetry
•
To better understand the bifurcation structure, we capitalize on
the symmetries of the function G(q)+D(q)
•
The “obvious” symmetry is that G(q)+D(q) is invariant to
relabelling of the N classes of YN
switch
labels
1 and 3
•
The symmetry group of all permutations on N symbols is SN.
Symmetry Breaking Bifurcations
q*
q 1 is fixed by S N  S 4
N
q1 
N
1

4

Symmetry Breaking Bifurcations
q is fixed by S N 1  S3
*
q* 
q*
q 1 is fixed by S N  S 4
N
q1 
N
1

4

Symmetry Breaking Bifurcations
q is fixed by S N 1  S3
*
q* 
q * is fixed by S N  2  S 2
q* 
q*
q 1 is fixed by S N  S 4
N
q1 
N
1

4

Symmetry Breaking Bifurcations
q*

Symmetry Breaking Bifurcations
q * is fixed by ( N  cycle ) p  (1324) 2  (12)(34)
q* 
q*

Existence Theorems for
Bifurcating Branches
q*
Given a bifurcation at a point fixed by SN ,

• Equivariant Branching Lemma
(Vanderbauwhede and Cicogna 1980-1)
•
There are N bifurcating branches, each which have symmetry SN-1 .
• The Smoller-Wasserman Theorem
(Smoller and Wasserman 1985-6)
•
There are bifurcating branches which have symmetry <(N-cycle)p> for every prime p|N, p<N.
Existence Theorems for
Bifurcating Branches
q*
Given a bifurcation at a point fixed by SN-1 ,

• Equivariant Branching Lemma
(Vanderbauwhede and Cicogna 1980-1)
•
Gives N-1 bifurcating branches which have symmetry SN-2 .
• The Smoller-Wasserman Theorem
(Smoller and Wasserman 1985-6)
•
Gives bifurcating branches which have symmetry <(M-cycle)p> for every prime p|N-1, p<N-1 .
When N = 4, N-1=3, there are no bifurcating branches given by SW Theorem.
Bifurcation Structure
corresponds with
Group Structure
A partial subgroup lattice for S4 and the corresponding bifurcating
directions given by the Equivariant Branching Lemma
S4
S3
 3v 
 
 v
 v
 v
 
0 
S2 S2 S2
 0 
 
 2v 
 v
 v
 
0 
 0 
 
 v
 2v 
 v
 
0 
 0 
 
 v
 v
 2v 
 
0 
S3
 v
 
 3v 
 v
 v
 
0 
S2 S2 S2
 2v 
 
 0 
 v
 v
 
0 
 v
 
 0 
 2v 
 v
 
0 
 v
 
 0 
 v
 2v 
 
0 
1
S3
 v
 
 v
 3v 
 v
 
0 
S2 S2 S2
 2v 
 
 v
 0 
 v
 
0 
 v
 
 2v 
 0 
 v
 
0 
 v
 
 v
 0 
 2v 
 
0 
S3
 v
 
 v
 v
 3v 
 
0 
S2 S2 S2
 2v 
 
 v
 v
 0 
 
 0 
 v
 
 2v 
 v
 0 
 
 0 
 v
 
 v
 2v 
 0 
 
 0 
A partial subgroup lattice for S4 and the corresponding bifurcating
directions given by the Smoller-Wasserman Theorem
S4
Fix ( (1234 ) )  0
A4 Fix ( A4 )  0
 (1324) 
 12, 34 
 v 
 
 v 
 v
 
 v
 
 13, 24 
 14, 23 
 v 
 
 v
 v
 
 v 
 
 v 
 
 v
 v 
 
 v
 
Conceptual Bifurcation
Structure
q*

The Equivariant Branching Lemma shows that the bifurcation structure from SM to SM-1 is …
Conceptual Bifurcation
Structure
q*

Group Structure
S4
S3
S2 S2 S2
S3
S2
S2
S3
S2
S2
1
S2 S2
S3
S2
S2
S2
The Equivariant Branching Lemma shows that the bifurcation structure from SM to SM-1 is …
Conceptual Bifurcation
Structure
q*
q*

Group Structure
S4
S3
S2 S2 S2
S3
S2
S2
S3
S2
S2
S2 S2
S3
S2
S2
S2
1

The Smoller-Wasserman Theorem shows additional structure …
Conceptual Bifurcation
Structure
q*
q*

Group Structure
S4
 (1324) 
A4
 12, 34 
 13, 24 
 14, 23 

The Smoller-Wasserman Theorem shows additional structure … 3 branches from the S4 to S3 bifurcation only.
Conceptual Bifurcation
Structure
q*
q*

Group Structure
S4
 (1324) 
A4
 12, 34 
 13, 24 
 14, 23 

If we stay on a branch which is fixed by SM , how many bifurcations are there?
Conceptual Bifurcation
Structure
q*
q*


Theorem: There are at exactly K/N bifurcations on the branch (q1/N ,  ) for the Information Distortion problem
Conceptual Bifurcation
Structure
q*
q*
There are 13
bifurcations
on the first
branch

Group Structure
S4
 (1324) 
A4
 12, 34 
 13, 24 
 14, 23 

Bifurcation theory
in the presence of symmetries
enables us to answer the
questions previously posed …
??????
Observed Bifurcations for the 4 Blob Problem
Conceptual Bifurcation Structure
q*

Why are there only 3 bifurcations observed? In general, are there only N-1 bifurcations?
What kinds of bifurcations do we expect: pitchfork-like, transcritical, saddle-node, or some other
type?
How many bifurcating solutions are there?
What do the bifurcating branches look like? Are they subcritical or supercritical ?
What is the stability of the bifurcating branches?
solutions of the optimization problem?
Is there always a bifurcating branch which contains
Are there bifurcations after all of the classes have resolved ?
Conceptual Bifurcation Structure
??????
Observed Bifurcations for the 4 Blob Problem
q*

Why are there only 3 bifurcations observed? In general, are there only N-1 bifurcations?
There are N-1 symmetry breaking bifurcations from SM to SM-1 for M  N.
What kinds of bifurcations do we expect: pitchfork-like, transcritical, saddle-node, or some other
type?
How many bifurcating solutions are there? There are at least N from the first bifurcation, at
least N-1 from the next one, etc.
What do the bifurcating branches look like? They are subcritical or supercritical depending on
the sign of the bifurcation discriminator (q*,*,uk) .
What is the stability of the bifurcating branches? Is there always a bifurcating branch which
contains solutions of the optimization problem? No.
Are there bifurcations after all of the classes have resolved ? In general, no.
We can explain the
bifurcation structure
of all problems of the form
maxq F(q, ) = maxq (G(q)+D(q))

where
•  [0,).
•  is a subset of RNK.
• G and D are sufficiently smooth in .
• G and D are invariant to relabelling of the classes of YN
• The blocks of the Hessian q(G+ D) at bifurcation satisfy a set of
generic conditions.
This class of problems includes the Information Distortion problem.
 q , L is the constraine d Hessian
 q F is unconstrai ned Hessian
 q , L is singular
 q F is singular
N M
B  R  MI K
B  Ri1  MI K
is nonsingula r
is singular
is nonsingula r
i 1
is singular
chapter 6
M 1
Non-generic
N M
B  R  MI K
Symmetry
breaking
bifurcation
i 1
1
i
i 1
M 1
Impossible
scenario
Saddle-node
bifurcation
chapter 4
i 1
N M
chapter 8
B  R  MI K
1
i
1
i
chapter 6
N M
 q F is nonsingula r
Impossible
scenario
Continuation techniques
provide
numerical confirmation
of the theory
Previously Observed Bifurcation Structure
for the Four Blob problem:
Equivariant Branching Lemma:
Previous vs. Actual Bifurcation Structure
Previous results:
Actual structure:
Singularity of F:
*
Singularity of L :
We used Continuation Techniques and the Theory of Bifurcations with Symmetries on the
4 Blob Problem using the Information Distortion method to get this picture.
q*

Smoller-Wasserman
Theorem: there are
bifurcating branches with
symmetry <(1324)2> = <(12)(34)>
q*

A closer look …
q*

Bifurcation from S4 to S3…
q*

The bifurcation from S4 to S3 is subcritical …
(the theory predicted this since the bifurcation discriminator (q1/4,*,u)<0 )
Bifurcation from S3 to S2…
q*

The bifurcation from S3 to S2 is subcritical …
Bifurcation from S2 to S1…
q*

The bifurcation from S2 to S1 …
q*

What are these branches ???
Conclusions
Theorem: In general, either symmetry breaking bifurcations or saddle-node
bifurcations can occur.
Outline of proof: The Equivariant Branching Lemma, Smoller-Wasserman
Theorem, and the following singularity structure:
 q , L is singular
 q F is singular
N M
N M
B
R
 Ri1  MI K
B
is singular
is non - singular
i 1
i 1
M 1
Non-generic
Symmetry
breaking
bifurcation
1
i
 MI K
 q F is non - singular
N M
N M
B
R
 Ri1  MI K
B
is singular
is non - singular
i 1
i 1
1
i
 MI K
M 1
Impossible
Scenario
Saddle-node
bifurcation
Impossible
scenario
Conclusions
q*
Theorem: All symmetry breaking bifurcations from
SM to SM-1 are pitchfork-like, and there exists M
bifurcating branches, for which we have explicit
directions.

Conclusions
Theorem: The bifurcation discriminator of the pitchfork-like branch
(q*,*,*) + (tu,0,(t)) is
 2L
 (q  , u )  3 u,  L[u, L E r ,s
[v]r [v]s  ( M 2  3M  3) v,  4 F [v, v, v]
qr qs
*
*
3
1
If (q*,*,uk) < 0, then the branch is subcritical. If (q*,*,uk) > 0, then
the branch is supercritical.
Conclusions
Theorem: Solutions of the optimization problem  do not always persist
from bifurcation.
Theorem: In general, bifurcations do not occur after all of the classes
have resolved.
A numerical algorithm to solve max(G(q)+D(q))
q

Let q0 be the maximizer of maxq G(q), 0 =1 and s > 0. For k  0, let (qk , k ) be a
solution to maxq G(q) +  D(q ). Iterate the following steps until
K =  max for
some K.
   qk 

1. Perform  -step: solve  q , L (qk , k ,  k )
        q , L (qk , k ,  k )
  k
   qk 
 and select  k+1 = k + dk where
for 

   k 
dk = (s sgn(cos )) /(||qk ||2 + ||k ||2 +1)1/2.
2.
The initial guess for (qk+1,k+1) at  k+1 is
(qk+1(0),k+1 (0)) = (qk ,k) + dk ( qk, k) .
3.
Optimization: solve maxq (G(q) +  k+1 D(q)) using pseudoarclength continuation
to get the maximizer qk+1, and the vector of Lagrange multipliers k+1 using initial
guess (qk+1(0),k+1 (0)).
4.
Check for bifurcation: compare the sign of the determinant of an identical block
of each of q [G(qk) +  k D(qk)] and q [G(qk+1) +  k+1 D(qk+1)]. If a bifurcation is
detected, then set qk+1(0) = qk + d_k u where u is bifurcating direction and repeat
step 3.
Details …
•
The Dynamical System
•
Types of Singularities
•
Continuation Techniques
•
The Explicit Group of Symmetries
•
Explicit Existence Theorems for bifurcating branches
A Class of Problems
max F(q, ) = max(G(q)+D(q))
q
q
• G and D are sufficiently smooth in .
• G and D must be invariant under relabelling of the classes.

The Dynamical System
Goal: To determine the bifurcation structure of solutions to
 maxq  (G(q) +  D(q)) for  [0,) .
Method: Study the equilibria of the of the flow

 q 



    q , L (q,  ,  ) :  q ,  G(q)   D(q)    y   q( z | y)  1  

yY
 z

 

•
q, L : n K    n K
•
The Jacobian wrt q of the K constraints {YNq(YN|y)-1} is J=(IK IK … IK).
•
If wT qF(q*,) w < 0 for every wker J, then q*() is a maximizer of .
•
The first equilibrium is q*(0 = 0)  1/N.
Properties of the Dynamical System
•
 q 
In our dynamical system      q , L (q,  ,  ) 
 
the hessian
 q F
 q , L (q,  ,  )   T
 J
J

0
determines the stability of equilibria and the location of
bifurcation.
.
 q , L is the constraine d Hessian
 q F is unconstrai ned Hessian
 q , L is singular
 q F is singular
N M
B  R  MI K
B  Ri1  MI K
is nonsingula r
is singular
is nonsingula r
i 1
is singular
chapter 6
M 1
Non-generic
N M
B  R  MI K
Symmetry
breaking
bifurcation
i 1
1
i
i 1
M 1
Impossible
scenario
Saddle-node
bifurcation
chapter 4
i 1
N M
chapter 8
B  R  MI K
1
i
1
i
chapter 6
N M
 q F is nonsingula r
Impossible
scenario
Investigating
the Dynamical
System
The Dynamical
System
How:
 Use numerical continuation in a constrained system to choose 
and to choose an initial guess to find the equilibria q*( ).
 Use bifurcation theory with symmetries to understand
bifurcations of the equilibria.
Continuation
•
•
•
•
A local maximum qk*(k) of  is an equilibrium of the gradient flow .
Initial condition qk+1(0)(k+1(0)) is sought in tangent direction qk, which is found
by solving the matrix system
   qk 
     q , L (qk , k ,  k )
 q , L (qk , k ,  k )
  
  k
The continuation algorithm used to find qk+1*(k+1) is based on Newton’s method.
Parameter continuation follows the dashed (---) path, pseudoarclength
continuation follows the dotted (…) path
(qk 1 , k 1 ,  k 1 )
( q,  )
(qk 1 , k 1 )
(qk 1 , k 1 )
( 0)
( 0)
( q k , k ,  k )
(qk 1 , k 1  k 1 )
( 0)
(qk , k )
k
( 0)
 k 1( 0)
( 0)

The Groups
• Let P be the finite group of n ×n “block” permutation matrices which
represents the action of SN on q and F(q,) . For example, if N=3,
0

 IK
0

IK
0
0
0

0   P permutes q(YN1|y) with q(YN2|y) for every y
I K 
• F(q,) is P -invariant means that for every   P,
F( q,) = F(q,)
• Let  be the finite group of (n+K) × (n+K) block permutation matrices
q 
which represents the action of SN on   and q, L(q,,):
 

 
 : 
 0

 K n

0 
 |   P
  the lagrange multiplier s and constraints are fixed !
I 

K K 

n K
• q, L(q, , ) is -equivariant means that for every  
q 
 q, L(q, , ) = q, L(   ,)
 
 
Notation and Definitions
•
q 
The symmetry of   is measured by its isotropy subgroup

 
 q ,

 q   q 
    |       
     

•
An isotropy subgroup  is a maximal isotropy subgroup of  if there does not
exist an isotropy subgroup  of  such that     .
•
 q*  *
At bifurcation ( ,  ) , the fixed point subspace of q*,* is
 * 
 

Fix ( q* ,* )  w  ker  q , L (q* , * ,  * ) | w  w,    q* ,*

Equivariant Branching Lemma
One of the Existence Theorems we use to describe a bifurcation in the
presence of symmetries is the Equivariant Branching Lemma
(Vanderbauwhede and Cicogna 1980-1).
Idea: The bifurcation structure of local solutions is described by the
isotropy subgroups of  which have dim Fix()=1.
• System:
x  r ( x,.  ), r : m    m
• r(x,) is G-equivariant for some compact Lie Group G
• r (0,0)  0,  x r (0,0)  0
• Fix(G)={0}
• Let H be an isotropy subgroup of G such that
dim Fix (H) = 1.
• Assume  r(0,0)  0 (crossing condition).
Then there is a unique smooth solution branch (tx0,(t)) to r = 0 such that
x0  Fix (H) and the isotropy subgroup of each solution is H.
Symmetry Breaking from SM to SM-1
From bifurcation, the Equivariant Branching Lemma shows that the
following solutions emerge:
An stationary point q* is M-uniform if there exists 1 M  N and a
K x 1 vector P such that q(yNi|Y)=P for M and only M classes, {yNi}Ni=1
of YN. These M classes of YN are unresolved classes. The classes of
YN that are not unresolved are called resolved.
The first equilibria, q*  1/N, is N-uniform.
Theorem: q* is M-uniform if and only if q* is fixed by SM.
Kernel of the Hessian at
Symmetry Breaking Bifurcation
Theorem: dim ker qF (q*,)=M with basis vectors {vi}Mi=1
v if  is the i th unresolved class
[vi ]  
0 otherwise
Theorem: dim ker q,L (q*,,)=M-1 with basis vectors
 vi   vM 
    
 0   0 
Point: Since the bifurcating solutions whose existence
is guaranteed by the EBL and the SW Theorem
are tangential to ker q,L (q*,,), then we know the
explicit form of the bifurcating directions.
Symmetry Breaking Bifurcation from
M-uniform solutions
Assumptions:
•
•
•
Let q* be M-uniform
Call the M identical blocks of qF (q*,): B. Call the other N-M blocks
of qF (q*,): {R}. We assume that B has a single nullvector v and that
R is nonsingular for every .
If M<N, then BR-1 + MIK is nonsingular.
Theorem:
Let (q*,*,*) be a singular point of the flow
 q 
    q , L (q,  ,  )

 

such that q* is M-uniform. Then there exists M bifurcating (M-1)uniform solutions (q*,*,*) + (tuk,0,(t)), where
( M  1)v if  is the k th unresolved class

[uk ]   v
if   k is any other unresolved class
0
otherwise

 q , L is the constraine d Hessian
 q F is unconstrai ned Hessian
 q , L is singular
 q F is singular
N M
B  R  MI K
B  Ri1  MI K
is nonsingula r
is singular
is nonsingula r
i 1
is singular
chapter 6
M 1
Non-generic
N M
B  R  MI K
Symmetry
breaking
bifurcation
i 1
1
i
i 1
M 1
Impossible
scenario
Saddle-node
bifurcation
chapter 4
i 1
N M
chapter 8
B  R  MI K
1
i
1
i
chapter 6
N M
 q F is nonsingula r
Impossible
scenario
Some of the bifurcating branches when N = 4 are given by the
following isotropy subgroup lattice for S4
S4
S3
 3v 
 
 v
 v
 v
 
0 
S2 S2 S2
 0 
 
 2v 
 v
 v
 
0 
 0 
 
 v
 2v 
 v
 
0 
 0 
 
 v
 v
 2v 
 
0 
S3
 v
 
 3v 
 v
 v
 
0 
S2 S2 S2
 2v 
 
 0 
 v
 v
 
0 
 v
 
 0 
 2v 
 v
 
0 
 v
 
 0 
 v
 2v 
 
0 
1
S3
 v
 
 v
 3v 
 v
 
0 
S2 S2 S2
 2v 
 
 v
 0 
 v
 
0 
 v
 
 2v 
 0 
 v
 
0 
 v
 
 v
 0 
 2v 
 
0 
S3
 v
 
 v
 v
 3v 
 
0 
S2 S2 S2
 2v 
 
 v
 v
 0 
 
 0 
 v
 
 2v 
 v
 0 
 
 0 
 v
 
 v
 2v 
 0 
 
 0 
For the 4 Blob problem:
The isotropy subgroups and bifurcating directions of the
observed bifurcating branches
isotropy group:
bif direction:
S4
S3
S2
(-v,-v,3v,-v,0)T (-v,2v,0,-v,0)T
1
(-v,0,0,v,0)T
… No more bifs!
Smoller-Wasserman Theorem
The other Existence Theorem:
Smoller-Wasserman Theorem (1985-6)
For variational problems where
r ( x,  )   x f ( x,  )
there is a bifurcating solution tangential to Fix(H) for
every maximal isotropy subgroup H, not only those with
dim Fix(H) = 1.
• dim Fix(H) =1 implies that H is a maximal isotropy subgroup
Other branches
The Smoller-Wasserman Theorem shows that (under
the same assumptions as before)
if M is composite, then there exists bifurcating
solutions with isotropy group <p> for every
element  of order M in  and every prime p|M,
p<M. Furthermore,
dim (Fix <p>)=p-1
Bifurcating branches from a 4-uniform solution are given by the
following isotropy subgroup lattice for S4
S4
Fix ( (1234 ) )  0
A4 Fix ( A4 )  0
 (1324) 
 12, 34 
 v 
 
 v 
 v
 
 v
 
 13, 24 
 14, 23 
 v 
 
 v
 v
 
 v 
 
 v 
 
 v
 v 
 
 v
 
Maximal isotropy subgroup for S4
S4
S3
S3
S3
S3
 12, 34 
A4
 13, 24 
 14, 23 
Issues: SM
• The full lattice of subgroups of the group SM is
not known for arbitrary M.
• The lattice of maximal subgroups of the group SM
is not known for arbitrary M.
More about the Bifurcation Structure
Theorem: All symmetry breaking bifurcations from SM to SM-1 are pitchfork-like.
Outline of proof: ’(0)=0 since 2xx r(0,0) =0.
Theorem: The bifurcation discriminator of the pitchfork-like branch
(q*,*,*) + (tuk,0,(t)) is
 2L
 (q  , u )  3 u,  L[u, L E r ,s
[v]r [v]s  ( M 2  3M  3) v,  4 F [v, v, v]
qr qs
*
*
3
1
If (q*,*,uk) < 0, then the branch is subcritical. If (q*,*,uk) > 0, then the branch
is supercritical.
Theorem: Generically, bifurcations do not occur after all of the classes have
resolved.
Theorem: If dim (ker q,L (q*,,)) = 1, and if a crossing condition is satisfied,
then saddle-node bifurcation must occur.