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1 (Zin ARAI) 1 dynamical systems ( mechanics ) dynamical systems 3 G X Ψ:G X X, (g, x) Ψ(g, x) =:Ψ g (x) Ψ id (x) =x, Ψ gh (x) =Ψ h (Ψ g (x)) ( x X g, h G) G X Ψ x X G O(x) := Ψ(G, x) G Z f =Ψ 1 : X X O(x) ={f k (x) k Z} x k f k (x) 1

2 f k f f k = f f }{{} k R Ψ R Z R 2 L λ (x) =λx L λ : R R λ 0 L λ x λ X = R G = Z Ψ k = L k λ 1 x λ =1.01 x <0 x λ λ Ψ(k, x) =L k λ (x) =λk x k λ > 1 λ < 1 Z k 2

3 3 λ 0 λ =0 L λ k <0 Ψ k Z Z Z 0 := {k Z k 0} 4 L λ Q λ (x) =λx(1 x) x Q λ L λ (1 x) x =1 Q λ (1) = 0 Q λ R Q λ ([0, 1]) [0, 1] Q λ :[0, 1] [0, 1] [0, 1] L λ 2 Q λ Q k λ (x) 3

4 1 5 1 dx dt = λx λ R x λ <0 t =0 x = x 0 x(t) =x 0 e λt Ψ:R R R Ψ(t, x) :=xe λt 4

5 M C 1 G = R X = M Ψ:R M M C 1 M ξ dψ(t, x) ξ x := dt Ψ Z Z 0 G R flow flow G = Z flow Ψ : R X X t R f(x) =Ψ(t, x) f : X X f Ψ -t f flow t t=0 S P(x) x 2 flow Ψ p X T T Ψ T (p) =p T>0 p p S p x S S T x S f(x) 5

6 p S U f : U S f flow S 2 flow S X flow flow f : X X X [0, 1] X {1} X {0} (x, 1) (f(x), 0) X = X [0, 1]/ (x, s) (x, s), (x, 1) (y, 0) y = f(x) flow Ψ Ψ t (x, s) =(f t+s (x), t+ s t + s ) f suspension flow x := max{k Z k x} f suspension flow Ψ S = X {0} f flow flow flow 6 G = Z, Z 0 R 100 6

7 3 F. Diacu and P. Holmes, Celestial Encounters, Springer L λ y X x X ω- t k Ψ t k (x) y R Z y X x X α- t k Ψ t k (x) y ω- α- ω(x) = Ψ t (x), α(x) = Ψ t (x) ω-α- T 0 t T T 0 t T 7

8 7 R 2. X = R 2 S 2 C 1 x X ω(x) α(x) flow α ω limit cycle 3 3 Van der Pol R 2 3 genus 3 genus T 2 flow x T 2 O(x) x T 2 O(x) =T 2 8

9 8 Lorenz ẋ = σx + σy ẏ = ρx y xz ż = βz + xy R 3 σ, ρ, β Lorenz (σ, ρ, β) =(10, 28, 8/3) z y x 4 Lorenz

10 [1] X. Ψ:G X X C>0 x X U y U t G t x y C x X U : x y U t G such that d(ψ t (x), Ψ t (y)) >C d X 9 ( ( ) x a x H a,b : R 2 R 2 : y) 2 + by x M. Hénon 10

11 a =1.4, b =0.3 5 amazon [6] [7] 11

12 [1],, [2] B. Hasselblatt and A. Katok, A First Course in Dynamics, Cambridge University Press, [3] R. L. Devaney, An Introduction to Chaotic Dynamical System, 2nd ed., Perseus Books Publishing, ,, [4] M. W. Hirsch, S. Smale and R. L. Devaney, Differential Equations, Dynamical Systems and an Introduction to Chaos, 2nd ed., Academic Press, 2004.,, [5]J.PalisandW.deMelo,Geometric Theory of Dynamical Systems, Springer, [6] C. Robinson, Dynamical Systems, CRC Press, 1999.,, [7] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, [10] 600 [8] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical System, and Bifurcations of Vector Fields, Springer, [9] K. T. Alligood, T. D. Sauer and J. A. Yorke, Chaos. An Introduction to Dynamical Systems Springer, , 2, 3, [10] P. Cvitanović, et al., Chaos: classical and quantum, Handbooks Handbook 3 Volume 1A, 1B Z R 12

13 random dynamical systems Volume 2 [11] B. Hasselblatt and A. Katok (ed.), Handbook of Dynamical Systems, Volume 1A, Elsevier Science, [12] B. Hasselblatt and A. Katok (ed.), Handbook of Dynamical Systems, Volume 1B, Elsevier Science, [13] B. Fiedler (ed.), Handbook of Dynamical Systems, Volume 2, Elsevier Science, S. Smale Differential Dynamical Systems [14] [6] [15] [16] [17] [14] S. Smale, The Mathematics of Time: Essays on Dynamical Systems, Economic Processes, and Related Topics, Springer, [15] M. Shub, Global Stability of Dynamical Systems, Springer, [16] J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, [17] C. Bonatti, L. Diaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Springer, [18] [19] [18] V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, ,, [19],, [11] 13

14 Franks Misiurewicz 2 [20] C. Conley, Isolated Invariant Sets and the Morse Index, American Mathematical Society, [21] J. Franks, Homology and Dynamical Systems, American Mathematical Society, [22] smooth ergodic theory [7] [25] [22] V. I. Arnold and A. Avez, Probrèmes Ergodiques de la Mécanique Classique, Gauthier-Villars, 1967.,, [23] M. Pollicott and M. Yuri, Dynamical Systems and Ergodic Theory, Cambridge University Press, [24] P. Walters, An Introduction to Ergodic Theory, Springer, [25] M. Pollicott, Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds, Cambridge University Press, [26] [27] [28] [29] [30] [26] A. F. Beardon, Iteration of Rational Functions, Springer, [27] J. Milnor, Dynamics in One Complex Variable, 3rd ed., Princeton University Press, 2006 [28] S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge University Press,

15 [29],, [30] M. Braverman and M. Yampolsky, Computability of Julia Sets, Springer, [31, 32] [33] [31] D. Lind and B. Marcus, Symbolic Dynamics and Coding, Cambridge University Press, [32] B. Kitchens, Symbolic Dynamics, Springer, [33] H. Xie, Grammatical Complexity and One-dimensional Dynamical Systems, World Scientific,

kokyuroku.dvi

kokyuroku.dvi On Applications of Rigorous Computing to Dynamical Systems (Zin ARAI) Department of Mathematics, Kyoto University email: [email protected] 1 [12, 13] Lorenz 2 Lorenz 3 4 2 Lorenz 2.1 Lorenz E. Lorenz