Dynamics of Networks 3 Bifurcations

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) )




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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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


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





Linearise about the trivial solutionLinearise about the trivial solution

(compute the Jacobian)(compute the Jacobian)

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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

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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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


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..


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 = - = -..


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..


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..


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.


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.


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


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.


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


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.


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


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.


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

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Dynamics of Networks 3 Bifurcations

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