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2 ? [5, 6] : phase slip [5, 8] [7, 9 11] [29]

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5 3 3.1? [2] 1 ( ) 1 (1) (2)2 1 (1) ( ( ) ; phase locking) (2) ( ; frequency locking) 0 ( ) π( ) (mutual entrainment) (intrinsic frequency, natural frequency) (observed frequency) ( ) 1 4

6 3.2 [5, 6] dx dt = F (X) X = (x 1, x 2,, x M ), M 2 (3.1) X (x 1, x 2,, x M ) mrna F (X) T X 0 = X 0 (t + T ) M 2 (3.1) 2 ( ) (x 1, x 2,, x M ) 2 5

7 M (3.1) X (3.1) ϕ = ϕ(x) ω 0 () dϕ dt = ω 0, (3.2) θ dθ dt = f(θ, t) ϕ (3.2) (3.2) θ dϕ = ω 0 dθ dθ dt θ 1 ϕ = ω 0 dθ dθ 0 dt θ 1 = ω 0 f(θ, t) dθ 0 (3.3) (isochrone) ϕ X dϕ(x) dt = ω 0 (3.4) ω 0 = 2π/T X 0 (ωt) = X 0 (ϕ) (3.5) ϕ X ϕ dϕ(x) dt = grad X ϕ(x) dx dt 6 (3.6)

8 3 (3.6),(3.1),(3.4) grad X ϕ(x) F (X) = ω 0 (3.7) (3.6) ( )p dx dt = F (X) + ϵp(x, t) (3.8) ϵ (3.7) dϕ(x) dt = grad X ϕ(x) dx dt = grad X ϕ(x) (F (X) + ϵp(x, t)) = ω 0 + ϵ grad X ϕ(x) p(x, t) (3.9) 2 ϵ X o (ϕ) dϕ(x) dt ω 0 + ϵ grad X ϕ(x) X0 (ϕ) p(x 0(ϕ), t) ϕ dϕ dt = ω 0 + ϵq(ϕ, t) Q(ϕ, t) grad X ϕ(x) X0 (ϕ) p(x 0(ϕ), t) (3.10) Q 1 2π (3.1) M ϕ 2 3 ϕ dϕ(x) = ϕ dx 1 + ϕ dx ϕ dx M x 1 x 2 x M dϕ(x) = grad X ϕ(x) dx 7

9 1. (3.1) 2. (3.1) : Landau-Stuart equation(λ ω system) Hopf A = R exp(iθ) da dt = (1 + iη)a (1 + iα) A 2 A (3.11) x = R cos θ, y = R sin θ x dx dt = x ηy (x2 + y 2 )(x αy) + ϵ cos(ωt) dy dt = y + ηx (x2 + y 2 )(y + αx) (3.12) ϕ ( 1-1) ϕ = θ α ln R θ = tan 1 (y/x), R = (x 2 + y 2 ) 1/2 ϕ = tan 1 y x α 2 ln(x2 + y 2 ) (3.13) (3.12) (3.10) dϕ dt = ω 0 + ϵq(ϕ, t) = ω 0 + ϵ ϕ x X0 (ϕ) p(x 0 (ϕ), t) = η α ϵ(sin ϕ + α cos ϕ) cos(ωt) = η α ϵ 1 + α 2 cos(ϕ ϕ 0 ) cos(ωt) (3.14) X 0 R = 1, θ = ϕ tan ϕ 0 = 1/α 2 (3.12) ϕ λ ω system θ R 2 ϕ ϕ = θ α ln R ϕ ( 3 ) 8

10 3.2.3 Q(ϕ, t) ϕ ()ω 0 t ( )ψ ϕ ω 0 t + ψ(= θ + ψ, θ ω 0 t ) ψ ϕ θ ψ ϕ ω 0 t(= ϕ θ) (3.10) dψ dt = dϕ dt ω 0 = ϵ grad X ϕ(x) X0 (ϕ) p(x 0(ϕ), t) (3.15) ψ ω dψ dt = 1 2π [ dθ ϵ grad 2π X ϕ(x) ] X0 (ϕ=θ+ψ) p(x 0(ϕ = θ + ψ), t) 0 ϵγ(ψ) (3.16) ϕ = θ + ψ θ ψ dϕ dt = ω 0 + ϵγ(ψ) (3.17) (3.8) Γ (3.10) Γ ψ 9

11 3.2.4 p(ω 1 t) = p(ω 1 t + 2π) (3.10) dϕ dt = ω 0 + ϵ grad X ϕ(x) X0 (ϕ) p(ω 1t) (3.18) ψ ψ = ϕ ω 1 t (3.19) (3.19) (3.18) dψ dt = dϕ dt ω 1 = (ω 0 ω 1 ) + ϵ grad X ϕ(x) X0 (ϕ=ψ+ω 1 t) p(ω 1t) (3.20) (3.20) 2 ϵ 1 δω (ω 0 ω 1 ) dψ dt ψ 2 θ = ω 1 t Γ(ψ) 1 2π [ dθ grad 2π X ϕ(x) ] X0 (ϕ=ψ+ω 1 t) p(ω 1t) (3.21) 0 (3.20) dψ dt = δω + ϵγ(ψ) (3.22) ψ dϕ dt = ω 0 + ϵγ(ψ) (3.23) (3.19) ψ = const.( :phase locking) dϕ dt = ω 1 1 : 1 n : m ω 0 n m ω 1, m, n ψ = mϕ nω 1 t dψ dt = mω 0 nω 1 + mϵγ(ψ) (3.24) 10

12 1-2 λ ω system 1:1 ωt 1-3 n : m (3.24) Γ (3.21) Q 2π Γ 2π Γ(ψ) = k a k e ikψ (3.25) Γ(ψ) = sin(ψ + α) + sin α (3.26) ( 3.1 ) 4 0 α π/2 (α = 0) 5 (3.22) Γ(ψ) = sin(ψ) (3.27) dψ dt = δω ϵ sin(ψ) (3.28) ψ = ϕ ω 1 t 4 [9] 5 () 11

13 3.1: Γ(ψ) = sin(ψ + α) + sin α () ( ) (3.28) 3.2 (3.28) ψ dψ/dt δω ϵ ψ ϕ dψ dt = dϕ dt ω 1 = 0 (3.29) dϕ/dt ω 1 (3.28) 0 δω ϵ sin ψ = 0 (3.30) ψ ψ dψ/dt (3.28) 3.2(a) ψ (3.30) 3.2(b) δω ϵ (3.31) 12

14 (a) (b) 0 0 (c) 0 3.2: : ψ dψ/dt, Γ(ψ) = ϵ sin(ψ), ϵ > 0. (a) δω ϵ, (b) δω > ϵ, (c) δω = ϵ (3.30) ψ ψ = sin 1 ( δω ϵ ) (3.32) 3.2(a) 0 < ψ < 2π 2 ψ dψ/dt dψ/dt > 0 ψ dψ/dt < 0 3.2(a) ψ 3.2(b) ψ (3.31) 3.2(c) 0 < ψ < 2π dψ/dt > 0 dψ/dt < 0 : δω ϵ c < ϵ (3.33) δω ϵ 13

15 0 δω 3.3: δω < ϵ. 3.3 δω δω ϵ = ϵ c δω / 3.2(a-c) 2 / saddle-node < ψ < 2π (ϵ > 0, δω 0) ψ 0 ϵ < 0 (3.32) : phase slip [5, 8] dψ/dt = 0 (3.22) dϕ/dt = ω 1 (3.19) ω 1 t ψ Ω Ω = dϕ dt = ω 1 + dψ dt (3.34) ω 1 + Ω b 14

16 Ω b (3.28) dψ dt = δω ϵ sin(ψ) F (ψ) (3.35) ϵ / ϵ = ϵ c 3.2(c) 1 F (ψ) F 1 (ψ) ψ 0 ϵ ϵ c 3.2(b) F 1 F (ψ) = ϵ c ϵ + F 1 (ψ) F 1 ψ 0 F 1 (ψ) = F 1 (ψ 0 ) + F 1(ψ 0 )(ψ ψ 0 ) + F 1 (ψ 0) (ψ ψ 0 ) 2 + (3.36) 2 F 1 = F, F 1 = F ψ 0 F 1 (ψ 0 ) = 0 ψ / F 1 (ψ 0) = 0 dψ dt = F (ψ) ϵ c ϵ + F (ψ 0 ) (ψ ψ 0 ) 2 (3.37) 2 dψ dt = ϵ c ϵ + F (3.38) (ψ 0 ) 2 (ψ ψ 0 ) 2 T b x T b = 2π 0 dψ ϵ c ϵ + F (3.39) (ψ 0 ) 2 (ψ ψ 0 ) 2 F (ψ 0 ) 2(ϵ c ϵ) (ψ ψ 0) ± Ω b = 2π T b ϵ c ϵ (3.40) ϕ = ω 1 t + Ω b t ϵ c ϵ () 3.4 ( 1, 2) ( 3-5) (2π/Ω b ) (phase slip) phase slip 2π/Ω b 15

17 0 δω 3.4: Phase slip. [5].(a) δω ϵ.(b).. (phase slip). ( ) (3.40) 1-4 F (ψ) (3.35) Ω b (= (3.39) (3.40) ) (3.28) 3.4 phase slip (2π/Ω b ) 16

18 y y x y x y (a) eta= 1 alpha= 0.1 epsilon= 50 delta= 0.01 tp= (b) eta= 1 alpha= 0.1 epsilon= 50 delta= 0.01 tp= time x time (c) eta= 1 alpha= 0.1 epsilon= 50 delta= 0.01 tp= 22 (d) eta= 1 alpha= 0.1 epsilon= 50 delta= 0.01 tp= time x time 3.5: (1). λ ω. (3.12) ϵ cos(ωt) (3.41).t p. η = 1.1,α = 0.1,ϵ = 50, σ = isochron,, (a),(b) 3π/2 < ϕ < 2π, t p = 20. (c), (d) 0 < ϕ < π/2,t p = [7, 9 11] Γ(ψ) 3.21 (phase response curve; PRC) λ ω (3.12) ϵ cos(ωt) (3.41).t p ϵ, t p t t p + σ p(t) = 0, otherwise (3.41) λ ω (Figs.3.5,3.6(a,c,e,g)) isochron α 17

19 y y x y x y (e) eta= 1 alpha= 0.1 epsilon= 50 delta= 0.01 tp= 21 (f) eta= 1 alpha= 0.1 epsilon= 50 delta= 0.01 tp= time x time (g) eta= 1 alpha= 0.1 epsilon= 50 delta= 0.01 tp= (h) eta= 1 alpha= 0.1 epsilon= 50 delta= 0.01 tp= time x time 3.6: (2)., Fig.3.5. (e),(f)ϕ 0, t p = 21. (g), (h)π < ϕ < 3π/2, t p = 25. (3.41) Figs.3.5,3.6 (a,b) (c,d) (e,f) (g,h) Figs.3.5,3.6(a,c,e,g) (3.41) x (a) () ( ) (c) ( ) (e) (g) ( 2 ) ϕ (ϕ) 18

20 3.7: (PRC)., Fig.3.5.(a) ϵ. (b) σ. 19

21 T τ (ϕ) = τ/t (3.42) Fig.3.7 PRC σ ϵ PRC σ ϵ Fig.(b) type1 type0 Fig.3.8 type1 (a) PRC Fig. (b) ϕ n+1 = F (ϕ n, p) Phase Transition Curve (PTC) type1 1 type1 type0 (c) π π PRC 1-5 Fig.3.8(a), (c) PRC 3.8(b), (d) (3.41) σ ϵ (type1 ) dϕ dt = ω 0 + (ϕ) p(t) (3.43) (3.10) (ϕ) grad X ϕ X0 λ ω Fig.3.9 Fig.3.7(a) ϵ PRC grad X ϕ X0 p(t) p(θ) = p(θ + 2π) (3.21) 20

22 3.8: type1 type0 (PRC).(a,b) type1. (c,d) type0. Γ Γ(ψ) 1 2π [ ] dθ (ψ + θ) p(θ) 2π 0 (3.44) λ ω Fig.3.9(b) 6 Γ(ψ) sin(ψ) λ ω λ ω B. Ermentraut G. Bowtell Adjoint [12] Ermentraut XPP-Aut [13] 21

23 3.9:.(a)grad X ϕ X0. Γ(ψ). (b) 22

24 X 1, X 2 dx 1 = F 1 (X 1 ) + ϵp(x 1, X 2 ) dt dx 2 = F 2 (X 2 ) + ϵp(x 2, X 1 ) dt (3.45) X 0 1, X0 2 ω 1, ω 2 ( ) ϕ 1, ϕ 2 ϵ (3.23) 7 dϕ 1 dt = ω 1 + ϵγ 1,2 (ϕ 1 ϕ 2 ) dϕ 2 dt = ω 2 + ϵγ 2,1 (ϕ 2 ϕ 1 ) (3.46) Γ 1,2, Γ 2,1 2 Γ(ψ) 1,2 = sin(ψ) Γ 1,2 (ψ) = sin(ψ + α) + sin(α) Γ(ψ) 1,2 = sin(ψ) ψ ϕ 1 ϕ 2 δω = ω 1 ω 2 (3.46) dψ dt = δω 2ϵ sin(ψ) (3.47) dψ dt = dϕ 1 dt dϕ 2 dt = 0 (3.48) (3.47) 0 δω 2ϵ sin ψ = 0 (3.49) ψ 3.2 δω 2 ϵ c < ϵ (3.50) 7 23

25 ( ) < ψ < 2π (ϵ > 0, δω 0) ψ 0 ψ = ψ 0 (3.49) ψ 0 = sin 1 ( ω 1 ω 2 ) (3.51) 2ϵ ω 1 ω 2 0 π ϵ 0 (ϵ > 0) ψ 0 π (ϵ < 0) Ω Ω = dϕ 1 dt = dϕ 2 dt = ω 1 ϵ sin(ψ 0 ) = ω 1 + ω 2 2 (3.52) Γ = sin Γ 1,2 (ψ) = sin(ψ + α) + sin(α)( (3.26)) ψ α Ω α ψ α = sin 1 ( ω 1 ω 2 2ϵ cos α ) (3.53) Ω α = ω 1 + ω ϵ(1 cos ψ α ) sin α (3.54) ϵ > 0, 0 < α < π/2 ( 2 ) 0 Ω α (ω 1 + ω 2 )/2 24

26 1-6 (3.53),(3.54) α = 0 α = π/4 2 ϵ (3.46) -ϵ Ω 1,2 Ω 1,2 1, 2 Ω 1 Ω Schuster Wagner [14] (3.46) τ dϕ 1 (t) = ω 1 ϵ sin(ϕ 1 (t) ϕ 2 (t τ)) dt dϕ 2 (t) = ω 2 ϵ sin(ϕ 2 (t) ϕ 1 (t τ)) dt (3.55) ψ ϕ 1,2 (t) = φ(t) ± α/2 (3.56) φ 1,2 α (3.55) ( [14] ) f(ω) ω 0 Ω ϵ tan(ωτ) cos 2 (Ωτ) ω 2 /ϵ 2 = 0, (3.57) sin 1 ( ω/ϵ cos(ωτ)) cos(ωτ) 0, ψ = (3.58) π sin 1 ( ω/ϵ cos(ωτ)) Ω ϕ 1,2 (t) = Ωt±ψ/2 ω 0 = (ω 1 +ω 2 )/2, ω = (ω 1 ω 2 )/2 (3.57) (3.58) cos(ωτ) ϵ (3.57),(3.58) 25

27 3.10 (3.57) τ ϵ 3.10(a) Ω 3.10(b,c) Ω 3.10(d) τ ϵ 3.11 (a) (3.55) (b) [15] ϵ L W ( ϵ ) L( τ ) 3.11 ( W ) ϵ < 0 L W ( I) L ( II) ϵ L( τ) W ( ϵ ) ( III) ( IV, V... ) I ( ) II 3.10 (3.57) Ω 26

28 (a) f(ω) (b) f(ω) ψ π ψ π ψ 0 0 Ω 0 Ω (c) f(ω) (d) f(ω) ψ π ψ 0 ψ π ψ 0 ψ π 0 Ω 0 Ω 3.10: Ω.. (3.58). ω 0 = 0.06, δω = (a)ϵ = 0.05, τ = 10, (b), ϵ = 0.15, τ = 10, (c), ϵ = 0.05, τ = 20, (d), ϵ = 0.15, τ =

29 (a) L (mm) L=4 W=0.30 (II) L=10 (I) (IV) (III) W (mm) (b) τ W=0.30 (II) (I) L=10 L= (V) (IV) (III) :.. (a), : (. ). : (), :, : (), :, :. (b) ( (3.55)). :, :, :. [15]. 28

30 (a) (b) (rad/sec) 0.10 UE Ω 0.10 AP UE AP IP AP W(mm) W(mm) K K UE AP UE AP IP (c) 0.09 AP 0.06 IP L(mm) τ(sec) 0.07 AP IP W(mm) AP W(mm) L(mm) 3.12:.,. (a) L=4 (mm), (b) L=10 (mm), (c) W=0.30 (mm) [15]. 8 τ Ω (3.57) τ 0 ω 0 tan Ωτ Ωτ, cos Ωτ 1 Ω ω ϵτ (3.59) (a) 3.12(b), (c) ( ) 8 (3.55) ( ) 29

31 (3.57) ( 3.10 ) 3.12(a) 10% oscillator death 0 [16, 17] [18, 19] 30

32 [20,21] Γ = sin ϕ 1 = ω 1 ϵ sin(ϕ 1 ϕ 2 ) ϕ i = ω i ϵ sin(ϕ i ϕ i 1 ) ϵ sin(ϕ i ϕ i+1 ) ϕ N = ω N ϵ sin(ϕ N ϕ N 1 ) 1 < i < N (3.60) (3.60) Γ = sin N ϕ i = i=1 N ω i (3.61) (3.60) i, j i=1 (3.61) N ϕ 1 = ϕ 2 = = ϕ i (3.62) ϕ i = 1 N k=1 ϕ i = Nϕ i N ω i ω (3.63) i=1 j (1 j < N) j (3.60) j j ϕ i = i=1 j ω i ϵ sin(ϕ j ϕ j+1 ) (3.64) i=1 (3.62) ψ j ϕ j+1 ϕ j j j ϕ k = jϕ i j ω k=1 ϵ sin(ψ j ) = j (ω i ω) X j (3.65) i=1 : X j ϵ (3.66) 31

33 X j i = 1 j ( ) ϵ j 1 j (3.66) 1 max X j ϵ (3.67) 1 j N 1-8 Γ i (ψ) = sin(ψ + α) + sin(α) α ϕ k = ω k ϵ N N sin(ϕ k ϕ j ) (3.68) j=1 ϵ/n N N z k = e iϕ k = cos ϕ k + i sin ϕ k Z Z 1 N N e iϕ k k=1 (3.69) Ke iθ K cos Θ = 1 N K sin Θ = 1 N N cos ϕ k k=1 (3.70) N sin ϕ k k=1 32

34 K Θ K Θ K 0 2π 0 K = 0 K 0 K (3.68) 2 ϵ sin(ϕ k ϕ j ) = ϵ (sin ϕ k cos ϕ j cos ϕ k sin ϕ j ) N N j j = ϵ N sin ϕ k cos ϕ j ϵ N cos ϕ k sin ϕ j = ϵk sin(ϕ k Θ) (3.68) j ϕ k = ω k ϵk sin(ϕ k Θ) (3.71) = ω Θ = ωt ψ k = ϕ k Θ = ϕ k ωt (3.71) ψ k = ω k ω ϵk sin ψ k (3.72) ψ k = 0 ω k ω ϵk sin ψ k = 0 ω k ω ϵk (3.73) ψ k = sin 1 ( ω k ω ϵk ) k Ω k = ϕ k = ω k ϵk sin ψ k = ω k + ( ω ω k ) = ω 33 j

35 (3.73) k 1 34

36 4 1 [26] [29] x,y x,y D x,d y x t = D x 2 x + f(x, y) y t = D y 2 y + g(x, y) (4.1) ẋ 1 = f(x 1, y 1 ) + d x (x 2 x 1 ) ẏ 1 = g(x 1, y 1 ) + d y (y 2 y 1 ) ẋ 2 = f(x 2, y 2 ) + d x (x 1 x 2 ) ẏ 2 = g(x 2, y 2 ) + d y (y 1 y 2 ) (4.2) f(x, y) = x x 3 4y,g(x, y) = x 3y 1 d u = d v = 0 35

37 (4.2) x = 0, y = 0 ẋ = x 4y ẏ = x 3y (4.3) λ = 1 < 0 (4.2) (4.2) 1 x 1 = y 1 = x 2 = y 2 = 0 2 (4.2) (0,0,0,0) ẋ 1 = x 1 4y 1 + d x (x 2 x 1 ) ẏ 1 = x 1 3y 1 + d y (y 2 y 1 ) ẋ 2 = x 2 4y 2 + d x (x 1 x 2 ) ẏ 2 = x 2 3y 2 + d y (y 1 y 2 ) (4.4) p 1 x 1 + x 2, q 1 y 1 + y 2, p 2 x 1 x 2, q 2 y 1 y 2 x 1 + x 2 y 1 + y 2 ṗ 1 = ẋ 1 + ẋ 2 = p 1 4q 1 q 1 = ẏ 1 + ẏ 2 = p 1 3q 1 (4.5) (4.3) p 1 = q 1 = 0 ṗ 2 = ẋ 1 ẋ 2 = p 2 4q 2 2d x p 2 h(p 2, q 2 ) q 2 = ẏ 1 ẏ 2 = p 2 3q 2 2d y q 2 k(p 2, q 2 ) (4.6) p 2 = q 2 = 0 (4.6) J det J = (1 2d x )(3 + 2d y ) + 4 tr J = 2(1 + d x + d y ) < 0 (4.7) 36

38 λ = (tr J ± tr J 2 4 det J)/2 tr J < 0 tr J + tr J 2 4 det J > 0 det J < 0 (1 2d x )(3 + 2d y ) + 4 < 0 (4.8) 2 (4.2) S 1 (x, y, x, y ), S 2 ( x, y, x, y ) 2 x = y = 3 + 2d y (1 2d x )(3 + 2d y ) 4, (3 + 2d y ) x (4.9) (4.8) x S 1,S x y (a) (4.2) US S 1,S 2 (b) (4.8) y d y x d x (4.3) activator-inhibitor x activator, y inhibitor inhibitor 37

39 4.1: 1.(a), (b). 4.1(a) (b) ( ) 4-1 (4.2) S 1 (x, y, x, y ), S 2 ( x, y, x, y ) 2 x, y (4.9) S 1 (x, y, x, y ), S 2 ( x, y, x, y ) (4.8) 4-3 (4.2) 1 38

40 (4.1) d = D y /D x, γ = 1/D x, t = τ/γ τ t x,y u, v (4.1) u t = 2 u + γf(u, v) v t = d 2 v + γg(u, v) (4.10) f(u, v) = 0, g(u, v) = 0 (u 0, v 0 ) ( 4.2.1) ( ) ( 4.2.2) (?) (4.10) (D x = 0, D y = 0 ) (u 0, v 0 ) ( ) f u f v J g u g v (4.11) tr J = f u + g v < 0 det J = f u g v + f v g u > 0 (4.12) (4.12) (u 0, v 0 ) w 1 39

41 w (u u 0, v v 0 ) (4.10) w t = γjw + D 2 w (4.13) ( ) 1 0 D 0 d (; 0; 0) 2 1 w x w (0, t) = (a, t) = 0 x x = 0 x = a ( 2 )1 w = c cos( nπx a ) (4.14) c n k nπ/a () k w x = cnπ a sin(nπx a ) x = 0 a w x = 0 2 w x 2 = c(nπ a )2 cos( nπx a ) = k 2 w 2 w + k 2 w = 0 (4.15) k w k w w(r, t) = C k e λt w k (r) (4.16) k 2 () w(0, t) = w(a, t) = w 0 40

42 r C k λ (4.16) (4.13) (4.15) λ k C k e λt w k (r) = γj k C k e λt w k (r) + D 2 w = γj k C k e λt w k (r) Dk 2 k C k e λt w k (r) C k k λw k = γjw k Dk 2 w k. (4.17) (λi γj + Dk 2 )w k = 0 (4.18) w k 0 λ 3 λi γj Dk 2 = 0 (4.19) λ 2 + λ(k 2 (1 + d) γ(f u + g v )) + h(k 2 ) = 0 (4.20) h(k 2 ) dk 4 γ(df u + g v )k 2 + γ 2 J (4.20) Re(λ) < 0 2 (4.16) (0,0) (4.1) D x = 0, D y = 0 D x > 0, D y > 0 (D x = 0, D y = 0) Re(λ(k = 0)) < (4.12) 3 w k = 0 (0,0) 41

43 D x > 0, D y > 0 Re(λ(k 0)) > 0 k 1, 0 B,C λ = B ± B 2 4C 2 (4.21) 1 Re(λ(k 0)) > 0 1) 2) 2) B + B 2 4C > 0 (4.12) B > 0 C = h(k 2 ) < 0 1) h(k 2 ) < 0 k h dk 4 > 0, γ 2 J > 0( (4.12) ) df u + g v > 0 h(k 2 ) k 2 2 k { h(kmin) 2 = γ 2 J (df u + g v ) 2 } 4d (4.22) k 2 min = γ(df u + g v )/2d (df u + g v ) 2 4d(f u g v f v g u ) > 0 (4.23) ( [27] ) df u + g v > 0 (df u + g v ) 2 4d(f u g v f v g u ) > 0 (4.24) (4.12) (4.24) k k k 2 h(k 2 ) d k c = γ J /d c d c 2 k 42

44 h(k 2 ) () - (4.12) (4.24) k 2 h(k 2 ) 4 (4.24) (4.12) 43

45 [1] - -,P., Y., Ch.,, (1992) [2] J. Pantaleone, Synchronization of metronomes, Am. J. Phys. 70, (2002) [3] Unserstanding Nonlinear Dynamics, Daniel Kaplan and Leon Glass, Springer (1995) [4] Nonlinear Dynamics and Chaos, Steven H. Strogatz, Addison Wesley (1994) [5] Synchronization: A universal concept in nonlinear sciences, Cambridge Nonlinear Science Series 12, A. Pikovsky, M. Rosenblum and J. Kurth, Cambridge (2001) [6], (1),, (2005) [7] Izhikevich EM, Kuramoto Y., Weakly coupled oscillators. In: Encyclopedia of mathematical physics, Vol 5, Ed 1 (Francoise J-P, Naber GL, Tsun TS, eds), pp 448. Boston: Elsevier (2006). [8] Z.Zheng, G. Hu, B. Hu, Phase slip and phase synchronization of coupled oscillators, Phys.Rev. Lett.,81, (1998) [9] G. B. Ermentraut, The Mathematics of Biological Oscillator, Methods Enzymol, 240, (1994) [10] A. T. Winfree, The geometryr of biological time 2nd edn., New York Springer (2000). [11] C. H. Johnson, Phase Response Curve: What can they tell us about circadian Clocks?, Circadian clocks from cell to human, edited by T. Hisohige and K. Honma, Hokkaido Univ. Press, Sapporo, (1992). 44

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ver.1 / c /(13)

ver.1 / c /(13) 1 -- 11 1 c 2010 1/(13) 1 -- 11 -- 1 1--1 1--1--1 2009 3 t R x R n 1 ẋ = f(t, x) f = ( f 1,, f n ) f x(t) = ϕ(x 0, t) x(0) = x 0 n f f t 1--1--2 2009 3 q = (q 1,..., q m ), p = (p 1,..., p m ) x = (q,