ver.1 / c /(13)

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1 c /(13)

2 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 q = (q 1,..., q m ), p = (p 1,..., p m ) x = (q, p) R n, n = 2m V(q) T(p) H(p, q) = T + V dq i dt = H p i, dp i dt = H q i, i = 1,..., m. H t dh/dt = 0 H (n 1) 2 3 c /(13)

3 0 div f = n i=1 f i x i discrete-time dynamical system continuous-time dynamical system k x(k) 0 x(0) {x(k), k = 0, 1,...} x(k) x(k + 1) x(k + 1) = T(x(k)), x(k) X, k = 0, 1,... (1 1) T : X X ; x(k) x(k + 1) = T(x(k)) (1 2) X logistic x(k + 1) = α(1 x(k))x(k), x [0, 1] (1 3) Hénon x 1 (k + 1) = 1 + x 2 (k) ax 1 (k) 2 x 2 (k + 1) = bx 1 (k), x 1, x 2 R (1 4) (1 3) α R (1 4) a, b R X c /(13)

4 dx dt = f (t, x), x X, t R (1 5) t x(t) autonomous system non-autonomous system (1 5) dx dt = g(x), x X, t R (1 6) g(x(t)) Poincaré R n 1 Π = {x R n q(x) = 0}, q : R n R ; x q(x) (1 7) x 0 ϕ(t, x 0 ) P : Π Π ; x ϕ(τ(x), x) (1 8) Poincaré map recurrence map τ Π Π Π Σ R n 1 T : Σ Σ P T c /(13)

5 2 (1 5) t f (t + L, x) = f (t, x), L > 0 (1 9) x 0 R n ϕ(t, x 0 ) L T : R n R n ; x ϕ(l, x) (1 10) stroboscopic map c /(13)

6 van der Pol van der Pol ẍ + µ(x 2 1)ẋ + x = (1 11) 1 1 van der Pol 1) µ > 0 Hopf µ van 2, der Pol 3) M L C R 1 1 van der Pol Dufng Duffing ẍ + δẋ + ω 2 0 x + βx3 = γ cos(ωt + φ) (1 12) Duffing β β Duffing δ = γ = 0 Jacobi 4, 3) 5) Mathieu Hill Hill ẍ + q(t)x = 0 (1 13) c /(13)

7 ẍ + (a + 2b cos 2t)x = 0 (1 14) Mathieu 2 ÿ + 2α(t)ẏ + β(t)x = 0 (1 15) x = ye αdt, q = β + α 2 α (1 16) Hill 6) F β>0 β=0 x β<0 L R C 1 3 Duffing 1 2 (F = ω 2 0 x + βx3 ) 1) B. van der Pol, A Theory of the Amplitude of Free and Forced Triode Vibrations, Radio Review, vol.1, pp , ) B. van der Pol and J. van der Mark, Frequency Demultiplication, Nature, vol.120, pp , ) E.A. Jackson, Perspectives of Nonlinear Dynamics, Cambridge University Press, 1991.,, ) C. Hayashi, Nonlinear Oscillations in Physical Systems, McGrawHill, New York, ),, ) W. Magnus and S. Winkler, Hill s Equation, Dover Publications, Inc., New York, c /(13)

8 dx/dt = f (x) f (x) = 0 equilibrium point x Lyapunov x Lyapunov ɛ δ δ ɛ Lyapunov ɛ ɛ x(t) lim x(t) = x t ξ(t) ξ (t) variation x(t) = x + ξ(t) ξ(t) dξ/dt = Aξ(t) A = D f ( x) = f x x(t)= x f / x f dξ/dt = Aξ(t) x dx/dt = f (x) A 0 hyperbolicsimple n n + 1 1) A f f / x (i, j) f i / x j (1 i, j n) n ds F F(ds) = Jds J J 0 1 critical curve basin of attraction 2) basin bifurcation ẋ = f(x) x x c /(13)

9 x U V(x). U { x} V(x) > 0 V( x) = 0; U V(x) 0 U { x} V(x) < 0 3, V(x) Lyapunov function 4) x V(x) V(x) 4, 5) ẋ(t) = f(x(t), t) t x(t) = x(t + T) T > 0 T x(t) T 0 T = 0 T x(t) C = 0 t T x(t) ɛ δ C δ τ t τ C ɛ x(t) C V x τ x C x(t) x(t) x = x(t) f A(t) = D f( x(t)) ẋ(t) = A(t)x(t) x = 0 x(t) T T A(t) n ẋ(t) = A(t)x(t) n X(t) X(t + T) X(t + T) = X(t)S S S = e TR R t T P(t) X(t) = P(t)e tr y(t) = P(t) 1 x(t) y ẏ = Ry S λ R µ = (1/T) ln λ 1 Lyapunov c /(13)

10 ẋ(t) = A(t)x(t) ẋ(t) = f(x(t), t) x(t) 1 x(t) ẋ(t) = f(x(t), t) 1 n 1 1 x(t) ẋ(t) = f(x(t), t) H. Bohr x(t) ɛ T(ɛ) = {τ R : x(t + τ) x(t) ɛ ( t R)} R L(ɛ) t R T(ɛ) [t, t + L(ɛ)] 6) ɛ L(ɛ) T ɛ L(ɛ) = T quasi-periodic solution 1),, ) C. Mira, Chaotic Dynamics, World Scientific, Singapore, ),, ) H.K. Khalil, Nonlinear systems, Prentice Hall, ) J.K. Hale and S.M.V. Lunel, Introduction to functional differential equations, Springer, ) A.M. Fink, Almost Periodic Differential Equations, Springer-Verlag, c /(13)

11 ) 2) 1 1 3) 1) Q ) Q 4) A B c /(13)

12 bifurcation local bifurcation global bifurcation 0 saddle-node bifurcation Hopf period-doubling bifurcation Neimark-Sacker pitchfork bifurcation transcritical bifurcation supercritical subcritical separatrix loop saddle connection homoclinic bifurcation heteroclinic bifurcation 2 doubly asymptotic point c /(13)

13 structural stability 5) A region of initial conditions basin of attraction 6) : basin boundary 1) SYNC:,, ),, ) R.C. Mackey, Injection locking of klystron oscillator, IRE Trans. Microwave Theory & Tech., vol.mtt-10, no.7, p.228, ) J.M.T. Thompson and H.B. Stewart, Nonlinear dynamics and chaos, John Wiley and Sons, ) C. Pugh and M.M. Peixoto, Structural Stability, Scholarpedia, vol.3, no.9, p.4008, ) E. Ott, Basin of attraction, Scholarpedia, vol.1, no.8, p.1701, c /(13)

ohp_06nov_tohoku.dvi

ohp_06nov_tohoku.dvi 2006 11 28 1. (1) ẋ = ax = x(t) =Ce at C C>0 a0 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)