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1 (Zin ARAI) Lorenz

2 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) z y x 1 Lorenz Lorenz 2

3 Hilbert 23 S. Smale Lorenz J. Guckenheimer Lorenz Lorenz Lorenz W. Tucker 1 (Tucker [21]). (σ, ρ, β) =(10, 28, 8/3) Lorenz robust strange attractor Lorenz Lorenz k Σ k := {1, 2,...,k} i=0 k 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. f : X X N X Inv(N,f) :={x N n Z f n (x) N} 2 (Mishaikow-Mrozek[15, 16, 17]). Lorenz (σ, ρ, β) (10, 28, 8/3) I {z =27} P well-defined π :Inv(I,P) Σ 6 π P = s π 3

4 Σ A π(inv(i,p)) A A = (Galias-Zgliczyński[12]). 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 Tucker Mischaikow-Mrozek Galias-Zgliczyński 0 Mischaikow-Mrozek Galias-Zgliczyński Tucker Normal Form 1. Tucker Normal Form Poincaré 2. Poincaré 2 4

5 2.2 Tucker Euler Galias-Zgliczyński 4 Taylor Mischaikow-Mrozek 4 Runge-Kutta 4 Runge-Kutta Galias-Zgliczyński 4 Runge-Kutta 4 Taylor Taylor 2.3 F I := {I =[a, b] R : a, b F} I 4. I,J I {x y x I, y J} K 5

6 K I (+,,,/) CPU IEEE754 CPU CPU f : R 2 R 2 X, Y I X Y R 2 f(x Y ) f(x Y ) int(x Y ) X,Y I ( ) Y Y X X wrapping effect 2.4 Wrapping Effect 2 X X 6

7 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 7

8 2 (x, y) (x + y, x y) X Lohner [23] 3 X R n X n Ω n d i (i =1...n) { n } Ω:= [k i d i, (k i +1)d i ]:k i Z i=1 B Ω B B f ω Ω f( ω ) ω Ω f( ω ) int F (ω) F (ω) X 3 6 CAPD f( ω ) Ω f( ω ) Ω F(ω) 4 F :Ω Ω F(ω) ={ω Ω: f( ω ) ω } f( ω ) int F(ω) G Ω ω F(ω) ω ω 4 f : B B f G 5. f B k G k k k 4.7 8

9 3 : ω f( ω ) : f( ω ) F (ω) 4 : F (ω) Ω F(ω) : G ω 6. x y n ε- {x = x 0,...,x n = y} 1 j n d(f(x j 1 ),x j ) <ε f R(f) :={x X ε >0 x x ε- } d X [19] G Inv G := {ω G bi-infinitely long path through ω} Scc G := {ω G path from ω to itself } G G G 9

10 G := ω G ω 7. R(f) Scc G, Inv(X, f) Inv G. R(f) Inv(X, f) M. Dellnitz O. Junge GAIO [8, 9] Henri Poincaré [18] [13] 4.2 X f : X X 10

11 4.2.1 Lefschetz-Hopf Lefschetz-Hopf. f : P P P f : H (P ) H (P ) Lefschetz 0 f 1 Lefschetz 8. E = {E n } n Z L = {L n } n Z Lefschetz λ(l) := n Z( 1) n tr(l n ) tr(l n ) L n : E n E n 9. C = {C n } n Z L H (L) : H (C) H (C) λ(l) =λ(h (L)) L H(L) Lefschetz-Hopf P δ>0 x P f δ P δ/2 P P g : P P f : P P α : C(P ) C(P ) α : H (P ) H (P ) C(P ) g α C(P ) α g C(P ) 11

12 f =(g α) λ((g α) ) 0 9 g α : C(P ) C(P ) Lefschetz 0 P g α δ Hopf Lefschetz P P f {(x, f(x)) x P } {(x, x) x P } Lefschetz Hopf g α f f f f f f 2 f f f f g f = g g f : X X g : Y Y h : X Y g h = h f f g h f g f g h 2 g f 1 f g 12

13 11. X f : X X h top (f) [0, ] (1) f h top (f) <. (2) k 1 h top (f k ) = k h top (f) f h top (f 1 ) = h top (f) (3) f g h top (f) =h top (g). (4) f g h top (f) h top (g). 12 (Yomdin [22]). M C f : M M h top (f) log s(f ). s(f ) f : H (M,R) H (M,R) f : R n R n R n log s(f )=0 M. Shub 1970 C

14 0 1 f : X X X f f [19, 10.1] f : X X R(f) X R(f) 13. S X S N S N S =Inv(N,f) int N 14

15 int N N N S N Inv(N,f) N int N Inv(N,f) N f f g N Inv(N,f) Inv(N,g) N 14. S index pair P 0 P 1 P =(P 1,P 0 ) (1) P 1 \ P 0 S (2) f(p 0 ) P 1 P 0 (3) f(p 1 \ P 0 ) P 1 P 1 /P 0 P 1 P 0 P 0 [P 0 ] P 1 /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 f P S index pair 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 ]) 15

16 H (P 1 /P 0, [P 0 ]) (P 1 /P 0, [P 0 ]) H (P 1 /P 0, [P 0 ]) = H (P 1,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 ]) H (P 1 /P 0, [P 0 ]) f P k S index pair H (P 1 /P 0, [P 0 ]) f P 15. f : X X g : Y Y m 1 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 Q S 2 index pair f P f Q 16. S P S index pair f P 17 (Ważewski principle [13, 18]). P =(P 1,P 0 ) S index pair f P 0:{0} {0} S S 0 S Lefschetz index pair f P [13] 18. P =(P 1,P 0 ) index pair λ(f P ) 0 S := Inv(P 1 \ P 0 ) λ(fp n ) 0 S f n 16

17 I Ω I I 2 2. I B Ω o(b) :={ω Ω ω B } d(b) :=o(b) \B Inv(B, F) :={ω B γ : Z Bs.t. γ(0) = ω, γ(k +1) F(γ(k)) for all k Z} 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 f 1 I o(inv(i, F)) I I [13] 3. I index pair I f B =Inv(I, F) (P 1, P 0 )= ( (d(b) F(B)) B, d(b) F(B) ) 17

18 P =( P 1, P 0 ) Inv( I,f) index pair [13] 4. CHomP ( f P f P : H ( P 1 / P 0, [ P 0 ]) H ( P 1 / P 0, [ P 0 ]) f F F CHomP 19. CHomP example repeller 1 f : R R repeller.map f 1 cnvmap (1) -> {(0)} (2) -> {(0) (1) (2)} (3) -> {(2) (3)} (4) -> {(3) (4) (5)} (5) -> {(5) (6)} (6) -> {(6) (7) (8)} (7) -> {(8)} (1) (0) (2) (0), (1), (2) index pair indxpair repeller.cub repeller.mp repeller.q1 repeller.q0 repeller.q1 Q 1 = {(2), (3), (4), (5), (6)} 5 repeller.q0 Q 0 = {(1), (7)} 2 index pair P =(Q 0 Q 1,Q 0 ) 5 18

19 5 repeller.q0, repeller.q1 homcubes -i repeller.map repeller.q1 repeller.q0 H_0 = 0 H_1 = Z... The composition of F and the inverse of the map induced by the inclusion: Dim 0: 0 Dim 1: F (x1) = x1 { Z (k =1) H k (Q 0 Q 1,Q 0 )= 0 (k 1) H 1 (Q 0 Q 1,Q 0 ) x1 f P (x1) =x1 { id (k =1) f P = 0 (k 1) Lefschetz 5 k- k- 1 19

20 y x 6 7 index pair 20. H a,b : R 2 R 2 : ( ) x y ( ) a x 2 + by x 7 a =1.4, b = index pair P 1 \ P 0 P 0 CHomP λ(fp 7 )=7 18 Inv(P 1 \ P 0 ) f 7 P 1 f Inv(P 1 \ P 0 ) Hartman-Grobman [4]

21 Day-Frongillo-Trevino [7] V G X G := {(v i ) i Z v i V i v i v i+1 } X G X G σ G : X G X G σ G ((v i ) i Z )=(v i+1 ) i Z σ G h top (σ G )=logs(t G ) T G G s(t G ) 22. N X N 1,...,N k G =({1,...,k},E) G γ = a 1 a m f γ := f f Na m N a1 Inv(N a1,f γ ) index pair P γ λ(f Pγ ) 0 G G ρ :Inv(N) X G (ρ(x)) i = j f i (x) N j ρ 11 (4) h top (f) log s(t G ) 21

22 f 15 ρ σ G ρ = ρ f ρ X G P G P X G Lefschetz P ρ f 18 P im ρ im ρ X G im ρ = cl(im ρ) cl(p )=X G. 22 Katok [14] [1] 23. (a, b) 7 R(H a,b ) 24. R- [19] R(H a,b ) R(H a,b )= [10] Davis-MacKay- [6] 22

23 b a 7 [1] 5.1 M f M Λ f T Λ TM Λ 25. f Λ Λ T Λ Tf- T Λ=E s E u c>0 0 <λ<1 Tf n E s <cλ n and Tf n E u <cλ n n 0 M 2 c λ cone fields 23

24 M T Λ c =1 f Tf Tf : TM TM Λ f- T Λ Tf- Tf : T Λ T Λ Tf TM f Λ Tf : T Λ T Λ f Λ R(f Λ )=Λ 27 ([5, 20]). f Λ f Λ f Λ N f [18] Inv f N := {x N f n (x) N for all n Z} N int N f S N Inv f N = S Tf : T Λ T Λ 1 Tf T Λ 0- T Λ 0- Tf : T Λ T Λ 0- N 1 Inv Tf N Tf : T Λ T Λ N T Λ T Λ 0- Λ 24

25 Choose K such that R(f) K holds, and put N := K D where D is the unit square in the tangent space. 2. Compute Scc G(K). 3. Replace K with Scc G(K). 4. Replace N with N (K D). 5. Compute Inv G(N). 6. if Inv G(N) int N 0 then the algorithm stops. else subdivide K, N and goto 2 29., R(f) 27 R(f) Devaney-Nitecki [10] R(a, b) := 1 2 (1 + b + (1 + b ) 2 +4a), S(a, b) :={(x, y) R 2 : x R(a, b), y R(a, b)}. 30. R(H a,b ) S(a, b) H a,b R(H a,b ) 23 [1] Algorithm 15 25

26 8 [11] 5.3 Frongillo[11] R [2] keading theory pruning front theory [3] 6 Software Packages 26

27 6.1 GAIO (Global Analysis of Invariant Objects) M. Dellnitz and O. Junge Python MATLAB MATLAB GAIO 3 GAIO Boost CAPD C/C++ MATLAB 6.2 Boost interval arithmetic library Boost C++ Boost interval arithmetic library C++ double GMP 6.3 CAPD (Computer Assisted Proofs in Dynamics) [12] Z. Galias P. Zgliczyński CHomP P. Pilarczyk C Lohner 27

28 6.4 INTLAB (INTerval LABoratory) Siegfried M. Rump Matlab Matlab C CHomP (Computational HOMology Project) P. Pilarczyk [1] Z. Arai, On Hyperbolic Plateaus of the Hénon Maps, Experimental Mathematics, 16:2 (2007), [2] Z. Arai, On Loops in the Hyperbolic Locus of the Complex Hénon Map and Their Monodromies, preprint. [3] Z. Arai, Monodromy and the Pruning Front, in preparation. [4] Z. Arai and K. Mischaikow, Rigorous computations of homoclinic tangencies, SIAM Journal on Applied Dynamical Systems 5 (2006), [5] R. C. Churchill, J. Franke and J. Selgrade, A geometric criterion for hyperbolicity of flows, Proc. Amer. Math. Soc., 62 (1977), [6] M. J. Davis, R. S. MacKay and A. Sannami, Markov shifts in the Hénon family, Physica D, 52 (1991), [7] S. Day, R. Frongillo and R. Trevino, Algorithms for rigorous entropy bounds and symbolic dynamics, preprint,

29 [8] M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, Handbook of dynamical systems II, North-Holland, 2002, [9] M. Dellnitz and O. Junge, The web page of GAIO project, [10] R. Devaney and Z. Nitecki, Shift automorphisms in the Hénon mapping, Commun. Math. Phys., 67 (1979), [11] R. M. Frongillo, Topological Entropy Bounds for Hyperbolic Dynamical Systems, [12] Z. Galias and P. Zgliczyński, Computer assisted proof of chaos in the Lorenz equations, Physica D, 115 (1998), [13] T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology, Applied Mathematical Sciences, 157, Springer-Verlag, [14] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980), [15] K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: a computerassisted proof, Bull.Amer.Math.Soc.(N.S.), 3 (1995), [16] K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: a computerassisted proof. II. Details, Mathematics of Computation, 67 (1998), [17] K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: a computerassisted proof. III. Classical parameter vallues, J. Differential Equations, 169 (2001), [18] K. Mischaikow and M. Mrozek, The Conley index theory, Handbook of Dynamical Systems II, North-Holland, 2002, [19] C. Robinson, Dynamical systems; stability, symbolic dynamics, and chaos, 2nd ed., CRC Press, Boca Raton, FL, [20] R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splitting for linear differential systems I, J. Differential Equations, 27 (1974) [21] W. Tucker, A rigorous ODE solver and Smale s 14th problem, Found. Comput. Math., 2 (2002), [22] Y. Yomdin, Volume growth and entropy, Israel J. Math. 57 (1987), [23] P. Zgliczyński, C 1 Lohner algorithm, Fuound. Comput. Math., 2 (2002),

kokyuroku.dvi

kokyuroku.dvi On Applications of Rigorous Computing to Dynamical Systems (Zin ARAI) Department of Mathematics, Kyoto University email: arai@math.kyoto-u.ac.jp 1 [12, 13] Lorenz 2 Lorenz 3 4 2 Lorenz 2.1 Lorenz E. Lorenz