Pattern Formation in Coupled Systems

Transcription

1 . p.1/35 Pattern Formation in Coupled Systems Martin Golubitsky Ian Stewart Houston Warwick Fernando Antoneli Ana Dias Toby Elmhirst Maria Leite Matthew Nicol Marcus Pivato Andrew Török Yunjiao Wang Sao Paulo Porto Houston Houston Houston Trent Houston Houston

2 . p.2/35 Two Identical Cells 1 2 σ(x 1, x 2 ) = (x 2, x 1 ) is a symmetry ẋ 1 = f(x 1, x 2 ) ẋ 2 = f(x 2, x 1 )

3 . p.2/35 Two Identical Cells 1 2 σ(x 1, x 2 ) = (x 2, x 1 ) Fix(σ) = {x 1 = x 2 } is a symmetry is flow invariant ẋ 1 = f(x 1, x 2 ) ẋ 2 = f(x 2, x 1 )

4 . p.2/35 Two Identical Cells 1 2 σ(x 1, x 2 ) = (x 2, x 1 ) Fix(σ) = {x 1 = x 2 } is a symmetry is flow invariant ẋ 1 = f(x 1, x 2 ) ẋ 2 = f(x 2, x 1 ) Synchrony is robust If x 1 (0) = x 2 (0), then x 1 (t) = x 2 (t)

5 . p.3/35 Symmetry Overview A symmetry of a DiffEq ẋ = f(x) is a linear map γ where γ(sol n) = sol n f(γx) = γf(x)

6 . p.3/35 Symmetry Overview A symmetry of a DiffEq ẋ = f(x) is a linear map γ where γ(sol n) = sol n f(γx) = γf(x) Fix(Σ) = {x R n : σx = x σ Σ} is flow invariant Proof: f(x) = f(σx) = σf(x)

7 . p.3/35 Symmetry Overview A symmetry of a DiffEq ẋ = f(x) is a linear map γ where γ(sol n) = sol n f(γx) = γf(x) Fix(Σ) = {x R n : σx = x σ Σ} is flow invariant Proof: f(x) = f(σx) = σf(x) Network symmetries are permutation symmetries Synchrony is robust in symmetric coupled systems

8 . p.3/35 Symmetry Overview A symmetry of a DiffEq ẋ = f(x) is a linear map γ where γ(sol n) = sol n f(γx) = γf(x) Fix(Σ) = {x R n : σx = x σ Σ} is flow invariant Proof: f(x) = f(σx) = σf(x) Network symmetries are permutation symmetries Synchrony is robust in symmetric coupled systems Symmetry group Γ is a modeling assumption

9 . p.3/35 Symmetry Overview A symmetry of a DiffEq ẋ = f(x) is a linear map γ where γ(sol n) = sol n f(γx) = γf(x) Fix(Σ) = {x R n : σx = x σ Σ} is flow invariant Proof: f(x) = f(σx) = σf(x) Network symmetries are permutation symmetries Synchrony is robust in symmetric coupled systems Symmetry group Γ is a modeling assumption Network architecture is also a modeling assumption

10 . p.4/35 Three-Cell Bidirectional Ring ẋ 1 = f(x 1, x 2, x 3 ) ẋ 2 = f(x 2, x 3, x 1 ) f(x 2, x 1, x 3 ) = f(x 2, x 3, x 1 ) ẋ 3 = f(x 3, x 1, x 2 )

11 . p.4/35 Three-Cell Bidirectional Ring ẋ 1 = f(x 1, x 2, x 3 ) ẋ 2 = f(x 2, x 3, x 1 ) f(x 2, x 1, x 3 ) = f(x 2, x 3, x 1 ) ẋ 3 = f(x 3, x 1, x 2 ) S 3 -symmetry

12 . p.4/35 Three-Cell Bidirectional Ring ẋ 1 = f(x 1, x 2, x 3 ) ẋ 2 = f(x 2, x 3, x 1 ) f(x 2, x 1, x 3 ) = f(x 2, x 3, x 1 ) ẋ 3 = f(x 3, x 1, x 2 ) S 3 -symmetry {x 1 = x 2 = x 3 } and {x i = x j } are synchrony subspaces

13 . p.5/35 Coupled Cell Overview Coupled cell system: discrete space, continuous time system symmetry robust synchrony Stewart, G., and Pivato (2003); G., Stewart, and Török (2005)

14 . p.5/35 Coupled Cell Overview Coupled cell system: discrete space, continuous time system symmetry network architecture robust synchrony input sets, balanced relations, quotient networks Stewart, G., and Pivato (2003); G., Stewart, and Török (2005)

15 . p.5/35 Coupled Cell Overview Coupled cell system: discrete space, continuous time system symmetry network architecture robust synchrony input sets, balanced relations, quotient networks Primary Question Which aspects of coupled cell dynamics are due to network architecture? Stewart, G., and Pivato (2003); G., Stewart, and Török (2005)

16 . p.5/35 Coupled Cell Overview Coupled cell system: discrete space, continuous time system symmetry network architecture robust synchrony input sets, balanced relations, quotient networks Primary Question Which aspects of coupled cell dynamics are due to network architecture? Beginner Question: Are all synchrony spaces fixed-point spaces? Answer: No Stewart, G., and Pivato (2003); G., Stewart, and Török (2005)

17 . p.6/35 Asymmetric Three-Cell Network ẋ 1 = f(x 1, x 2, x 3 ) x 1 R k ẋ 2 = f(x 2, x 1, x 3 ) x 2 R k ẋ 3 = g(x 3, x 1 ) x 3 R l Y = {x : x 1 = x 2 } is flow-invariant Restrict equations ẋ 1, ẋ 2 to Y : ẋ 1 = f(x 1, x 1, x 3 ) ẋ 2 = f(x 1, x 1, x 3 )

18 . p.6/35 Asymmetric Three-Cell Network ẋ 1 = f(x 1, x 2, x 3 ) x 1 R k ẋ 2 = f(x 2, x 1, x 3 ) x 2 R k ẋ 3 = g(x 3, x 1 ) x 3 R l Y = {x : x 1 = x 2 } is flow-invariant Restrict equations ẋ 1, ẋ 2 to Y : ẋ 1 = f(x 1, x 1, x 3 ) ẋ 2 = f(x 1, x 1, x 3 ) Robust synchrony exists in networks without symmetry

19 . p.6/35 Asymmetric Three-Cell Network ẋ 1 = f(x 1, x 2, x 3 ) x 1 R k ẋ 2 = f(x 2, x 1, x 3 ) x 2 R k ẋ 3 = g(x 3, x 1 ) x 3 R l Y = {x : x 1 = x 2 } is flow-invariant Restrict equations ẋ 1, ẋ 2 to Y : ẋ 1 = f(x 1, x 1, x 3 ) ẋ 2 = f(x 1, x 1, x 3 ) Robust synchrony exists in networks without symmetry Cells 1 and 2 are identical within the network

20 . p.7/35 Input Sets Input set of cell j: the arrows that connect to cell j Key idea: cells 1, 2 have isomorphic input sets

21 . p.8/35 Coupled Cell Network Definition A set of cells C = {1,..., N} Each cell has its own phase space

22 . p.8/35 Coupled Cell Network Definition A set of cells C = {1,..., N} Each cell has its own phase space An equivalence relation on cells Equivalent cells have the same phase space

23 . p.8/35 Coupled Cell Network Definition A set of cells C = {1,..., N} Each cell has its own phase space An equivalence relation on cells Equivalent cells have the same phase space Each cell c has set of input arrows I(c) Arrows represent coupling

24 . p.8/35 Coupled Cell Network Definition A set of cells C = {1,..., N} Each cell has its own phase space An equivalence relation on cells Equivalent cells have the same phase space Each cell c has set of input arrows I(c) Arrows represent coupling An equivalence relation on arrows Equivalent arrows represent same coupling

25 . p.8/35 Coupled Cell Network Definition A set of cells C = {1,..., N} Each cell has its own phase space An equivalence relation on cells Equivalent cells have the same phase space Each cell c has set of input arrows I(c) Arrows represent coupling An equivalence relation on arrows Equivalent arrows represent same coupling Equivalent arrows have equivalent tail and head cells

26 . p.9/35 Local Network Symmetry coupled cell networks represented by directed graphs For each class of cells choose node symbol,, For each class of arrows choose arrow symbol,,

27 . p.9/35 Local Network Symmetry coupled cell networks represented by directed graphs For each class of cells choose node symbol,, For each class of arrows choose arrow symbol,, Input isomorphism is arrow type preserving bijection β : I(c) I(d) Input isomorphic cells have same equations

28 . p.9/35 Local Network Symmetry coupled cell networks represented by directed graphs For each class of cells choose node symbol,, For each class of arrows choose arrow symbol,, Input isomorphism is arrow type preserving bijection β : I(c) I(d) Input isomorphic cells have same equations B G = groupoid of all input isomorphisms

29 . p.9/35 Local Network Symmetry coupled cell networks represented by directed graphs For each class of cells choose node symbol,, For each class of arrows choose arrow symbol,, Input isomorphism is arrow type preserving bijection β : I(c) I(d) Input isomorphic cells have same equations B G = groupoid of all input isomorphisms Coupled cell systems: ODEs that commute with B G

30 . p.10/35 Asymmetric Three-Cell Network (2) ẋ 1 = f(x 1, x 2, x 3 ) x 1 R k ẋ 2 = f(x 2, x 1, x 3 ) x 2 R k ẋ 3 = g(x 3, x 1 ) x 3 R l Two cell types: Three arrow types: Equivalent cells 1 and 2 have same phase space R k Cells 1 and 2 are input isomorphic Have same systems of differential equations f

31 . p.11/35 Synchrony Subspaces A polydiagonal is a subspace {x : x c = x d for some subset of cells}

32 . p.11/35 Synchrony Subspaces A polydiagonal is a subspace {x : x c = x d for some subset of cells} Σ Γ = permutation group of network symmetries Fix(Σ) is a polydiagonal and is flow-invariant

33 . p.11/35 Synchrony Subspaces A polydiagonal is a subspace {x : x c = x d for some subset of cells} Σ Γ = permutation group of network symmetries Fix(Σ) is a polydiagonal and is flow-invariant A synchrony subspace is a flow-invariant polydiagonal

34 . p.11/35 Synchrony Subspaces A polydiagonal is a subspace {x : x c = x d for some subset of cells} Σ Γ = permutation group of network symmetries Fix(Σ) is a polydiagonal and is flow-invariant A synchrony subspace is a flow-invariant polydiagonal Synchrony subspaces are coupled cell analogs of fixed-point subspaces

35 . p.11/35 Synchrony Subspaces A polydiagonal is a subspace {x : x c = x d for some subset of cells} Σ Γ = permutation group of network symmetries Fix(Σ) is a polydiagonal and is flow-invariant A synchrony subspace is a flow-invariant polydiagonal Synchrony subspaces are coupled cell analogs of fixed-point subspaces Synchrony subspaces are patterns of synchrony

36 . p.12/35 Synchrony Subspaces (2) Let be a polydiagonal Stewart, G., and Pivato (2003); G., Stewart, and Török (2005)

37 . p.12/35 Synchrony Subspaces (2) Let be a polydiagonal Color cells the same color if cell coord s in are equal Stewart, G., and Pivato (2003); G., Stewart, and Török (2005)

38 . p.12/35 Synchrony Subspaces (2) Let be a polydiagonal Color cells the same color if cell coord s in are equal Coloring is balanced if every pair of cells with same color has an input isomorphism that preserves tail colors Stewart, G., and Pivato (2003); G., Stewart, and Török (2005)

39 . p.12/35 Synchrony Subspaces (2) Let be a polydiagonal Color cells the same color if cell coord s in are equal Coloring is balanced if every pair of cells with same color has an input isomorphism that preserves tail colors Theorem: synchrony subspace balanced Stewart, G., and Pivato (2003); G., Stewart, and Török (2005)

40 . p.13/35 1D-Lattice Dynamical Systems Assume nearest neighbor coupling i 2 i 1 i i+1 i+2 ẋ i = f(x i, x i 1, x i+1 ) where f(x, y, z) = f(x, z, y) Antoneli, Dias, G. and Wang (2004)

41 . p.13/35 1D-Lattice Dynamical Systems Assume nearest neighbor coupling i 2 i 1 i i+1 i+2 ẋ i = f(x i, x i 1, x i+1 ) where f(x, y, z) = f(x, z, y) There are four balanced k colorings (when k 3) ABC ABC ABC ABCB ABCB ABCB ABCBA ABCBA CBA ABCCBA ABCCBA Antoneli, Dias, G. and Wang (2004)

42 . p.13/35 1D-Lattice Dynamical Systems Assume nearest neighbor coupling i 2 i 1 i i+1 i+2 ẋ i = f(x i, x i 1, x i+1 ) where f(x, y, z) = f(x, z, y) There are four balanced k colorings (when k 3) ABC ABC ABC ABCB ABCB ABCB ABCBA ABCBA CBA ABCCBA ABCCBA Every synchrony subspace is a fixed-point subspace Antoneli, Dias, G. and Wang (2004)

43 . p.13/35 1D-Lattice Dynamical Systems Assume nearest neighbor coupling i 2 i 1 i i+1 i+2 ẋ i = f(x i, x i 1, x i+1 ) where f(x, y, z) = f(x, z, y) There are four balanced k colorings (when k 3) ABC ABC ABC ABCB ABCB ABCB ABCBA ABCBA CBA ABCCBA ABCCBA Every synchrony subspace is a fixed-point subspace Every balanced coloring is periodic Antoneli, Dias, G. and Wang (2004)

44 . p.14/35 2D-Lattice Dynamical Systems Consider square lattice with nearest neighbor coupling Form a two-color balanced relation Each black cell connected to two black and two white Each white cell connected to two black and two white Stewart, G. and Nicol (2004) Owen, Sherratt, and Wearing. Lateral induction by juxtacrine is a new mechanism for pattern formation. Devel. Biol. 217 (2000)

45 . p.15/35 Lattice Dynamical Systems (2) On Black/White diagonal interchange black and white Result is balanced

46 . p.15/35 Lattice Dynamical Systems (2) On Black/White diagonal interchange black and white Result is balanced Continuum of different synchrony subspaces

47 . p.16/35 Lattice Dynamical Systems (2) There are eight isolated balanced two-colorings on square lattice with nearest neighbor coupling 4B W ; 4W B 2B W ; 4W B 1B W ; 4W B 3B W ; 3W B 2B W ; 3W B 2B W ; 1W B 2B W ; 1W B 1B W ; 1W B Wang and G. (2004) indicates nonsymmetric solution

48 . p.17/35 Symmetries (i, j) (i, j + 4) (i, j) (i + 3, j + 2) (i, j) ( i, j) (i, j) (i, j)

49 . p.18/35 Lattice Dynamical Systems (3) There are two infinite families of balanced two-colorings 2B W ; 2W B 1B W ; 3W B Up to symmetry these are all balanced two-colorings

50 . p.19/35 Lattice Dynamical Systems (4) Architecture is really important Antoneli, Dias, G., and Wang (2004)

51 . p.19/35 Lattice Dynamical Systems (4) Architecture is really important For square (and hexagonal) lattices with nearest and next nearest neighbor coupling Antoneli, Dias, G., and Wang (2004)

52 . p.19/35 Lattice Dynamical Systems (4) Architecture is really important For square (and hexagonal) lattices with nearest and next nearest neighbor coupling No infinite families For each k a finite number of balanced k colorings All balanced colorings are doubly-periodic Antoneli, Dias, G., and Wang (2004)

53 Windows 1 W 0 = {0} and W i+1 = I(W i ) NEAREST NEIGHBOR NEXT NEAREST NEIGHBOR Input set of U = I(U) = {c C : c connects to cell in U} Input set contains lattice generators: L = W 0 W 1 W k 1 contains all k colors of a balanced k-coloring. p.20/35

54 . p.21/35 Windows 2 bd(u) = I(U) U c bd(u) is 1-determined if the color of c is determined by the colors of cells in U

55 . p.21/35 Windows 2 bd(u) = I(U) U c bd(u) is 1-determined if the color of c is determined by the colors of cells in U Define p-determined inductively

56 . p.21/35 Windows 2 bd(u) = I(U) U c bd(u) is 1-determined if the color of c is determined by the colors of cells in U Define p-determined inductively U determines its boundary if all c bd(u) are p-determined for some p

57 Windows 2 bd(u) = I(U) U c bd(u) is 1-determined if the color of c is determined by the colors of cells in U Define p-determined inductively U determines its boundary if all c bd(u) are p-determined for some p Square lattice Nearest neighbor coupling W 2 is not 1-determined. p.21/35

58 Windows 3 Square lattice Nearest and next nearest neighbor coupling indicates 1-determined cells of W 2. p.22/35

59 Windows 3 Square lattice Nearest and next nearest neighbor coupling indicates 1-determined cells of W 2 Three cells in corners of square are 2-determined. p.22/35

60 Windows 3 Square lattice Nearest and next nearest neighbor coupling indicates 1-determined cells of W 2 Three cells in corners of square are 2-determined W i determines its boundary for all i 2. p.22/35

61 . p.23/35 Windows 4 W i0 is a window if W i determines its boundary i i 0 Recall a balanced k-coloring restricted to W k 1 contains all k colors. Hence k-coloring is uniquely determined on whole lattice by its restriction to W k 1 Thm: Suppose lattice network has window. Fix k. Then Finite number of balanced k-colorings on L Each balanced k-coloring is multiply-periodic Antoneli, Dias, G., and Wang (2004)

62 . p.24/35 Synchronous Dynamics & Quotients Given cell network C and balanced coloring G., Stewart, and Török (2005)

63 . p.24/35 Synchronous Dynamics & Quotients Given cell network C and balanced coloring Define quotient network: C = {c : c C} = C/ G., Stewart, and Török (2005)

64 . p.24/35 Synchronous Dynamics & Quotients Given cell network C and balanced coloring Define quotient network: C = {c : c C} = C/ Quotient cells equivalent if C cells equivalent Quotient arrows are projections of C arrows Quotient arrows equivalent if C arrows equivalent G., Stewart, and Török (2005)

65 . p.24/35 Synchronous Dynamics & Quotients Given cell network C and balanced coloring Define quotient network: C = {c : c C} = C/ Quotient cells equivalent if C cells equivalent Quotient arrows are projections of C arrows Quotient arrows equivalent if C arrows equivalent Thm: C-admissible DE restricted to is C -admissible Every C -admissible DE on lifts to C-admissible DE G., Stewart, and Török (2005)

66 . p.25/35 Quotients: Self-Coupling & Multiarrows Balanced two-coloring of bidirectional ring ẋ 1 = f(x 1, x 2, x 3 ) ẋ 2 = f(x 2, x 3, x 1 ) where f(x, y, z) = f(x, z, y) ẋ 3 = f(x 3, x 1, x 2 )

67 . p.25/35 Quotients: Self-Coupling & Multiarrows Balanced two-coloring of bidirectional ring ẋ 1 = f(x 1, x 2, x 3 ) ẋ 2 = f(x 2, x 3, x 1 ) where f(x, y, z) = f(x, z, y) ẋ 3 = f(x 3, x 1, x 2 ) Quotient network: ẋ 1 = f(x 1, x 1, x 3 ) ẋ 3 = f(x 3, x 1, x 1 ) where f(x, y, z) = f(x, z, y)

68 . p.26/35 Asym Network; Symmetric Quotient = {(x, y, x, y, z) : x, y, z R k } Quotient is bidirectional 3-cell ring with D 3 symmetry

69 . p.27/35 Networks and Motifs Two-cell homogeneous networks Leite and G. (2005) R. Milo, S. Shen-Orr,..., and U. Alon. Network motifs: simple building blocks of complex networks. Science 298 (2002) 824.

70 . p.27/35 Networks and Motifs Two-cell homogeneous networks Motifs: feedforward, three chain, uplinked mutual dyad X X X Y Y Y Z Z Z Leite and G. (2005) R. Milo, S. Shen-Orr,..., and U. Alon. Network motifs: simple building blocks of complex networks. Science 298 (2002) 824.

71 . p.27/35 Networks and Motifs Two-cell homogeneous networks Motifs: feedforward, three chain, uplinked mutual dyad X X X Y Y Y Z Z Z 34 homogeneous three-cell networks Leite and G. (2005) R. Milo, S. Shen-Orr,..., and U. Alon. Network motifs: simple building blocks of complex networks. Science 298 (2002) 824.

72 . p.28/35 Three-Cell Feed-Forward Network ẋ 1 = f(x 1, x 1 ) ẋ 2 = f(x 2, x 1 ) ẋ 3 = f(x 3, x 2 ) J = α + β 0 0 β α 0 0 β α G., Nicol, and Stewart (2004); Elmhirst and G. (2005)

73 . p.28/35 Three-Cell Feed-Forward Network ẋ 1 = f(x 1, x 1 ) ẋ 2 = f(x 2, x 1 ) ẋ 3 = f(x 3, x 2 ) J = α + β 0 0 β α 0 0 β α Network supports solution by Hopf bifurcation where x 1 (t) equilibrium x 2 (t), x 3 (t) time periodic G., Nicol, and Stewart (2004); Elmhirst and G. (2005)

74 . p.28/35 Three-Cell Feed-Forward Network ẋ 1 = f(x 1, x 1 ) ẋ 2 = f(x 2, x 1 ) ẋ 3 = f(x 3, x 2 ) J = α + β 0 0 β α 0 0 β α Network supports solution by Hopf bifurcation where x 1 (t) equilibrium x 2 (t), x 3 (t) time periodic x 2 (t) λ 1/2 x 3 (t) λ 1/6 4 x x x x t G., Nicol, and Stewart (2004); Elmhirst and G. (2005)

75 . p.29/35 f(u, v) = (λ + i u 2 )u v; u, v C ẋ 1 = f(x 1, x 1 ) = (λ + i x 1 2 )x 1 x 1 ẋ 2 = f(x 2, x 1 ) = (λ + i x 2 2 )x 2 x 1 ẋ 3 = f(x 3, x 2 ) = (λ + i x 3 2 )x 3 x 2

76 . p.29/35 f(u, v) = (λ + i u 2 )u v; u, v C ẋ 1 = f(x 1, x 1 ) = (λ + i x 1 2 )x 1 x 1 x 1 = 0 is a stable equilibrium for λ < 1 ẋ 2 = f(x 2, x 1 ) = (λ + i x 2 2 )x 2 x 1 ẋ 3 = f(x 3, x 2 ) = (λ + i x 3 2 )x 3 x 2

77 . p.29/35 f(u, v) = (λ + i u 2 )u v; u, v C x 1 = 0 is a stable equilibrium for λ < 1 ẋ 2 = f(x 2, x 1 ) = (λ + i x 2 2 )x 2 x 1 ẋ 2 = f(x 2, 0) = (λ + i x 2 2 )x 2 ẋ 3 = f(x 3, x 2 ) = (λ + i x 3 2 )x 3 x 2

78 . p.29/35 f(u, v) = (λ + i u 2 )u v; u, v C x 1 = 0 is a stable equilibrium for λ < 1 ẋ 2 = f(x 2, x 1 ) = (λ + i x 2 2 )x 2 x 1 ẋ 2 = f(x 2, 0) = (λ + i x 2 2 )x 2 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 ẋ 3 = f(x 3, x 2 ) = (λ + i x 3 2 )x 3 x 2

79 . p.29/35 f(u, v) = (λ + i u 2 )u v; u, v C x 1 = 0 is a stable equilibrium for λ < 1 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 ẋ 3 = f(x 3, x 2 ) = (λ + i x 3 2 )x 3 x 2 ẋ 3 = f(x 3, λe it ) = (λ + i x 3 2 )x 3 λe it

80 . p.30/35 F.F. Ex. f(u, v) = (λ + i u 2 )u v x 1 = 0 is a stable equilibrium for λ < 1 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 ẋ 3 = (λ + i x 3 2 )x 3 λe it

81 . p.30/35 F.F. Ex. f(u, v) = (λ + i u 2 )u v x 1 = 0 is a stable equilibrium for λ < 1 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 Set x 3 = ye it ẋ 3 = (λ + i x 3 2 )x 3 λe it

82 . p.30/35 F.F. Ex. f(u, v) = (λ + i u 2 )u v x 1 = 0 is a stable equilibrium for λ < 1 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 x 3 (t) = y(t)e it ẋ 3 = (λ + i x 3 2 )x 3 λe it ẏe it + yie it = (λ + i y 2 )ye it λe it

83 . p.30/35 F.F. Ex. f(u, v) = (λ + i u 2 )u v x 1 = 0 is a stable equilibrium for λ < 1 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 x 3 (t) = y(t)e it ẋ 3 = (λ + i x 3 2 )x 3 λe it ẏe it + yie it = (λ + i y 2 ) ye it λe it

84 . p.30/35 F.F. Ex. f(u, v) = (λ + i u 2 )u v x 1 = 0 is a stable equilibrium for λ < 1 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 x 3 (t) = y(t)e it ẋ 3 = (λ + i x 3 2 )x 3 λe it ẏe it + yie it = (λ + i y 2 ) ye it λe it ẏe it = (λ y 2 )ye it λe it

85 . p.30/35 F.F. Ex. f(u, v) = (λ + i u 2 )u v x 1 = 0 is a stable equilibrium for λ < 1 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 x 3 (t) = y(t)e it ẋ 3 = (λ + i x 3 2 )x 3 λe it ẏe it + yie it = (λ + i y 2 ) ye it λe it ẏ e it = (λ y 2 )y e it λ e it

86 . p.30/35 F.F. Ex. f(u, v) = (λ + i u 2 )u v x 1 = 0 is a stable equilibrium for λ < 1 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 x 3 (t) = y(t)e it ẋ 3 = (λ + i x 3 2 )x 3 λe it ẏe it + yie it = (λ + i y 2 ) ye it λe it ẏ e it = (λ y 2 )y e it λ e it ẏ = (λ y 2 )y λ

87 . p.31/35 F.F. Ex. f(u, v) = (λ + i u 2 )u v x 1 = 0 is a stable equilibrium for λ < 1 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 x 3 (t) = y(t)e it ẏ = (λ y 2 )y λ

88 . p.31/35 F.F. Ex. f(u, v) = (λ + i u 2 )u v x 1 = 0 is a stable equilibrium for λ < 1 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 x 3 (t) = y(t)e it Set y = λ 1/6 u ẏ = (λ y 2 )y λ

89 . p.31/35 F.F. Ex. f(u, v) = (λ + i u 2 )u v x 1 = 0 is a stable equilibrium for λ < 1 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 x 3 (t) = y(t)e it y(t) = λ 1/6 u(t) ẏ = (λ y 2 )y λ λ 1/6 u = (λ 7/6 λ u 2 )u λ

90 . p.31/35 F.F. Ex. f(u, v) = (λ + i u 2 )u v x 1 = 0 is a stable equilibrium for λ < 1 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 x 3 (t) = y(t)e it y(t) = λ 1/6 u(t) ẏ = (λ y 2 )y λ λ 1/6 u = (λ 7/6 λ u 2 )u λ u = (λ λ 1/3 u 2 )u λ 1/3

91 . p.31/35 F.F. Ex. f(u, v) = (λ + i u 2 )u v x 1 = 0 is a stable equilibrium for λ < 1 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 x 3 (t) = y(t)e it y(t) = λ 1/6 u(t) ẏ = (λ y 2 )y λ λ 1/6 u = (λ 7/6 λ u 2 )u λ u = (λ λ 1/3 u 2 )u λ 1/3 u = λ 1/3 ( u 2 u + 1) + λu

92 . p.32/35 F.F. Ex. f(u, v) = (λ + i u 2 )u v x 1 = 0 is a stable equilibrium for λ < 1 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 x 3 (t) = y(t)e it y(t) = λ 1/6 u(t) u = λ 1/3 ( u 2 u + 1) + λu

93 . p.32/35 F.F. Ex. f(u, v) = (λ + i u 2 )u v x 1 = 0 is a stable equilibrium for λ < 1 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 x 3 (t) = y(t)e it y(t) = λ 1/6 u(t) Solve u = 0 for equilibria u = λ 1/3 ( u 2 u + 1) + λu ( u 2 u + 1) + λ 2/3 u = 0

94 . p.32/35 F.F. Ex. f(u, v) = (λ + i u 2 )u v x 1 = 0 is a stable equilibrium for λ < 1 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 x 3 (t) = y(t)e it y(t) = λ 1/6 u(t) Solve u = 0 for equilibria u = λ 1/3 ( u 2 u + 1) + λu ( u 2 u + 1) + λ 2/3 u = 0 Use IFT to obtain branch of (stable) equilibria u 0 (λ) = 1 + O(λ 2/3 )

95 . p.32/35 F.F. Ex. f(u, v) = (λ + i u 2 )u v x 1 = 0 is a stable equilibrium for λ < 1 x 2 (t) = λe it is stable periodic solution for 0 < λ < 1 x 3 (t) = y(t)e it y(t) = λ 1/6 u(t) u = λ 1/3 ( u 2 u + 1) + λu u 0 (λ) = 1 + O(λ 2/3 ) Thus x 3 (t) is periodic with same period as x 2 (t) x 3 (t) = y(t)e it = λ 1/6 u(t)e it λ 1/6 u 0 (λ)e it = λ 1/6 e it +O(λ 5/6 )

96 Feed Forward Simulation t. p.33/35

97 . p.34/35 Nilpotent Hopf Three-cell network: nilpotent Hopf generically leads to two or four periodic solutions with amplitude growth of λ 1/2 1 Cell System

98 . p.34/35 Nilpotent Hopf Three-cell network: nilpotent Hopf generically leads to two or four periodic solutions with amplitude growth of λ 1/2 1 Cell System Five-cell network: nilpotent Hopf generically leads to two periodic solutions with amplitude growth of λ

99 . p.35/35 Network Summary synchrony iff polydiagonal flow-invariant iff balanced iff quotient network

100 . p.35/35 Network Summary synchrony iff polydiagonal flow-invariant iff balanced iff quotient network different kind of pattern formation patterns of synchrony

101 . p.35/35 Network Summary synchrony iff polydiagonal flow-invariant iff balanced iff quotient network different kind of pattern formation patterns of synchrony genericity in quotient network implies genericity in original network

102 . p.35/35 Network Summary synchrony iff polydiagonal flow-invariant iff balanced iff quotient network different kind of pattern formation patterns of synchrony genericity in quotient network implies genericity in original network codimension one bifurcations are different