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, 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 2010 2/(13)
0 div f = n i=1 f i x i 0 0 1--1--3 2009 3 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 2010 3/(13)
dx dt = f (t, x), x X, t R (1 5) t x(t) 1--1--4 2009 3 autonomous system non-autonomous system (1 5) dx dt = g(x), x X, t R (1 6) g(x(t)) 0 1--1--5 2009 3 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 2010 4/(13)
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 1--1--6 c 2010 5/(13)
1 -- 11 -- 1 1--2 1--2--1 van der Pol van der Pol ẍ + µ(x 2 1)ẋ + x = 0 2009 4 (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 1--2--2 Dufng Duffing ẍ + δẋ + ω 2 0 x + βx3 = γ cos(ωt + φ) (1 12) Duffing β 1 2 1 3 β Duffing δ = γ = 0 Jacobi 4, 3) 5) 1--2--3 Mathieu Hill Hill ẍ + q(t)x = 0 (1 13) c 2010 6/(13)
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.701-710, 1920. 2) B. van der Pol and J. van der Mark, Frequency Demultiplication, Nature, vol.120, pp.363-364, 1927. 3) E.A. Jackson, Perspectives of Nonlinear Dynamics, Cambridge University Press, 1991.,, 1994. 4) C. Hayashi, Nonlinear Oscillations in Physical Systems, McGrawHill, New York, 1964. 5),, 2006. 6) W. Magnus and S. Winkler, Hill s Equation, Dover Publications, Inc., New York, 1979. c 2010 7/(13)
1 -- 11 -- 1 1--3 1--3--1 2009 3 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 1--3--2 2009 3 1-3-1 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 1--3--3 2009 3 ẋ = f(x) x x c 2010 8/(13)
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) 1--3--4 2009 3 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 2010 9/(13)
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) 1--3--5 2009 3 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),, http://cms.db.tokushima-u.ac.jp/dav/person/s10723/lecturenote/2007/nonlinearphenomena.pdf, 2005. 2) C. Mira, Chaotic Dynamics, World Scientific, Singapore, 1987. 3),, 2003. 4) H.K. Khalil, Nonlinear systems, Prentice Hall, 2001. 5) J.K. Hale and S.M.V. Lunel, Introduction to functional differential equations, Springer, 1993. 6) A.M. Fink, Almost Periodic Differential Equations, Springer-Verlag, 1974. c 2010 10/(13)
1 -- 11 -- 1 1--4 1--4--1 2009 2 1629-1695 1) 2) 1 1 3) 1) Q 1--4--2 2009 2 2) Q 4) 1 4 1 4 A B c 2010 11/(13)
1 4 1--4--3 2009 3 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 1--4--4 2009 3 c 2010 12/(13)
structural stability 5) A 1--4--5 2009 3 region of initial conditions basin of attraction 6) : basin boundary 1) SYNC:,, 2005. 2),, 1987 3) R.C. Mackey, Injection locking of klystron oscillator, IRE Trans. Microwave Theory & Tech., vol.mtt-10, no.7, p.228, 1962. 4) J.M.T. Thompson and H.B. Stewart, Nonlinear dynamics and chaos, John Wiley and Sons, 1986. 5) C. Pugh and M.M. Peixoto, Structural Stability, Scholarpedia, vol.3, no.9, p.4008, 2008. 6) E. Ott, Basin of attraction, Scholarpedia, vol.1, no.8, p.1701, 2006. c 2010 13/(13)