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Transcription:

C 2. / 24 8 5-6 9-2

i.................................................................................2.................................. 3..3................................. 5..4..................................... 6..5........................................ 7..6.................................. 7.2......................................... 9.2.................................... 9.2.2 DYNASCAL....................................... 2 2 22 2............................................ 22 2......................................... 22 2..2.................................. 23 2..3............................. 24 2..4................................. 26 2..5..................... 26 2..6....................................... 27 2.2............................................ 29 2.2.................. 29 2.2.2................................... 29 2.2.3......................... 29 A 3

... (differential equation (solution curve dx = ax, (. dt (a solution k x = f(t =e ta k, (.2 (. k (. (an initial value problem x = ax, x( = k, (.3 (. (a unique solution. a =. (.2 k 2,,, 2 Maple 9 (uniqueness of solutions t (local existence t (existence in the large Hartman (982 Hartman [ ] y = y 2, y( =. (.4 c t c = t c =(t c 2 /4 [.] (. (.2

2 4 2 - -5 t 5-2 -4.: The simplest differential equations with initial values (.3 (.2. a> k > x k < x 2. a = x = k(constant 3. a< x Fig. a =. k =2,, 2 (. (linear homogeneous first-order differential equations dx/dt = A(t x (homogeneous equation (. x (nonhomogeneous equation (. x B(t dx/dt = A(t x + B(t (higher-order differential equations d 2 x dt 2 + a dx + b(t = dt

3..2 (state space (a system of linear first-order differential equations dx dt = Ax,,x( = k Rn. (.5 (.5 x = f(t =e ta k = e ta x. (.6 e ta = k= k! Ak (Hirsch & Smale, 974, p.86. dx dt = g(x, x R n g(x = x = x (an equilibrium point x = φ t (x t φ t (x =x. (.7 x (a stationary point (a fixed point (a zero point (a singular point, or singularity 2 - -.5.5 - -2.2: The vector field and some solution curves near a saddle with λ < <λ 2 (.5 A dx/dt = Ax x = Py P (.5 dy dt = By, B = P AP, y( = P x(. (.8

4 P x( = k [.2] dx/dt = Ax (.8 P A A A (real canonical form.3.2. -2 - -. 2 -.2 -.3.3: The vector field and some solution curves near a saddle with λ 2 < <λ.5 - -.5.5 -.5 -.4: The vector field and some solution curves near an inward node with λ <λ 2 <

5 n A J O J 2... P AP = F F 2. O.., (.9 J J 2 (elementary Jordan matrices λ (elementary λ blocks λ O... J l =, l =, 2,...,p. (....... O λ F F 2 D O. I.. 2 F m =...... O I 2 D, D, m =, 2,...,q. (. ( D = a b b a (, I 2 = (.2 D λ = a ± bi.5 - -.5.5 -.5 -.5: The vector field and some solution curves near an inward node with λ 2 <λ < A A

6 (a ( (b ( ( λ λ 2 λ λ λ λ,,, (c ( a b b a, b > ( a b b a, b > 2 -.4 -.2.2.4 - -2.6: The vector field and some solution curves near an outward node with <λ <λ 2 (.8 =e tb k, (.3 ( λ Case (a B =, λ 2 (.8 ( λ y = λ 2 y 2 ( dy dt dy 2 dt, (.4

7 (.2 ( ( y (t e λt k =. (.5 y 2 (t e λ2t k 2 (.5 y 2 (t y (t = e(λ2 λ t k 2. (.6 k (.5 (.6 λ λ 2 λ λ 2 (saddle (.2.3 (inward node.4.5 (outward node.6.7 λ = λ 2 (.6 y 2 (t y (t = k 2 k, (.7 t y (t y 2 (t k k 2.8.9.4.2-2 - 2 -.2 -.4.7: The vector field and some solution curves near an outward node with <λ 2 <λ ( ( λ λ Case (b B =, B =, λ λ ( λ B =, =e tb k λ ( ( y (t = y 2 (t e λt k e λt tk + e λt k 2. (.8 (.8

8.5 - -.5.5 -.5 -.8: The vector field and some solution curves near an inward focus with λ = λ 2 <.4.2 -.4 -.2.2.4 -.2 -.4.9: The vector field and some solution curves near an outward focus with λ = λ 2 > ( tλ Case (b B tb = t tλ ( tb =(tλi + N, N =. t. e tb = e tλi e N. e X = r= X r r!,

9 e tλi = e tλ I N 2 = N 3 =...= O e N = I + N ( e tb = e tλ I (I + N =e tλ (I + N =e tλ. t ( e tb = e tλ te tλ e tλ..5 - -.5.5 -.5 -.: The vector field and some solution curves near an inward improper node with eqs.(.2,λ <.5 - -.5.5 -.5 -.: The vector field and some solution curves near an inward improper node with eqs.(.8,λ < (.8 y 2 (t y (t = t + k 2 k. (.9

( λ B = λ ( λ λ ( e tb = e tλ te tλ e tλ. (.2 ( y (t y 2 (t ( e λt k + e λt tk 2 =. (.2 e λt k 2 y (t y 2 (t = t + k k 2. (.22 (.9 (.22 t y (t y 2 (t, y 2 (t. (.23 y (t.. λ >.2.3..5 -. -.5.5. -.5 -..2: The vector field and some solution curves near an outward improper node with eqs.(.2,λ > ( ( a b a b Case (c B =, B =, b > b a b a ( a b B =, b a ( ta tb tb = = tai + tbl. (.24 tb ta L = (. (.25

(.24 e tb = e tai e tbl = e ta e tbl. (.26 e tbl L ( e tbl cos tb sin tb =. (.27 sin tb cos tb e tbl = I + tb! L + (tb2 L 2 +, 2! L 2 = I, L 3 = L, L 4 = I, L 5 = L, (.28 L cos(tb sin(tb.5 - -.5.5 -.5 -.3: The vector field and some solution curves near an outward improper node with eqs.(.8,λ > (.26 (.27 e tb = e ta ( cos tb sin tb sin tb cos tb. (.29 ( y (t y 2 (t ( e ta (k cos tb k 2 sin tb = e ta (k sin tb + k 2 cos tb. (.3 y 2 (t y (t = k sin tb + k 2 cos tb k cos tb k 2 sin tb. (.3 (.3 cos tb sin tb a < lim t =

2 y ( a b = b a ( dy dt dy 2 dt ( y ( ay = by (.32 y 2 dy 2 dt = by. (.33 b > y > dy 2 /dt > y < dy 2 /dt < spiral.4 b < spiral.5 y 2 = dy 2 /dt = by a > lim t =.6.7.8.4 -.8 -.6 -.4 -.2.2.4 -.4.4: The vector field and some solution curves near a (counterclockwise spiral sink.8.4 -.4 -.2.2.4.6.8 -.4.5: The vector field and some solution curves near a (clockwise spiral sink

3 ( ( b b Case (c B =, B =, b > b b Case (c a = (.3.8.4 -.4 -.2.2.4.6.8 -.4.6: The vector field and some solution curves near a (counterclockwise spiral source.8.4 -.8 -.6 -.4 -.2.2.4 -.4.7: The vector field and some solution curves near a (clockwise spiral source ( y (t y 2 (t ( = (k cos tb k 2 sin tb (k sin tb + k 2 cos tb. (.34 (.34 y 2(t+y2 2 (t y 2 (t+y2 2 (t =k2 + k2 2 = r2 (k,k 2. (.35 (k,k 2 Case (b.8.9

4 A (negative real parts sink inward node inward improper node inward focus spiral sink sink A (positive real parts source outward node outward improper node outward focus spiral source source.5.5 -.5 - -.5.5.5 -.5 - -.5.8: The vector field and some solution curves near a (counterclockwise center.5.5 -.5 - -.5.5 -.5.5 - -.5.9: The vector field and some solution curves near a (clockwise center 99, Chino, 987, 99.2.3 saddle

5.4.5 spiral sink.8 focus.6.7.9 focus focus.. improper node.2.3 improper node.4.5 spiral sinks.6.7 spiral source.8.9 centers center..3..2 (.5 A (a system of nonlinear first-order differential equations (linearize Guckenheimer & Holmes, 983, pp.2-3 dx dt = f(x, x Rn, x( = k. (.36 (.36 x f(x = (.36 f(x x (the Jacobian matrix Jx dx dt = J x x, (.37 Jx { } = fi = x j x=x f f x f 2 f 2 x x 2 f x n f 2 x n x 2... f n x f n x 2 f n x n. (.38 x=x (non-zero real part

6 center.8.9..4 (limit cycles spiral sink spiral source (closed orbit (α limit cycle (ω limit cycle spiral sink spiral source.2.5.5 -.5.5 -.5 -.2: An ω-limit cycle Hirsch & Smale (974, pp.227-228 µ =.5 Maple ( ( µx x 3 + y =. (.39 x source µ

7 8 Heath, 2 (99 Chino (987, 99.2 spiral sink spiral source..5 (Hartman & Grobman s theorem (non-zero real part (structurally stable (structurally unstable.8.9 center..6 (bifurcation (bifurcation parameters c dx dt = f(x, c. (.4 (.39 c = µ (local bifurcation Guckenheimer & Holmes, 983 (saddle-node bifurcation (pitchfork bifurcation

8.4.2 -.2 -...2.3.4.5 -.2.2: A spiral sink before an ω-limit cycle appears (transcritical bifurcation (Hopf bifurcation saddle node spiral sink spiral source center center spiral sink spiral source spiral source spiral sink.2.2 spiral sink Maple (.39 µ =.5.2 µ =.5 µ =. (saddle loop (saddle connection spiral sink saddle spiral sink center center source spiral sink saddle saddles saddle saddle saddles (99 Chino (987, 99

9.2.2 spiral sink sink source spiral source spiral sink.2 PsycINFO dynamical system Tobler (976-977 (the wind model (flow (curl free part (divergence free part (gradient vector field (scalar potential 8 (bifurcation theory DYNASCAL (Dynamical System Scaling Chino & Nakagawa, 983, 99 DYNASCAL Yadohisa and Niki (999 DYNASCAL.2. (997 t jk = d jk r + c jk. (.4 t jk j k d jk r c jk

2 (.4 c jk = c kj t kj = d kj /(r + c kj c jk = r t kj t jk t jk + t kj. (.42 t jk (j,k (997.2.2 DYNASCAL DYNASCAL Guttman (968 SSA-II (nonautonomous system (autonomous system 4 2-8 -4 4 8-2.22: Illustration of the estimated vector field at a time by DYNASCAL.22 spiral sinks saddles DYNASCAL DYNASCAL (.4 x t dx dt = f(x,t, (.43

2 dx /dt f (x,x 2,x 3 dx 2 /dt = f 2 (x,x 2,x 3, (.44 dx 3 /dt DYNASCAL (.4 c DYNASCAL 23. 2. DYNASCAL 3. (limit cycles DYNASCAL Newcomb (96 Chino and Nakagawa (99 DYNASCAL...3 DYNASCAL. saddle.2 saddle 4 spiral source.3

22 2 2. 2.. (difference equation x(n +=a(n x(n, x(n =x, n n, (2. (a linear homogeneous first-order difference equation ( Elaydi (999 ( n x(n = a(i x, (2.2 i=n x(n +=a(n x(n+g(n, x(n =x, n n, (2.3 (a linear nonhomogeneous first-order difference equation ( n ( n n x(n = a(i x + a(i g(r, (2.4 i=n i=r+ k- Elaydi, 999, p.54 r=n y(n + k+p (n y(n + k + + p k (n y(n =g(n. (2.3 a(n =a g(n =b x(n +=ax(n+b, x(n =x, n n, (2.5 { ( a n a x + b n a, a x(n = x + bn, a = (2.6 (2.6 n. a> = x.

2 23 2. a = = b> x, b = x = x, b< x. 3. <a< = x b a. 4. a = = x x b x 5. a< = x ±. 2..2 k (a system of linear first-order difference equations x(n +=Ax(n, x(n =x. (2.7 A = a a 2 a k a 2 a 22 a 2k.... (2.8 a k a k2 a kk (real nonsingular matrix (2.7 x(n, n, x =A n n x. (2.9 y(n x =x(n y(n +=Ay(n y( = x(n (2.9 y(n =A n y(, (2. (2.7 A A n (the Putzer algorithm Elaydi, 999, p.6 (2.7 x(n =(x (n,x 2 (n,...,x k (n t k (states s,s 2,...,s k x(n =p(n =(p (n,p 2 (n,...,p k (n t A = P = {p ij = p(s i s j } p ij s j s i (the Markov chain p(n +=Sp(n, (2. S = {p ij }, (2.2 (the transition matrix A = {a ij } A a ij k k A j =, 2,...,k k i= a ij = (Markov (stochastic λ λ Elaydi, 999, p.42 λ = λ ρ(a ρ(a k k (the spectral radius ρ(a =max i k λ i (the Perron s theorem (Elaydi, 999, p.42

2 24 2..3 x(n +=f(x(n, x( = x, (2.3 x f (the fixed point f(x =x, (2.4 (2.3 (the equilibrium point (2.4 (.7 x(n x(n + x(n (2.4 x(n + x(n = (a x ɛ > x x <δ(δ > n> f n (x x <ɛ δ (stable (b x x x <η(η > lim n x n = x η (attracting x η = (a global attractor (globally attracting (c x (asymptotically stable equilibrium η = (globally asymptotically stable x x x δ> n> x n x ɛ> x n x x x x η> x n n x x x x n n x x x x η> n> x n x ɛ> n x Elaydi, 999 x x(n +=f(x(n, (2.5 f x (i f (x < x

2 25 (ii f (x > x x f (x (hyperbolic (2.5 x f (x = (i f (x x (ii f (x = f (x > x (iii f (x = f (x < x (2.5 x f (x = (i Sf(x < x (ii Sf(x > x Sf(x (the Schwarzian derivative of a function f Sf(x = f (x f (x 3 ( f 2 (x 2 f. (2.6 (x k k (a periodic point Elaydi, 999 b f (i b k f k (b =b f (2.5 x(n +=g(x(n, g = f k, (2.7 f k k- (k-periodic b O(b = { b, f(b,...,f k (b } k- (k-cycle (ii b m f m (b k- k- (eventually k-periodic (995 k- b (988, p.54 k- b f k- b (i f k (ii f k (iii f k

2 26 b k- { x( = b, x( = f(b,...,x(k = f k (b } k- O(b ={b = x(,x(,...,x(k } f k- k- (i f (x(f (x( f (x(k < (ii f (x(f (x( f (x(k > 2..4 dx/dt = f(x,t,b dx/dt = f(x, b b x(n +=f(x(n,n,b b Peitgen and Richter (986 (Verhulst dynamics x( n x(n R R = r( x(n x(n + = ( + Rx(n x(n + = ( + rx(n rx 2 (n. r r r r n r (chaos 2..5 x(n +=f(n, x(n, x(n =x, (2.8 x(n R k f : z + R k R k f(n, x(n x (2.8 n (nonaoutonomous (2.8 n x(n +=f(x(n (autonomous (2.8 n Z N f(n + N,x =f(n, x, (2.9

2 27 (periodic R k x f(n, x =x n n (2.8 (an equilibrium point x y(n =x(n x (2.8 y(n +=f(n, y(n+x x = g(n, y(n, y = x = x (2.8 Elaydi (999, p.57 (2.8 x (i ɛ > n n n x x < δ x(n, n, x x <ɛ δ = δ(ɛ, n (stable δ n (uniformly stable (unstable (ii x x <µ lim n x(n, n, x =x µ = µ(n (attractive µ n (uniformly attractive ɛ n x x <µ n n + N x(n, n, x x <ɛ n N = N(ɛ µ> (iii x (asymptotically stable ( (uniformly asymptotically stable (iv x x <δ x(n, n, x x M x x η n n δ> M> η (, (exponentially stable (v M n n x(n, n, x M x(n, n, x (bounded (ii (iii µ = (iv δ = (global 2..6 2..3 k- 2 x x(n +=Ax(n, x(n =x. (2.2 A

2 28 Ax = x (A Ix = (2.2 A I x = A I = c x = c y = x(n x (2.2 y(n +=Ay(n x x = (2.2 A x(n =Py(n P (2.2 y(n +=By(n, B = P AP, y( = P x(. (2.2 y( = k =(k,k 2 t (..2 A B (..2 (a (b (c ( λ Case (a B =, λ 2 (2.2 ( ( y (n + λ y (n =, (2.22 y 2 (n + λ 2 y 2 (n (2.6 b = ( ( y (n λ n = k y 2 (n λ n 2 k. (2.23 2 (2.23 ( n y 2 (n y (n = λ2 k 2. (2.24 λ k (2.23 (2.24 λ λ 2 λ λ 2 (saddle (asymptotically stable node (unstable node

2 29 2.2 (the Julia set (the filled Julia set (the Fatou set 2.2. f(z p 2 f(z =a n z p + a p z p + + a z + a, a n, (2.25. K f z z C 2. J K (the boundary K 3. F f n : S S n f n (the domain of normality Maple 2.2.2 (2.25 p =2 z(n +=z 2 (n+c, (2.26 c (the Mandelbrot set 2.2.3 Chino (22, 23 MDS HFM

2 3 References Chino, N. (987. A bifurcation model of changes in interdependence structure among objects. Bulletin of The Faculty of Humanities of Aichigakuin University, 7, 85-9. (99. II (pp.385-43 Chino, N. (22. Complex space models for the analysis of asymmetry. In S. Nishisato, Y. Baba, H. Bozdogan, & K. Kanefuji (Eds. Measurement and Multivariate Analysis, Tokyo: Springer. pp.7-4. Chino, N. (23. Complex difference system models for the analysis of asymetry. In H. Yanai, A. Okada, K. Shigemasu, Y. Kano, & J. J. Meulman (Eds. New Developments in Psychometrics, Tokyo: Springer. pp.479-486. (23. Chino, N., & Nakagawa, M. (983. A vector field model for sociometric data. Proceedings of the th annual meeting of the Behaviormetric Society of Japan (pp.9-, Kyoto, September. Chino, N., & Nakagawa, M. (99. A bifurcation model of change in group structure. The Japanese Journal of Experimental Social Psychology, 29, No.3, 25-38. Chino, N., & Shiraiwa, K. (993. Geometrical structures of some non-distance models for asymmetric MDS. Behaviormetrika, 2, 35-47. Elaydi, S. N. (999. An introduction to differential equations. 2nd Ed., New York: Springer-Verlag. Guckenheimer, J. & Holmes, P. (983. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Berlin: Springer-Verlag. Guttman, L. (968. A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika, 33, 469-56. Hartman, P. (982. Ordinary Differential Equations, Second Edition, Boston: Birkhäuser. Hirsch, M. W., & Smale, S. (974. Differential equations, dynamical systems, and linear algebra. New York: Academic Press. Milnor, J. (2. Dynamics in One Complex Variable. 2nd edition. Wiesbaden:Vieweg & Sohn. Newcomb, T. M. (96. The acquaintance process. New York: Holt. Rinehart and Winston. Peitgen, H.-O., & Richter, P. H. (986. The Beauty of Fractals. New York: Springer-Verlag. Tobler, W. (976-77. Spatial interaction patterns. Journal of Environmental Systems, 6, 27-3. (995. Yadohisa, H., & Niki, N. (999. Vector field representation of asymmetric proximity data. Communications in Statistics, 28, 35-48.

3 A. g(t d dt (g(t e at =g (t e at + g(t( ae at =ag(t e at ag(t e at =. (A. g(t e at k g(t =e at k.2 dx/dt = Ax x = Py P dx/dt = d(py/dt = P dy/dt dx/dt = P dy/dt = A(Py dy/dt = P AP y

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