ohp_06nov_tohoku.dvi

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2 1. (1) ẋ = ax = x(t) =Ce at C C>0 a<0 x(t) (t ) a>0 x(t) 0(t )!! (1) a =2 C =1

3 1. (1) τ>0 (2) ẋ(t) = ax(t τ) (2) a =2 τ =1!!

4 1. (2) A. (2) 0 <aτ< π 2 B. (2) aτ > 1 e (2) λ = ae λτ A B

5 1. A B 0 1/e ¼/2 a =

6 1. Mackey-Glass (3) ẋ(t) = ax(t) + bx(t τ) 1+x(t τ) m τ (3) a =2 b =3.54 m =10 τ =1

7 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)

8 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 ) Ex. 3: k =0.2, τ=11.75 = periodic (T ) Ex. 4: k =0.2, τ =17.5 = chaotic

9 2-1. DF by Pyragas (6) k =0 0 t 2000

10 2-1. DF by Pyragas (6) k =0.2, τ =5.9 0 t 1000

11 2-1. DF by Pyragas (6) k =0.2, τ = t 1000

12 2-1. DF by Pyragas (6) k =0.2, τ = t 2000

13 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

14 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 )

15 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

16 2-2. DF by Atay (8) ε =0.1 k =0 2

17 2-2. DF by Atay (9) ε =0.1 τ =1 k = 1 (3/2)2 sin

18 2-2. DF by Atay (9) ε =0.1 τ =1 k = 1 (1.5/2)2 sin

19 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 ( A2 + )

20 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

21 1 A k k 2A 2 «sin τ = Atay T 2π» j 3 8 A k k 2A 2 «A 2 = 4k 1 /(3k 2 ) 1 T 2π 1 k «1 12k 2 ff cos τ (1/6)

22 2-3. Atay (11) k 1 =6k 2 (2π 7)/π k 2 =1 t 1 = , t 2 = = x(0) = 0.1, ẋ(0) = 0 x(0) = 0.2, ẋ(0) = 0 t =

23 by Sasaki & Miyazaki (12) ẋ(t) = αx(t) + f(x(t τ)) α f (13) ẋ(t) = αx(t) {f(x(t τ)) + f(x(t 1))} 2

24 by Sasaki & Miyazaki f f(u) := { c u [0,b] 0 u (b, ) f u c 0 b u

25 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 u

26 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 ) (13) α =1/2 b = c = τ =1

27 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 ).

28 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 =?

29 References [1] Atay, F. M., Delayed-feedback control of oscillations in non-linear planar systems, Int. J. Control, (2002), 75, [2] Pyragas, K., Continuous control of chaos by self-controlling feedback, Phys. Lett. A, 170 (1992),

1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0

1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx