2006 11 28
1. (1) ẋ = ax = x(t) =Ce at C C>0 a<0 x(t) (t ) a>0 x(t) 0(t )!! 1 0.8 0.6 0.4 0.2 2 4 6 8 10-0.2 (1) a =2 C =1
1. (1) τ>0 (2) ẋ(t) = ax(t τ) 4 2 2 4 6 8 10-2 -4 (2) a =2 τ =1!!
1. (2) A. (2) 0 <aτ< π 2 B. (2) aτ > 1 e (2) λ = ae λτ A B
1. A B 0 1/e ¼/2 a =
1. Mackey-Glass (3) ẋ(t) = ax(t) + bx(t τ) 1+x(t τ) m τ 0 1.2 0.8 1 0.6 0.4 0.2 50 100 150 200 (3) a =2 b =3.54 m =10 τ =1
2-1. DF by Pyragas (4) ẋ = f(x) f :Ω R n, Ω R n, f: C 1 (4) ω x (t) Pyragas DF (5) ẋ(t) =f(x(t)) K(x(t) x(t ω)) K x (t)
2-1. DF by Pyragas Rossler DF ẋ = dy ez (6) ẏ = hx + fy k(y(t) y(t τ)) ż = b + gz(x c) d = e = h = g = 1, f = b = 0.2, c = 5.7 Ex. 1: k =0( ) = chaotic Ex. 2: k =0.2, τ=5.9 = periodic (T 5.88867) Ex. 3: k =0.2, τ=11.75 = periodic (T 11.7536) Ex. 4: k =0.2, τ =17.5 = chaotic
2-1. DF by Pyragas (6) k =0 0 t 2000
2-1. DF by Pyragas (6) k =0.2, τ =5.9 0 t 1000
2-1. DF by Pyragas (6) k =0.2, τ =11.75 0 t 1000
2-1. DF by Pyragas (6) k =0.2, τ =17.5 0 t 2000
2-2. DF by Atay (7) ẍ + ω 2 x + εg(x, ẋ, ε) =0 ω, ε g : R 3 R C 2 ε g(0, 0,ε)=0 Atay DF (8) ẍ + ω 2 x + εg(x, ẋ, ε) =εf(x(t τ)) f : R R
2-2. DF by Atay van der Pol DF (9) ẍ + ε(x 2 1)ẋ + x = εkx(t τ) C. k sin τ<1 ε (9) x(t) =2 1 k sin τ cos (1 ε ) 2 k cos τ t + O(ε 2 )
2-2. DF by Atay A := 2 1 k sin τ k = 0 or sin τ =0= A =2 A 2 τ k sin τ 0 k = 1 (A/2)2 sin τ
2-2. DF by Atay 2 1 20 40 60 80 100-1 -2 (8) ε =0.1 k =0 2
2-2. DF by Atay 3 2 1 20 40 60 80 100-1 -2-3 (9) ε =0.1 τ =1 k = 1 (3/2)2 sin 1 3 1.48549
2-2. DF by Atay 1.5 1 0.5 20 40 60 80 100 120 140-0.5-1 -1.5 (9) ε =0.1 τ =1 k = 1 (1.5/2)2 sin 1 1.5 0.519923
2-3. Atay (10) ẍ +sinx =0 x A A T T =4 π 2 0 dψ 1 sin 2 (A/2) sin 2 ψ =2π ( 1+ 1 16 A2 + )
2-3. Atay DF (11) ẍ +sinx = 1 6 { k1 x(t τ)+k 2 x(t τ) 3} 1. k 2 sin τ > 0 k 1 k 2 < 0 (11) x(t) 4k 1 3k 2 cos ( 12k2 12k 2 k 1 ) t sin x x x 3 /6 ε = 1 6 Attay
1 A k 1 + 34 k 2A 2 «sin τ =0 2-3. Atay T 2π» 1+ 1 6 j 3 8 A2 + 1 2 k 1 + 34 k 2A 2 «A 2 = 4k 1 /(3k 2 ) 1 T 2π 1 k «1 12k 2 ff cos τ (1/6) 2 0.0277778
2-3. Atay 1.5 1 0.5 20 40 60 80 100-0.5-1 -1.5 (11) k 1 =6k 2 (2π 7)/π 1.36902 k 2 =1 t 1 = 390.199, t 2 = 397.297 = 7.09779 x(0) = 0.1, ẋ(0) = 0 x(0) = 0.2, ẋ(0) = 0 t = 390 7.0977897916112624 7.097789791656169 10
2-4. 2- by Sasaki & Miyazaki (12) ẋ(t) = αx(t) + f(x(t τ)) α f (13) ẋ(t) = αx(t) + 1 2 {f(x(t τ)) + f(x(t 1))} 2
2-4. 2- by Sasaki & Miyazaki f f(u) := { c u [0,b] 0 u (b, ) f u c 0 b u
2-4. 2- by Sasaki & Miyazaki Mackey-Glass (3) ẋ(t) = ax(t) + bx(t τ) 1+x(t τ) m f f(u) = bu, (b =3.54, m = 10) 1+um f u 2.5 2 1.5 1 0.5 0.5 1 1.5 2 u
2-4. 2- by Sasaki & Miyazaki D (Heiden & Mackey). γ := c/α > b (13) T 1 A 1 T 1 = 1 α log (γeα + b γ)(γe α b) b(γ b), A 1 = γ(1 e α ). 1.5 1.25 1 0.75 0.5 0.25 5 10 15 20 (13) α =1/2 b = c = τ =1
2-4. 2- by Sasaki & Miyazaki 1. 0 <τ<1 2b(e α 1)/(e α + e ατ 2) γ 2b(e α 1)/(e α e ατ ) (13) T 2 T 2 = 1 α α e (γ +e ατ + b γ)(γ eα +e ατ b) 2 2 log, b(γ b) α log 2 A 2 A 2 = γ(1 e ατ ).
2-4. 2- by Sasaki & Miyazaki C 1 τ <1 T 1 = 1 α log (γeα + b γ)(γe α b) b(γ b) T 2 = 1 e (γ α +e ατ 2 + b γ)(γ eα +e ατ 2 b) log α b(γ b) A 1 = γ(1 e α ), A 2 = γ(1 e ατ ). T 1 >T 2, A 1 >A 2 2 =?
References [1] Atay, F. M., Delayed-feedback control of oscillations in non-linear planar systems, Int. J. Control, (2002), 75, 297 304. [2] Pyragas, K., Continuous control of chaos by self-controlling feedback, Phys. Lett. A, 170 (1992), 421 428.