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1 On Applications of Rigorous Computing to Dynamical Systems (Zin ARAI) Department of Mathematics, Kyoto University 1 [12, 13] Lorenz 2 Lorenz Lorenz 2.1 Lorenz E. Lorenz R 3 ẋ = σx + σy ẏ = ρx y xz ż = βz + xy Lorenz σ, ρ, β Lorenz (σ, ρ, β) =(10, 28, 8/3)

2 z y x 1 Lorenz Lorenz Hilbert 23 S. Smale Lorenz 1990 W. Tucker k Σ k := i=0 {1, 2,...,k} k full-shift s :Σ k Σ k s(x 0,x 1,...)=(x 1,x 2,...) k k A =(a ij ) Σ A := {(s n ) Σ k a sn s n+1 0} s(σ A )=Σ A. 1 (Mishaikow-Mrozek[8, 9, 10]). Lorenz (σ, ρ, β) (10, 28, 8/3) I {z =27} P well-defined π :Inv(I,P) Σ 6 π P = s π 2

3 Σ A π(inv(i,p)) A A = (Galias-Zgliczyński[6]). Lorenz (σ, ρ, β) (10, 28, 8/3) I {z =27} P well-defined π :Inv(I,P 2 ) Σ 2 π P 2 = s π π Lorenz 3 (Tucker[14]). (σ, ρ, β) =(10, 28, 8/3) Lorenz robust starange attractor robust strange attractor [4] Tucker Mischaikow-Mrozek Galias-Zgliczyński 0 Mischaikow-Mrozek Galias-Zgliczyński Tucker Normal Form 1. Tucker Normal Form Poincaré 2. Poincaré 3

4 Tucker Euler Galias-Zgliczyński 4 Taylor Mischaikow-Mrozek 4 Runge-Kutta 4 Runge-Kutta Galias-Zgliczyński 4 Runge-Kutta 4 Taylor Taylor wrapping effect 2.3 Wrapping Effect 2 X X 4

5 X X 2 wrapping effect X wrapping effect Lorenz wrapping effect Tucker X wrapping effect Galias-Zgliczyński wrapping effect. x ɛ B(x, ɛ) h x h P P logarithmic norm P P B(x, ɛ) P Mischaikow-Mrozek wrapping effect {z =27} {z =27} wrapping effect wrapping effect 2 (x, y) (x + y, x y) X Lohner [16] 5

6 3 Results for Discrete Dynamical Systems [11] 3.1 X f : X X X f f(s) =S S X f int N N 4. S N S N, S =Inv(N,f) :={x N {x i } i Z N s.t. x 0 = x, f(x i )=x i+1 for all i Z} S =Inv(N,f) int N N S N f f C 0 g N Inv(N,f) Inv(N,g) N 6

7 5. S index pair P =(P 1,P 0 ) P 1 \P 0 S f(p 0 ) P 1 P 0 f(p 1 \P 0 ) P 1 3 P 1 /P 0 P 1 P 0 P 0 [P 0 ] f P : P 1 /P 0 P 1 /P 0 { [f(x)] f(x) P 1 f P ([x]) := [P 0 ] f P index map index map S f [7] P 1 /P 0 H (P 1 /P 0, [P 0 ]) f P f P : H (P 1 /P 0, [P 0 ]) H (P 1 /P 0, [P 0 ]) H (P 1 /P 0, [P 0 ]) (P 1 /P 0, [P 0 ]) H (P 1 /P 0, [P 0 ]) H k (P 1 /P 0, [P 0 ]) f P 0 k f P k : H k (P 1 /P 0, [P 0 ]) H k (P 1 /P 0, [P 0 ]) S index pair H (P 1 /P 0, [P 0 ]) f P 6. f : X X g : Y Y m r : X Y, s : Y X r f = g r, s g = f s, r s = g m,s r = f m S index pair P = (P 1,P 0 ) Q =(Q 1,Q 0 ) S index pair f P f Q 7

8 7. S P =(P 1,P 0 ) S index pair f P 8(Ważewski principle [7, 11]). P =(P 1,P 0 ) S index pair f P 0:{0} {0} S 9 (Index pair Lefschetz [7]). P = (P 1,P 0 ) S index pair L(f P ):= k ( 1)k tr f P k 0 S k ( 1)k tr f n P k 0 S f n [7, 15] connecting orbit [3] index pair index map 3.2 X = R n R n R n n n d i (i =1...n) { n } Ω:= [k i d i, (k i +1)d i ]:k i Z i=1 R n Ω B Ω B B R n f f ω Ω f( ω ) f(ω) f( ω ) Ω f( ω ) Ω F(ω) F :Ω {Ω } : ω {ω Ω: f( ω ) ω } f( ω ) int F(ω) f( ω ) F(ω) 8

9 I Ω I I index pair 1. I 2 2. I Inv( I,f) int I I B Ω o(b) :={ω Ω: ω B }, d(b) :=o(b) \B o(b) Ω B Inv(B, F) {ω B γ : Z B γ(0) = ω γ(k +1) F(γ(k)) } f( ω ) int F(ω) Inv( I,f) Inv(I, F) o(inv(i, F)) I Inv( I,f) Inv(I, F) int o(inv(i, F)) int I I o(inv(i, F)) I I [7] I I 3. I f B =Inv(I, F) (P 1, P 0 )= ( (d(b) F(B)) B, d(b) F(B) ) P =( P 1, P 0 ) Inv( I,f) index pair [7] 4. [7] H ( P 1 / P 0, [ P 0 ]) f P f F 4 CHomP F 9

10 y x 3 7 index pair H a,b : R 2 R 2 :(x, y) (a x 2 + by, x) 9 Hénon Hénon Lorenz a =1.4, b =0.3 3 index pair P 1 \ P 0 P 0 CHomP f P 1 = : Z 7 Z f P 0 tr((f P 1 ) 7 )=7 9 Inv(P 1 \ P 0 ) f 7 P 1 f Inv(P 1 \ P 0 ) 7 Hénon [3] [2] [1] 10

11 4 Software Packages 4.1 GAIO (Global Analysis of Invariant Objects) M. Dellnitz and O. Junge Python MATLAB MATLAB GAIO 3 GAIO PROFIL C/C++ MATLAB 4.2 CHomP (Computational HOMology Project) Conley P. Pilarczyk 4.3 BIAS (Basic Interval Arithmetic Subroutines) 11

12 O. Knüppel C PROFIL BIASINTERVAL C PROFIL b4m BIAS BiasF.c BIAS sin, cos, exp BiasExp libm exp BIAS 4.4 b4m (BIAS for MATLAB) J. Zemke MATLAB BAIS x = interval(1,2) x [1, 2] MATLAB BIAS BIAS 4.5 PROFIL (Programmer s Runtime Optimized Fast Interval Library) BIAS O. Knüppel C++ BIAS BIAS 4.6 CAPD (Computer Assisted Proofs in Dynamics) 2 [6] Z. Galias P. Zgliczyński CHomP P. Pilarczyk 2 Lohner 12

13 5 [1],,, 15 (2005), [2] Z. Arai, On Hyperbolic Plateaus of the Hénon Maps, preprint. [3] Z. Arai and K. Mischaikow, Rigorous Computations of Homoclinic Tangencies, preprint. [4] C. Bonatti, L. Díaz and M. Viana, Dyamics Beyound Uniform Hyperbolicity, Encyclopaedia of Mathematical Sciences, 102, Springer-Verlag, [5] M. Dellnitz and O. Junge, The algorithms behind GAIO - set oriented numerical methods for dynamical systems, Ergodic theory, analysis, and efficient simulation of dynamical systems, Springer, Berlin, 2001, , [6] Z. Galias and P. Zgliczyński, Computer assisted proof of chaos in the Lorenz equations, Physica D, 115 (1998), [7] T. Kaczynski, K. Mischaikow andm. Mrozek, Computational Homology, Applied Mathematical Sciences, 157, Springer-Verlag, [8] K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: a computer-assisted proof, Bull.Amer.Math.Soc.(N.S.), 3 (1995), [9] K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: a computer-assisted proof. II. Details, Mathematics of Computation, 67 (1998), [10] K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: a computer-assisted proof. III. Classical parameter vallues, J. Differential Equations, 169 (2001), [11] K. Mischaikow and M. Mrozek, The Conley index theory, Handbook of Dynamical Systems II, North-Holland, 2002, [12],,, [13],,, [14] W. Tucker, A rigorous ODE solver and Smale s 14th problem, Found. Comput. Math., 2 (2002), [15] A. Szymczak, The Conley index and symbolic dynamics, Topology, 35 (1996), [16] P. Zgliczyński, C 1 Lohner algorithm, Fuound. Comput. Math., 2 (2002),

sakigake2.dvi

sakigake2.dvi (Zin ARAI) arai@cris.hokudai.ac.jp http://www.cris.hokudai.ac.jp/arai/ 1 2.1 Lorenz 3 4 5 6 1 2 2.1 Lorenz E. Lorenz R 3 ẋ = σx + σy ẏ = ρx y xz ż = βz + xy Lorenz σ, ρ, β Lorenz (σ, ρ, β) =(10, 28, 8/3)