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Dynamics of Networks 3BifurcationsBifurcations
Dynamics of Networks 3BifurcationsBifurcations
UK-Japan Winter SchoolDynamics and Complexity
Ian StewartIan Stewart
Mathematics InstituteUniversity of Warwick
NetwoNetworkrk In this talk I will ignore distinctions between In this talk I will ignore distinctions between
different types of cells and arrows, and consider different types of cells and arrows, and consider only only regular homogeneous networksregular homogeneous networks. .
These have one type of cell, one type of arrow, These have one type of cell, one type of arrow, and the number of arrows entering each cell is and the number of arrows entering each cell is the same.the same.
This number is the This number is the valencyvalency of the network. of the network.
Arrows may form Arrows may form loopsloops (same head and tail), (same head and tail), and there may be and there may be multiple arrowsmultiple arrows (connecting (connecting the same pair of cells).the same pair of cells).
NetwoNetworkrkThis is a regular homogeneous network of valency 3. This is a regular homogeneous network of valency 3.
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Network EnumerationNetwork Enumeration
NN vv=1=1 vv=2=2 vv=3=3 vv=4=4 vv=5=5 vv=6=6
11 11 11 11 11 11 11
22 33 66 1010 1515 2121 2828
33 77 4444 180180 590590 15821582 37243724
44 1919 475475 69156915 6342063420 412230412230 20808272080827
55 4747 68746874 444722444722 104072268104072268 265076184265076184 34056654123405665412
66 130130 126750126750 4324260443242604 55696772105569677210 355906501686355906501686 1350853483470413508534834704
Number of topologically distinct regular homogeneous Number of topologically distinct regular homogeneous networks on networks on NN cells with valency cells with valency vv
BifurcatiBifurcationon A A qualitative changequalitative change in the behaviour of a in the behaviour of a
dynamical systemdynamical system which occurs as which occurs as parametersparameters are varied.are varied.
The bifurcation is The bifurcation is locallocal if the change if the change occurs in some small neighbourhood of state occurs in some small neighbourhood of state variables and parameters.variables and parameters.
•• steady statesteady state bifurcation—the number of bifurcation—the number of equilibria changesequilibria changes
BifurcatiBifurcationon •• HopfHopf bifurcation—a steady state becomes bifurcation—a steady state becomes
unstable and throws off a limit cycle (time-periodic unstable and throws off a limit cycle (time-periodic behaviour).behaviour).
GeneriGenericc A property of vector fields (or parametrised A property of vector fields (or parametrised
families of them) is families of them) is genericgeneric or or typicaltypical if it persists if it persists after any sufficiently small perturbation. after any sufficiently small perturbation.
That is, the set of vector fields (or families of That is, the set of vector fields (or families of them) with that property contains an open them) with that property contains an open neighbourhood of the vector field (or family) under neighbourhood of the vector field (or family) under discussion.discussion.
In a dynamical system without special constraints, In a dynamical system without special constraints, generic local bifurcation from a steady state is generic local bifurcation from a steady state is either steady-state or Hopf.either steady-state or Hopf.
In this talk we consider only In this talk we consider only steady statessteady states..
BifurcatiBifurcationon
How do we associateHow do we associate
bifurcationsbifurcations
with networks?with networks?
Network Network DynamicsDynamicsTo any network we associate a class of To any network we associate a class of
admissible vector fieldsadmissible vector fields, defining , defining admissible admissible ODEsODEs, which consists of those vector fields, which consists of those vector fields
FF((xx))
That respect the network structure, and the That respect the network structure, and the corresponding ODEscorresponding ODEs
ddxx/d/dtt = = FF((xx))
Admissible Admissible ODEsODEsAdmissible ODEsAdmissible ODEs are defined in terms of the are defined in terms of the input input
structurestructure of the network. of the network.
The The input setinput set II((cc)) of a cell of a cell cc is the set of all arrows is the set of all arrows whose head is whose head is cc. .
Choose coordinates Choose coordinates xxcc R Rkk for each cell for each cell cc. (We . (We
use use RRkk because we consider only because we consider only locallocal bifurcation). bifurcation). ThenThen
ddxxcc/d/dtt = = ff((xxcc,,xxTT((II ( (cc))))))
where where TT((II((cc)))) is the tuple of tail cells of is the tuple of tail cells of II((cc)). .
Admissible Admissible ODEsODEsAdmissible ODEs for the example network:Admissible ODEs for the example network:
ddxx11/d/dtt = = ff((xx11, , xx22,, xx22,, xx33) )
ddxx22/d/dtt = = ff((xx22, , xx33,, xx44,, xx55) )
ddxx33/d/dtt = = ff((xx33, , xx11,, xx33,, xx44) )
ddxx44/d/dtt = = ff((xx44, , xx22,, xx33,, xx55) )
ddxx55/d/dtt = = ff((xx55, , xx22,, xx44,, xx44))Where Where ff satisfies the symmetry condition satisfies the symmetry condition
ff((xx,,uu,,vv,,ww)) is symmetric in is symmetric in uu, , vv, , ww
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Admissible Admissible ODEsODEsBecause the network is regular and homogeneous, Because the network is regular and homogeneous,
the condition “respect the network structure” implies the condition “respect the network structure” implies that in any admissible ODEthat in any admissible ODE
ddxxcc/d/dtt = = ff((xxcc,,xxTT((II ( (cc))))))
the the samesame function function ff occurs in each equation. occurs in each equation.
Moreover, Moreover, ff is is symmetricsymmetric in the variables in the variables xxTT((II ( (cc))))..
However, the first variable is distinguished, so However, the first variable is distinguished, so ff is is not required to be symmetric in that variable.not required to be symmetric in that variable.
SynchronySynchrony
A A synchronous equilibriumsynchronous equilibrium is an equilibrium state in is an equilibrium state in which which
xxcc = = xxdd
For all cells For all cells cc, , dd. If this common value is . If this common value is xx** then the then the network state isnetwork state is
((xx*, *, xx*, *, xx*, …, *, …, xx*)*)
By a change of coordinates (respecting the network By a change of coordinates (respecting the network structure) we may assume that structure) we may assume that x*x* = 0 = 0. Equivalently,. Equivalently,
f f (0,0,0,…,0) = 0(0,0,0,…,0) = 0
Synchrony-Synchrony-BreakingBreakingConsider a parametrised family of admissible ODEsConsider a parametrised family of admissible ODEs
ddxxcc/d/dtt = = ff((xxcc,,xxTT((II ( (cc)))),,))
Where Where is a is a bifurcation parameterbifurcation parameter. Here all . Here all variables are close to 0.variables are close to 0.
Suppose that the Suppose that the synchronous statesynchronous state
(0,0,0,…,0)(0,0,0,…,0)
is is stablestable for for < 0 < 0, but , but loses stabilityloses stability at at = 0 = 0, so that , so that it is it is unstableunstable for for > 0 > 0..
Then the ODE undergoes a Then the ODE undergoes a synchrony-breaking synchrony-breaking bifurcationbifurcation at at = 0 = 0..
Critical Critical EigenvaluesEigenvaluesIf a If a synchrony-breaking bifurcationsynchrony-breaking bifurcation occurs at occurs at = 0 = 0
then the then the Jacobian matrixJacobian matrix
DDff|| = 0 = 0
has a real zero eigenvalue (steady state bifurcation) has a real zero eigenvalue (steady state bifurcation) or a purely imaginary eigenvalue (Hopf bifurcation).or a purely imaginary eigenvalue (Hopf bifurcation).
Generically, this critical eigenvalue is Generically, this critical eigenvalue is simplesimple..
In a symmetric system, multiple eigenvalues may In a symmetric system, multiple eigenvalues may occur generically, and the situation can be more occur generically, and the situation can be more complicated.complicated.
Generic Generic BifurcationBifurcationIn a general dynamical system, local steady-state In a general dynamical system, local steady-state
bifurcation from a trivial solution is generically bifurcation from a trivial solution is generically transcriticaltranscritical..
Normal Form: dx/dt = x-x2 = 0
x
Generic Generic BifurcationBifurcationIf the system has some kind of symmetry, another If the system has some kind of symmetry, another
type of bifurcation becomes generic: the type of bifurcation becomes generic: the pitchforkpitchfork..
Normal Form: ddxx/d/dtt = = xx--xx33 = 0 = 0
x
Degenerate Degenerate BifurcationBifurcation
Normal Form: ddxx/d/dtt = = xx--xxpp = 0 = 0
p even p odd
Generic Bifurcation in Generic Bifurcation in NetworksNetworks
An admissible ODE for a regular homogeneous An admissible ODE for a regular homogeneous network can benetwork can be more degeneratemore degenerate. That is, a generic . That is, a generic local synchrony-breaking bifurcation local synchrony-breaking bifurcation
•• May not occur at a simple eigenvalueMay not occur at a simple eigenvalue
•• Even when the critical eigenvalue is simple, Even when the critical eigenvalue is simple, the bifurcation need not be transcritical or the bifurcation need not be transcritical or pitchfork.pitchfork.
CombinatoricsCombinatorics
Why am I telling you all this stuff when the topic is Why am I telling you all this stuff when the topic is discrete+continuous mathematics?discrete+continuous mathematics?
Because ODEs are continuous, but the problem Because ODEs are continuous, but the problem reduces to a rather curious question in discrete reduces to a rather curious question in discrete mathematics about the eigenvectors of certain mathematics about the eigenvectors of certain integer (or rational) matrices.integer (or rational) matrices.
Namely, the Namely, the adjacency matricesadjacency matrices of regular of regular homogeneous networks. These are homogeneous networks. These are non-negative non-negative integer matrices with constant row-sumsinteger matrices with constant row-sums..
So here So here bifurcation theorybifurcation theory reduces reduces to to combinatoricscombinatorics
Adjacency Adjacency MatrixMatrix
In the regular homogeneous case the In the regular homogeneous case the adjacency matrix isadjacency matrix is
AA = ( = (aaijij))
Where Where aaijij is the number of arrows from cell is the number of arrows from cell jj
to cell to cell ii..
Adjacency MatrixAdjacency Matrix— — ExampleExample
AA = =
00 22 11 00 00
00 00 11 11 11
11 00 11 11 00
00 11 11 00 11
00 11 00 22 00
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Adjacency MatrixAdjacency Matrix— — ExampleExample
ddxx11/d/dtt = = ff((xx11, , xx22,, xx22,, xx33) )
ddxx22/d/dtt = = ff((xx22, , xx33,, xx44,, xx55) )
ddxx33/d/dtt = = ff((xx33, , xx11,, xx33,, xx44) )
ddxx44/d/dtt = = ff((xx44, , xx22,, xx33,, xx55) )
ddxx55/d/dtt = = ff((xx55, , xx22,, xx44,, xx44))
Where for simplicity we Where for simplicity we assume assume xxcc R. R.
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00 22 11 00 00
00 00 11 11 11
11 00 11 11 00
00 11 11 00 11
00 11 00 22 00
Adjacency MatrixAdjacency Matrix— — ExampleExample
00 22 11 00 00
00 00 11 11 11
11 00 11 11 00
00 11 11 00 11
00 11 00 22 00
ddxx11/d/dtt = = axax11++bxbx22++bxbx22++bxbx33
ddxx22/d/dtt = = axax22++bxbx33++bxbx44++bxbx55
ddxx33/d/dtt = = axax33++bxbx11++bxbx33++bxbx44
ddxx44/d/dtt = = axax44++bxbx22++bxbx33++bxbx55
ddxx55/d/dtt = = axax55++bxbx22++bxbx44++bxbx44
Linearise about the trivial solutionLinearise about the trivial solution
(compute the Jacobian)(compute the Jacobian)
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Adjacency MatrixAdjacency Matrix— — ExampleExample
00 22 11 00 00
00 00 11 11 11
11 00 11 11 00
00 11 11 00 11
00 11 00 22 00
ddxx11/d/dtt = = axax11++22bxbx22++bxbx33 d dxx22/d/dtt
= = axax22++bxbx33++bxbx44++bxbx55 d dxx33/d/dtt = =
axax33++bxbx11++bxbx33++bxbx44 d dxx44/d/dtt = =
axax44++bxbx22++bxbx33++bxbx55 d dxx55/d/dtt = =
axax55++bxbx22++22bxbx44
Linearise about the trivial solutionLinearise about the trivial solution
(compute the Jacobian)(compute the Jacobian)
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Adjacency MatrixAdjacency Matrix— — ExampleExample
00 22 11 00 00
00 00 11 11 11
11 00 11 11 00
00 11 11 00 11
00 11 00 22 00
ddxx/d/dtt = ( = (aIaI++bAbA))xx
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This is a general factThis is a general fact
Adjacency MatrixAdjacency Matrix— — ExampleExample
00 22 11 00 00
00 00 11 11 11
11 00 11 11 00
00 11 11 00 11
00 11 00 22 00
ddxx/d/dtt = ( = (aIaI++bAbA))xx
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Eigenvalues of Eigenvalues of aIaI++bAbA are are aa++bb
where where runs through the runs through the eigenvalues of eigenvalues of AA..
Same eigenvectors as Same eigenvectors as AA..
Adjacency MatrixAdjacency Matrix— — ExampleExample
00 22 11 00 00
00 00 11 11 11
11 00 11 11 00
00 11 11 00 11
00 11 00 22 00
ddxx/d/dtt = ( = (II++AA))xx
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Eigenvalues of Eigenvalues of II++AA are are ++
where where runs through the runs through the eigenvalues of eigenvalues of AA..
This vanishes when This vanishes when = - = -..
Adjacency MatrixAdjacency Matrix— — ExampleExample
00 22 11 00 00
00 00 11 11 11
11 00 11 11 00
00 11 11 00 11
00 11 00 22 00
Eigenstructure of Eigenstructure of AA
Eigenvalues Eigenvalues 33, , -1-1, , -1-1, , -1-1, , 11
Eigenvectors:Eigenvectors:
33 : (1,1,1,1,1) : (1,1,1,1,1)
-1-1: (3,-1,-1,-1,3) and no others: (3,-1,-1,-1,3) and no others
(3x3 Jordan block)(3x3 Jordan block)
11 : (-1,1,-3,1,3) : (-1,1,-3,1,3)
(simple, real)(simple, real)
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Liapunov-SchmidtLiapunov-SchmidtReductionReduction
Let Let JJ be the Jacobian. Then we can use the implicit be the Jacobian. Then we can use the implicit function theorem to restrict/project the bifurcation function theorem to restrict/project the bifurcation equation onto the kernel (and cokernel) of equation onto the kernel (and cokernel) of JJ, obtaining , obtaining a a reduced bifurcation equationreduced bifurcation equation
gg((xx,,) = 0) = 0
When the critical eigenvalue is real and simple, the When the critical eigenvalue is real and simple, the kernel of kernel of JJ has dimension 1 so we may assume that has dimension 1 so we may assume that xx R R..
Liapunov-SchmidtLiapunov-SchmidtReductionReduction
Some general nonsense relates the coefficients of Some general nonsense relates the coefficients of reduced bifurcation equationreduced bifurcation equation
gg((xx,,) = 0) = 0To those of the function To those of the function ff in the original network ODE. in the original network ODE.
When the critical eigenvalue is real and simple, the When the critical eigenvalue is real and simple, the crucial data are the associated eigenvector crucial data are the associated eigenvector vv of of AA, , and the corresponding eigenvector and the corresponding eigenvector uu of the of the transpose transpose AATT..
Liapunov-SchmidtLiapunov-SchmidtReductionReduction
In fact,In fact,
gg((xx,,) = ) = x + pxx + px22 + qx + qx33 + h.o.t + h.o.t
wherewhere
p = p = u u..vv[2] [2] ( ( R) R)
and—if this quadratic term vanishes—and—if this quadratic term vanishes—
q = q = u u..vv[3] [3] ++uu.(.(vv**AvAv[2][2]) +) +uu.(.(vv**((AA--II))-1-1vv[2][2]))
for coefficientsfor coefficients R. R.
Liapunov-SchmidtLiapunov-SchmidtReductionReduction
HereHere
vv[2][2]jj = = vvjj
22 vv[3][3]jj = = vvjj
33
((vv**ww))jj = = vvjjwwjj
(componentwise multiplication)(componentwise multiplication)
and and ((AA--II))-1-1 is restricted to the orthogonal complement is restricted to the orthogonal complement of of uu, where the inverse makes sense., where the inverse makes sense.
Basic TheoremBasic TheoremConsider a regular homogeneous network, whose Consider a regular homogeneous network, whose adjacency matrix adjacency matrix AA has a simple real eigenvalue. Let has a simple real eigenvalue. Let vv be be the associated eigenvector, and let the associated eigenvector, and let uu be the corresponding be the corresponding eigenvector of eigenvector of AATT..
The associated bifurcation is transcritical if and only if The associated bifurcation is transcritical if and only if
uu..vv[2] [2] 0 0
It is a pitchfork if and only if It is a pitchfork if and only if uu..vv[2] [2] = 0= 0 but at least one ofbut at least one of
uu..vv[3] [3] 0 0
uu.(.(vv**AvAv[2][2]) ) 0 0
uu.(.(vv**((AA--II))-1-1vv[2][2]) ) 0 0
ExampleExampleFor this network For this network AA has a simple real has a simple real
eigenvalue eigenvalue = 1 = 1, with associated eigenvector, with associated eigenvector
vv = (-1, 1, -3, 1, 3) = (-1, 1, -3, 1, 3)
And the corresponding eigenvector of And the corresponding eigenvector of AATT is is
uu = (-1, 0, -1, 1, 1) = (-1, 0, -1, 1, 1)NowNow
vv[2][2] = (1, 1, 9, 1, 9) = (1, 1, 9, 1, 9)soso
uu. . vv[2][2] = -1 +0 - 9 + 1 + 9 = 0 = -1 +0 - 9 + 1 + 9 = 0ButBut
uu. . vv[3][3] = 56 = 56 uu. . vv**AvAv[2][2] = 72 = 72so we have a so we have a pitchforkpitchfork..
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00 22 11 00 00
00 00 11 11 11
11 00 11 11 00
00 11 11 00 11
00 11 00 22 00
The Symmetric CaseThe Symmetric CaseFor ODEs with symmetry, it can be proved that the cubic For ODEs with symmetry, it can be proved that the cubic term in the Liapunov-Schmidt reducted equation is term in the Liapunov-Schmidt reducted equation is generically nonzerogenerically nonzero. So here we always (generically) get . So here we always (generically) get either a transcritical bifurcation or a pitchfork.either a transcritical bifurcation or a pitchfork.
We might hope that something similar occurs for networks. We might hope that something similar occurs for networks. The surprise (???) is that it does not. The surprise (???) is that it does not.
As a warning, there exist networks whereAs a warning, there exist networks where
uu..vv[2] [2] = 0= 0uu..vv[3] [3] = 0= 0uu.(.(vv**AvAv[2][2]) = 0) = 0
Degenerate 5-Cell Degenerate 5-Cell NetworkNetwork5777537766057775377660 21057771639819202105777163981920 1914447922488400 1914447922488400 00 00
126680954400126680954400 376936977576159 376936977576159 342669979614690 342669979614690 126680954400 126680954400 3300422542748331 3300422542748331
12668095440000 12668095440000 126680954400000 126680954400000 115164504000000 115164504000000 12668095440000 12668095440000 3753101212567980 3753101212567980
71216870030460 71216870030460 00 63340477200 63340477200 71182032768000 71182032768000 387782061857232 387782061857232
00 00 00 00 40202828618479804020282861847980
Valency: Valency: 40202828618479804020282861847980
Degenerate 5-Cell Degenerate 5-Cell NetworkNetwork
Eigenvalues: Eigenvalues: 40202828618479804020282861847980
00
~ 5.4396 x 10~ 5.4396 x 101414
~ 1.9172 x 10~ 1.9172 x 101313
~ ~ 1.9911 x 101.9911 x 101111
(the last three are the roots of an irreducible cubic)(the last three are the roots of an irreducible cubic)
Degenerate 5-Cell Degenerate 5-Cell NetworkNetworkEigenvector for eigenvalue 0: Eigenvector for eigenvalue 0:
vv = = (-20, -10, 11, 20, 0 )(-20, -10, 11, 20, 0 )
Eigenvector of transpose for eigenvalue 0: Eigenvector of transpose for eigenvalue 0:
uu = = (-1701, 14880, -16000, 2821, 0)(-1701, 14880, -16000, 2821, 0)
Direct calculation shows that: Direct calculation shows that:
uu..vv[2][2] = = uu..vv[3][3] = = uu.(.(vvAvAv[2][2]) = 0) = 0
So the reduced bifurcation equation has no So the reduced bifurcation equation has no quadratic or cubic terms. It does have a nonzero quadratic or cubic terms. It does have a nonzero quartic term, hence is quartic term, hence is 4-determined4-determined. .
ImprovementsImprovementsdatedate #cells#cells valencyvalency
28.1.0828.1.08 5 5 40202828618479804020282861847980
15.2.0815.2.08 55 429792429792
15.2.0815.2.08 55 10051005
15.2.0815.2.08 55 750750
15.2.0815.2.08 55 475475
19.2.0819.2.08 55 6868
21.2.0821.2.08 44 4176041760
21.2.0821.2.08 44 98409840
21.2.0821.2.08 44 800800
21.2.0821.2.08 44 736736
But we have so far But we have so far ignored...ignored...The “troublesome” cubic termThe “troublesome” cubic term
uu.(.(vv**((AA--II))-1-1vv[2][2]) ) 0 0
Which vanishes for some of those Which vanishes for some of those examples, but not all of them.examples, but not all of them.
Degenerate 4-Cell Degenerate 4-Cell NetworkNetworkOne such matrix is the simplest 4-cell One such matrix is the simplest 4-cell
example we knew until a few months example we knew until a few months ago:ago:
00 2525 171171 540540
6464 9696 576576 00
6464 00 3232 640640
00 00 00 736736
Valency:Valency: 736 736
Eigenvalues:Eigenvalues: 736, 136, 16, 736, 136, 16, -64-64
vv = = (3, 6, -2, 0)(3, 6, -2, 0)
uu = = (-32, 5, 27, 0)(-32, 5, 27, 0)
ConstructionConstruction
Choose Choose vv and and uu so that so that
uu11++uu22++uu33 = 0 = 0
uu11vv1122++uu22vv22
22++uu33vv3322=0=0
uu11vv1133++uu22vv22
33++uu33vv3333=0=0
Ignore ‘constant row-sum’ condition.Ignore ‘constant row-sum’ condition.
Assume an eigenvalue Assume an eigenvalue 00. Check simplicity later.. Check simplicity later.
Solve the Solve the linearlinear equations (for equations (for AA=(=(aaijij))))
AvAv = 0 = 0 AATTuu = 0 = 0 uu.(.(vv**AvAv[2][2]) = 0) = 0
in in non-negativenon-negative rational numbers. rational numbers.
Ignore the troublesome term at this point.Ignore the troublesome term at this point.
Fix up the row-sums by Fix up the row-sums by bordering.bordering.
ConstructionConstruction
Let Let nn = 3 = 3..
The general solution for The general solution for vv and and uu is is
vv = ( = (ss+1, +1, ss22++ss, -, -ss))
u u = (-= (-ss33((ss+2), 2+2), 2ss+1, (+1, (ss-1)(-1)(ss+1)+1)33))
Where Where ss is a rational parameter. is a rational parameter.
There is a constraint:There is a constraint:
Finite Determinacy TheoremFinite Determinacy Theorem
If If vv has exactly has exactly kk distinct nonzero entries, distinct nonzero entries, thenthen
uu..vv[[rr]] 00 for some for some rr such that such that 2 ≤ 2 ≤ rr ≤ ≤ kk+1+1
ConstructionConstruction
In particular, if In particular, if vv has at most 2 distinct nonzero has at most 2 distinct nonzero
entries, then entries, then uu..vv[3][3] 00. So we want all three entries . So we want all three entries
of of vv to be distinct, and nonzero. to be distinct, and nonzero.
Take Take ss = 2 = 2. Then. Then
vv = (3, 6, -2) = (3, 6, -2)
uu = (-32, 5, 27) = (-32, 5, 27)
Which seems to be the simplest solution of this Which seems to be the simplest solution of this kind.kind.
ConstructionConstruction
The conditions on The conditions on AA then become: then become:
aa1111 = 5 = 5aa2323/18-9/18-9aa3232/2-/2-aa3333
aa1212 = -25 = -25aa2323/288+9/288+9aa3232/32+25/32+25aa3333 /32 /32
aa1313 = 5 = 5aa2323/32+27/32+27aa3333 /32 /32
aa2121 = -16 = -16aa2323/9-18/9-18aa3232-10-10aa3333
aa2222 = -5 = -5aa2323/9+9/9+9aa3232+5+5aa3333
aa3131 = -2 = -2aa3232/32+2/32+2aa3333/3/3
ConstructionConstruction
There are solutions—a simple one is:There are solutions—a simple one is:
aa2323 = 24 = 24 aa3232 = 0 = 0 aa3333 = 4 = 4
A A ==
Multiply by 96 to remove denominators:Multiply by 96 to remove denominators:
64 25 171
64 160 576
64 0 96
Eigenvalues are 240, 80, Eigenvalues are 240, 80, 00, so 0 is simple., so 0 is simple.
ConstructionConstruction
However, row-sums are 260, 800, 160.However, row-sums are 260, 800, 160.
A A ==
BorderBorder the matrix to fix up the row-sums: the matrix to fix up the row-sums:
64 25 171 540
64 160 576 0
64 0 96 640
0 0 0 800
ConstructionConstruction
Now there is an extra eigenvalue 800. The Now there is an extra eigenvalue 800. The
eigenvectors for eigenvalue 0 areeigenvectors for eigenvalue 0 are
VV = (3, 6, -2, 0) = (3, 6, -2, 0)
UU = (-32,5,27,0) = (-32,5,27,0)
And they have the same properties as And they have the same properties as uu and and vv::
UU..VV[2][2] = = UU..VV[3][3] = = UU.(.(VVAvAv[2][2]) = 0) = 0
There is a general theorem to this effect.There is a general theorem to this effect.
Miraculously, the troublesome term also vanishes Miraculously, the troublesome term also vanishes
by direct computation.by direct computation.
Troublesome termTroublesome term
This happens becauseThis happens because
VV[2][2] = (9, 36, 4, 0) = (9, 36, 4, 0)
is is alsoalso an eigenvector, which is a an eigenvector, which is a sufficientsufficient condition condition
for the troublesome term to vanish, provided the for the troublesome term to vanish, provided the
other two cubic terms vanish.other two cubic terms vanish.
ConstructionConstruction
Since the diagonal entries are nonzero, we can Since the diagonal entries are nonzero, we can
subtract subtract 6464II to get to get
0 25 171 540
64 96 576 0
64 0 32 640
0 0 0 736
With valency 736. The eigenvalue is now -64.With valency 736. The eigenvalue is now -64.
TheoremsTheorems
No such degeneracy occurs for regular No such degeneracy occurs for regular
homogeneous 2-cell networks.homogeneous 2-cell networks.
No such degeneracy occurs for regular No such degeneracy occurs for regular
homogeneous 3-cell networks.homogeneous 3-cell networks.
TheoremsTheorems
Regular homogeneous Regular homogeneous nn-cell networks cannot -cell networks cannot
have simple eigenvalue bifurcations that are have simple eigenvalue bifurcations that are
degenerate to orderdegenerate to order n n+1.+1.
Regular homogeneous Regular homogeneous nn-cell networks cannot -cell networks cannot
have simple eigenvalue bifurcations that are have simple eigenvalue bifurcations that are
degenerate to orderdegenerate to order n n..
Path-connectedPath-connected regular homogeneous regular homogeneous nn-cell -cell
networks (networks (nn ≥ 4) cannot have simple eigenvalue ≥ 4) cannot have simple eigenvalue
bifurcations that are degenerate to orderbifurcations that are degenerate to order n n-1.-1.
Higher DegeneracyHigher Degeneracy
All terms of order 2, 3, and 4 vanish forAll terms of order 2, 3, and 4 vanish for
15 1 0 17 63
32 12 4 48 0
48 4 12 32 0
17 0 1 15 63
0 0 0 0 96
With valency 96. The eigenvalue here is 0.With valency 96. The eigenvalue here is 0.
(Can subtract 12(Can subtract 12II to get valency 84.) to get valency 84.)
ConjectureConjecture
With enough cells, there exist With enough cells, there exist
adjacency matrices for regular adjacency matrices for regular
homogeneous networks such that all homogeneous networks such that all
terms up to any required order vanish.terms up to any required order vanish.
However, these are very unusual.However, these are very unusual.
Even Higher Even Higher DegeneracyDegeneracyAll terms of order 2, 3, 4, and 5 vanish forAll terms of order 2, 3, 4, and 5 vanish for
71179687117968 84656008465600 2155507221555072 1843296018432960 39203920 1441440014414400
32741283274128 75096457509645 162162162162 95945859594585 00 18018001801800
382668382668 731445731445 522522522522 11861851186185 00 480480480480
408408408408 645645645645 882882882882 11861851186185 00 360360360360
5269492852694928 1029600010296000 612323712612323712 423783360423783360 720720720720 00
00 00 00 00 00 10969959001096995900
With valency 1096995900. The eigenvalue here is 0.With valency 1096995900. The eigenvalue here is 0.
Path-Connected Path-Connected NetworksNetworksCan all terms of order 2, 3 vanish for a path-connected Can all terms of order 2, 3 vanish for a path-connected
network?network?
No — for ≤ 4 cells.No — for ≤ 4 cells. Yes — for ≥ 5 cells.Yes — for ≥ 5 cells.
11 1313 153153 66 33
3232 00 144144 00 00
1616 88 152152 00 00
1616 3232 9696 1616 1616
116116 4646 66 00 88
Valency 176, eigenvalue -16
[with M.Golubitsky and M.Pivato] Symmetry groupoids and [with M.Golubitsky and M.Pivato] Symmetry groupoids and patterns of synchrony in coupled cell networks, patterns of synchrony in coupled cell networks, SIAM J. SIAM J. Appl. Dyn. Sys.Appl. Dyn. Sys. 22 (2003) 609-646. DOI: (2003) 609-646. DOI: 10.1137/S111111110341989610.1137/S1111111103419896
[with M.Golubitsky and A.Török] Patterns of synchrony in [with M.Golubitsky and A.Török] Patterns of synchrony in coupled cell networks with multiple arrows, coupled cell networks with multiple arrows, SIAM J. Appl. SIAM J. Appl. Dyn. Sys.Dyn. Sys. 44 (2005) 78-100. [DOI: 10.1137/040612634] (2005) 78-100. [DOI: 10.1137/040612634]
[with M.Golubitsky] Nonlinear dynamics of networks: the [with M.Golubitsky] Nonlinear dynamics of networks: the groupoid formalism, groupoid formalism, Bull. Amer. Math. SocBull. Amer. Math. Soc. . 4343 (2006) (2006) 305-364.305-364.
[with M.Golubitsky] Synchrony-breaking bifurcations[with M.Golubitsky] Synchrony-breaking bifurcations at at simple eigenvalues for regular networks, in preparation.simple eigenvalues for regular networks, in preparation.
ReferencesReferences