1. Bifurcations Dynamics of Networks 3 Bifurcations UK-Japan Winter School Dynamics and Complexity Ian Stewart Mathematics Institute University of Warwick

2. Networ k In this talk I will ignore distinctions between different types of cells and arrows, and consider only regular homogeneous networks. These have one type of cell, one type of arrow, and the number of arrows entering each cell is the same. This number is the valency of the network. Arrows may form loops (same head and tail), and there may be multiple arrows (connecting the same pair of cells).

3. Networ k This is a regular homogeneous network of valency 3.

4. Network Enumeration N v=1 v=2 v=3 v=4 v=5 v=6 1111111 23610152128 374418059015823724 4194756915634204122302080827 54768744447221040722682650761843405665412 613012675043242604556967721035590650168613508534834704 Number of topologically distinct regular homogeneous networks on N cells with valency v

5. Bifurcatio n A qualitative change in the behaviour of a dynamical system which occurs as parameters are varied. The bifurcation is local if the change occurs in some small neighbourhood of state variables and parameters. steady state bifurcationthe number of equilibria changessteady state bifurcationthe number of equilibria changes

6. Bifurcatio n Hopf bifurcationa steady state becomes unstable and throws off a limit cycle (time-periodic behaviour).Hopf bifurcationa steady state becomes unstable and throws off a limit cycle (time-periodic behaviour).

7. Generi c A property of vector fields (or parametrised families of them) is generic or typical if it persists after any sufficiently small perturbation. That is, the set of vector fields (or families of them) with that property contains an open neighbourhood of the vector field (or family) under discussion. In a dynamical system without special constraints, generic local bifurcation from a steady state is either steady-state or Hopf. In this talk we consider only steady states.

8. Bifurcatio n How do we associate bifurcations with networks?

9. Network Dynamics To any network we associate a class of admissible vector fields, defining admissible ODEs, which consists of those vector fields F(x) That respect the network structure, and the corresponding ODEs dx/dt = F(x)

10. Admissible ODEs Admissible ODEs are defined in terms of the input structure of the network. The input set I(c) of a cell c is the set of all arrows whose head is c. Choose coordinates x c R k for each cell c. (We use R k because we consider only local bifurcation). Then dx c /dt = f(x c,x T(I (c)) ) where T(I(c)) is the tuple of tail cells of I(c).

11. Admissible ODEs Admissible ODEs for the example network: dx 1 /dt = f(x 1, x 2, x 2, x 3 ) dx 2 /dt = f(x 2, x 3, x 4, x 5 ) dx 3 /dt = f(x 3, x 1, x 3, x 4 ) dx 4 /dt = f(x 4, x 2, x 3, x 5 ) dx 5 /dt = f(x 5, x 2, x 4, x 4 ) Where f satisfies the symmetry condition f(x,u,v,w) is symmetric in u, v, w f(x,u,v,w) is symmetric in u, v, w

12. Admissible ODEs Because the network is regular and homogeneous, the condition respect the network structure implies that in any admissible ODE dx c /dt = f(x c,x T(I (c)) ) the same function f occurs in each equation. Moreover, f is symmetric in the variables x T(I (c)). However, the first variable is distinguished, so f is not required to be symmetric in that variable.

13. Synchrony A synchronous equilibrium is an equilibrium state in which x c = x d For all cells c, d. If this common value is x* then the network state is (x*, x*, x*, …, x*) By a change of coordinates (respecting the network structure) we may assume that x* = 0. Equivalently, f (0,0,0,…,0) = 0

14. Synchrony- Breaking Consider a parametrised family of admissible ODEs dx c /dt = f(x c,x T(I (c)), ) Where is a bifurcation parameter. Here all variables are close to 0. Suppose that the synchronous state (0,0,0,…,0) is stable for 0. Then the ODE undergoes a synchrony-breaking bifurcation at = 0.

15. Critical Eigenvalues If a synchrony-breaking bifurcation occurs at = 0 then the Jacobian matrix Df| = 0 has a real zero eigenvalue (steady state bifurcation) or a purely imaginary eigenvalue (Hopf bifurcation). Generically, this critical eigenvalue is simple. In a symmetric system, multiple eigenvalues may occur generically, and the situation can be more complicated.

16. Generic Bifurcation In a general dynamical system, local steady-state bifurcation from a trivial solution is generically transcritical. Normal Form: dx/dt = x-x 2 = 0

17. Generic Bifurcation If the system has some kind of symmetry, another type of bifurcation becomes generic: the pitchfork. dx/dt = x-x 3 = 0 Normal Form: dx/dt = x-x 3 = 0

18. Degenerate Bifurcation dx/dt = x-x p = 0 Normal Form: dx/dt = x-x p = 0 p even p odd

19. Generic Bifurcation in Networks An admissible ODE for a regular homogeneous network can be more degenerate. That is, a generic local synchrony-breaking bifurcation May not occur at a simple eigenvalueMay not occur at a simple eigenvalue Even when the critical eigenvalue is simple, the bifurcation need not be transcritical or pitchfork.Even when the critical eigenvalue is simple, the bifurcation need not be transcritical or pitchfork.

20. Combinatorics Why am I telling you all this stuff when the topic is discrete+continuous mathematics? Because ODEs are continuous, but the problem reduces to a rather curious question in discrete mathematics about the eigenvectors of certain integer (or rational) matrices. Namely, the adjacency matrices of regular homogeneous networks. These are non-negative integer matrices with constant row-sums. So here bifurcation theory reduces to combinatorics

21. Adjacency Matrix In the regular homogeneous case the adjacency matrix is A = (a ij ) Where a ij is the number of arrows from cell j to cell i.

22. Adjacency Matrix Example Example A =A =A =A = 02100 00111 10110 01101 01020

23. Adjacency Matrix Example Example dx 1 /dt = f(x 1, x 2, x 2, x 3 ) dx 2 /dt = f(x 2, x 3, x 4, x 5 ) dx 3 /dt = f(x 3, x 1, x 3, x 4 ) dx 4 /dt = f(x 4, x 2, x 3, x 5 ) dx 5 /dt = f(x 5, x 2, x 4, x 4 ) Where for simplicity we assume x c R. 0210000111 10110 01101 01020

24. Adjacency Matrix Example Example 02100 00111 10110 01101 01020 dx 1 /dt = ax 1 +bx 2 +bx 2 +bx 3 dx 2 /dt = ax 2 +bx 3 +bx 4 +bx 5 dx 3 /dt = ax 3 +bx 1 +bx 3 +bx 4 dx 4 /dt = ax 4 +bx 2 +bx 3 +bx 5 dx 5 /dt = ax 5 +bx 2 +bx 4 +bx 4 Linearise about the trivial solution (compute the Jacobian)

25. Adjacency Matrix Example Example 02100 00111 10110 01101 01020 dx 1 /dt = ax 1 +2bx 2 +bx 3 dx 2 /dt = ax 2 +bx 3 +bx 4 +bx 5 dx 3 /dt = ax 3 +bx 1 +bx 3 +bx 4 dx 4 /dt = ax 4 +bx 2 +bx 3 +bx 5 dx 5 /dt = ax 5 +bx 2 +2bx 4 Linearise about the trivial solution (compute the Jacobian)

26. Adjacency Matrix Example Example 02100 00111 10110 01101 01020 dx/dt = (aI+bA)x This is a general fact

27. Adjacency Matrix Example Example 02100 00111 10110 01101 01020 dx/dt = (aI+bA)x Eigenvalues of aI+bA are a+b where runs through the eigenvalues of A. Same eigenvectors as A.

28. Adjacency Matrix Example Example 02100 00111 10110 01101 01020 dx/dt = ( I+A)x Eigenvalues of I+A are + where runs through the eigenvalues of A. This vanishes when = -.

29. Adjacency Matrix Example Example 02100 00111 10110 01101 01020 Eigenstructure of A Eigenvalues 3, -1, -1, -1, 1 Eigenvectors: 3 : (1,1,1,1,1) -1: (3,-1,-1,-1,3) and no others (3x3 Jordan block) 1 : (-1,1,-3,1,3) (simple, real)

30. Liapunov-SchmidtReduction Let J be the Jacobian. Then we can use the implicit function theorem to restrict/project the bifurcation equation onto the kernel (and cokernel) of J, obtaining a reduced bifurcation equation g(x, ) = 0 When the critical eigenvalue is real and simple, the kernel of J has dimension 1 so we may assume that x R.

31. Liapunov-SchmidtReduction Some general nonsense relates the coefficients of reduced bifurcation equation g(x, ) = 0 To those of the function f in the original network ODE. When the critical eigenvalue is real and simple, the crucial data are the associated eigenvector v of A, and the corresponding eigenvector u of the transpose A T.

32. Liapunov-SchmidtReduction In fact, g(x, ) = x + px 2 + qx 3 + h.o.t where p = u.v [2] ( R) andif this quadratic term vanishes q = u.v [3] + u.(v * Av [2] ) + u.(v * (A- I) -1 v [2] ) for coefficients R.

33. Liapunov-SchmidtReduction Here v [2] j = v j 2 v [3] j = v j 3 (v * w) j = v j w j (componentwise multiplication) and (A- I) -1 is restricted to the orthogonal complement of u, where the inverse makes sense.

34. Basic Theorem Consider a regular homogeneous network, whose adjacency matrix A has a simple real eigenvalue. Let v be the associated eigenvector, and let u be the corresponding eigenvector of A T. The associated bifurcation is transcritical if and only if u.v [2] 0 It is a pitchfork if and only if u.v [2] = 0 but at least one of u.v [3] 0 u.(v * Av [2] ) 0 u.(v * (A- I) -1 v [2] ) 0

35. Example For this network A has a simple real eigenvalue = 1, with associated eigenvector v = (-1, 1, -3, 1, 3) And the corresponding eigenvector of A T is u = (-1, 0, -1, 1, 1) Now v [2] = (1, 1, 9, 1, 9) v [2] = (1, 1, 9, 1, 9)so u. v [2] = -1 +0 - 9 + 1 + 9 = 0 But u. v [3] = 56 u. v * Av [2] = 72 u. v [3] = 56 u. v * Av [2] = 72 so we have a pitchfork. 0210000111 10110 01101 01020

36. The Symmetric Case For ODEs with symmetry, it can be proved that the cubic term in the Liapunov-Schmidt reducted equation is generically nonzero. So here we always (generically) get either a transcritical bifurcation or a pitchfork. We might hope that something similar occurs for networks. The surprise (???) is that it does not. As a warning, there exist networks where u.v [2] = 0 u.v [3] = 0 u.(v * Av [2] ) = 0

37. Degenerate 5-Cell Network 577753776602105777163981920191444792248840000 1266809544003769369775761593426699796146901266809544003300422542748331 12668095440000126680954400000115164504000000126680954400003753101212567980 7121687003046006334047720071182032768000387782061857232 00004020282861847980 Valency: 4020282861847980

38. Degenerate 5-Cell Network Eigenvalues: 4020282861847980 0 ~ 5.4396 x 10 14 ~ 1.9172 x 10 13 ~ 1.9911 x 10 11 (the last three are the roots of an irreducible cubic)

39. Degenerate 5-Cell Network Eigenvector for eigenvalue 0: v = (-20, -10, 11, 20, 0 ) Eigenvector of transpose for eigenvalue 0: u = (-1701, 14880, -16000, 2821, 0) Direct calculation shows that: u.v [2] = u.v [3] = u.(v Av [2] ) = 0 So the reduced bifurcation equation has no quadratic or cubic terms. It does have a nonzero quartic term, hence is 4-determined.

40. Improvements date#cells valency 28.1.085 4020282861847980 15.2.085429792 15.2.0851005 15.2.085750 15.2.085475 19.2.08568 21.2.08441760 21.2.0849840 21.2.084800 21.2.084736

41. But we have so far ignored... The troublesome cubic term u.(v * (A- I) -1 v [2] ) 0 Which vanishes for some of those examples, but not all of them.

42. Degenerate 4-Cell Network One such matrix is the simplest 4-cell example we knew until a few months ago: 025171540 64965760 64032640 000736 Valency: 736 Eigenvalues: 736, 136, 16, -64 v = (3, 6, -2, 0) u = (-32, 5, 27, 0)

43. Construction Choose v and u so that u 1 +u 2 +u 3 = 0 u 1 v 1 2 +u 2 v 2 2 +u 3 v 3 2 =0 u 1 v 1 3 +u 2 v 2 3 +u 3 v 3 3 =0 Ignore constant row-sum condition. Assume an eigenvalue 0. Check simplicity later. Solve the linear equations (for A=(a ij )) Av = 0 A T u = 0 u.(v * Av [2] ) = 0 in non-negative rational numbers. Ignore the troublesome term at this point. Fix up the row-sums by bordering.

44. Construction Let n = 3. The general solution for v and u is v = (s+1, s 2 +s, -s) u = (-s 3 (s+2), 2s+1, (s-1)(s+1) 3 ) Where s is a rational parameter. There is a constraint: Finite Determinacy Theorem If v has exactly k distinct nonzero entries, then u.v [r] 0 for some r such that 2 r k+1

45. Construction In particular, if v has at most 2 distinct nonzero entries, then u.v [3] 0. So we want all three entries of v to be distinct, and nonzero. Take s = 2. Then v = (3, 6, -2) u = (-32, 5, 27) Which seems to be the simplest solution of this kind.

46. Construction The conditions on A then become: a 11 = 5a 23 /18-9a 32 /2-a 33 a 12 = -25a 23 /288+9a 32 /32+25a 33 /32 a 13 = 5a 23 /32+27a 33 /32 a 21 = -16a 23 /9-18a 32 -10a 33 a 22 = -5a 23 /9+9a 32 +5a 33 a 31 = -2a 32 /32+2a 33 /3

47. Construction There are solutionsa simple one is: a 23 = 24 a 32 = 0 a 33 = 4 A = Multiply by 96 to remove denominators: 6425171 64160576 64096 Eigenvalues are 240, 80, 0, so 0 is simple.

48. Construction However, row-sums are 260, 800, 160. A = Border the matrix to fix up the row-sums: 6425171540 641605760 64096640 000800

49. Construction Now there is an extra eigenvalue 800. The eigenvectors for eigenvalue 0 are V = (3, 6, -2, 0) U = (-32,5,27,0) And they have the same properties as u and v: U.V [2] = U.V [3] = U.(V Av [2] ) = 0 There is a general theorem to this effect. Miraculously, the troublesome term also vanishes by direct computation.

50. Troublesome term This happens because V [2] = (9, 36, 4, 0) is also an eigenvector, which is a sufficient condition for the troublesome term to vanish, provided the other two cubic terms vanish.

51. Construction Since the diagonal entries are nonzero, we can subtract 64I to get 025171540 64965760 64032640 000736 With valency 736. The eigenvalue is now -64.

52. Theorems No such degeneracy occurs for regular homogeneous 2-cell networks. No such degeneracy occurs for regular homogeneous 3-cell networks.

53. Theorems Regular homogeneous n-cell networks cannot have simple eigenvalue bifurcations that are degenerate to order n+1. Regular homogeneous n-cell networks cannot have simple eigenvalue bifurcations that are degenerate to order n. Path-connected regular homogeneous n-cell networks (n 4) cannot have simple eigenvalue bifurcations that are degenerate to order n-1.

54. Higher Degeneracy All terms of order 2, 3, and 4 vanish for 15101763 32124480 412320 17011563 000096 With valency 96. The eigenvalue here is 0. (Can subtract 12I to get valency 84.)

55. Conjecture With enough cells, there exist adjacency matrices for regular homogeneous networks such that all terms up to any required order vanish. However, these are very unusual.

56. Even Higher Degeneracy All terms of order 2, 3, 4, and 5 vanish for 711796884656002155507218432960392014414400 32741287509645162162959458501801800 38266873144552252211861850480480 40840864564588288211861850360360 52694928102960006123237124237833607207200 000001096995900 With valency 1096995900. The eigenvalue here is 0.

57. Path-Connected Networks Can all terms of order 2, 3 vanish for a path-connected network? No for 4 cells. Yes for 5 cells. 11315363 32014400 16815200 1632961616 11646608 Valency 176, eigenvalue -16

58. [with M.Golubitsky and M.Pivato] Symmetry groupoids and patterns of synchrony in coupled cell networks, SIAM J. Appl. Dyn. Sys. 2 (2003) 609-646. DOI: 10.1137/S1111111103419896 [with M.Golubitsky and A.Török] Patterns of synchrony in coupled cell networks with multiple arrows, SIAM J. Appl. Dyn. Sys. 4 (2005) 78-100. [DOI: 10.1137/040612634] [with M.Golubitsky] Nonlinear dynamics of networks: the groupoid formalism, Bull. Amer. Math. Soc. 43 (2006) 305-364. [with M.Golubitsky] Synchrony-breaking bifurcations at simple eigenvalues for regular networks, in preparation. References