Chaos induced by coupled-expanding maps

Transcription

1 First Prev Next Last Seminar on CCCN, Jan. 26, 2006 Chaos induced by coupled-expanding maps Page 1 of 35 Yuming Shi Department of Mathematics Shandong University, China ymshi@sdu.edu.cn Collaborators: G. Chen, P. Yu, and H. Ju

2 First Prev Next Last Outline 1. Introduction Page 2 of Concept of coupled-expanding map 3. Relationships with subshifts of finite type 4. Chaos induced by coupled-expanding maps 5. Some applications 6. Examples with simulations

3 First Prev Next Last 1. Introduction A general discrete system x n+1 = f(x n ), n 0, Page 3 of 35 where f : D X X is a map and (X, d) is a metric space. Orbit: x 0, x 1 = f(x 0 ), x 2 = f(x 1 ),... What is chaos? No general definition of chaos Li and Yorke [1975], Devaney [1987], Wiggins [1990]

4 First Prev Next Last 1. Li-Yorke chaos The system has an uncountable scrambled set. Page 4 of 35 Scrambled set: Let S D, containing at least two distinct points. Then, S is called a scrambled set if x 0, y 0 S, x 0 y 0, (i) lim inf n d(x n, y n ) = 0; (ii) lim sup n d(x n, y n ) > 0.

5 First Prev Next Last 2. Devaney chaos [1987]: f : V D V satisfies (i) dense periodic points in V ; (ii) topologically transitive in V ; (iii) sensitive dependence on initial conditions in V. Page 5 of Wiggins chaos [1990]: (ii) + (iii)

6 First Prev Next Last Relationships: they are not equivalent in general. 1. Devaney chaos = Wiggins chaos. Page 6 of Let V be a compact set of X, containing infinitely many points, and let f : V V be continuous and surjective. (i) Devaney chaos = Li-Yorke chaos. (ii) Wiggins chaos + one periodic point in V = Li-Yorke chaos. 3. The converses are not true in general.

7 First Prev Next Last How to determine whether a given system is chaotic? 1. One-dimensional systems: period k 2 n, positive entropy, turbulence 2. Higher-dimensional systems: snap-back repeller Page 7 of Infinite-dimensional systems?

8 First Prev Next Last 2. Concept of coupled-expanding map Block LS and Coppel WA [1992] Lecture Notes in Mathematics, Vol Page 8 of 35 A C 0 map f : I I is said to be turbulent if closed and bounded subintervals J and K, with at most one common point, s.t. f(j) J K, f(k) J K. Further, it is said to be strictly turbulent if J K = φ. * J and K are compact and connected.

9 First Prev Next Last A turbulent map f is chaotic in the sense of Li-Yorke. Example 1. The logistic map: f(x) = µx(1 x), µ 4. Example 2. The tent map Page 9 of 35

10 First Prev Next Last Extended to maps in metric spaces, Turbulent map = Coupled-expanding map DEFINITION f : D X X. Assume that m ( 2) subsets V i D, 1 i m, s.t. Page 10 of 35 V i V j = D V i D V j, 1 i j m m f(v i ) V j, 1 i m. j=1 Then f is said to be coupled-expanding in V i, 1 i m. Strictly coupled-expanding: d(v i, V j ) > 0 for all 1 i j m. coupled-expanding CE strictly coupled-expanding SCE

11 First Prev Next Last DEFINITION [Shi, Ju and Chen, 2006] Let f : D X X and A = ((A) ij ) be an m m transitive matrix (m 2). Assume that m subsets V i D s.t. V i V j = D V i D V j, 1 i j m, Page 11 of 35 f(v i ) j (A) ij =1 V j, 1 i m Then f is said to be CE for matrix A in V i, 1 i m. SCE for matrix A: d(v i, V j ) > 0 for all 1 i j m. * A transitive matrix A = ((A) ij ) m m (m 2) (A) ij = 0 or 1 for all i, j; m j=1 (A) ij 1 for all i; m i=1 (A) ij 1 for all j.

12 First Prev Next Last 3. SCE maps and subshifts of finite type 3.1. One-sided symbolic dynamical systems Page 12 of 35 S := {1, 2,..., m}, m 2, + m := {α = (a 0, a 1, a 2,...) : a i S, i 0} ρ(α, β) := i=0 d(a i, b i ) 2 i, d(a i, b i ) := 0 if a i = b i, d(a i, b i ) := 1 if a i b i, where α = (a 0, a 1, a 2,...) and β = (b 0, b 1, b 2,...). ( + m, ρ) is a complete metric space and a Cantor set (compact, perfect, and totally disconnected).

13 First Prev Next Last The shift map σ : + m + m, σ(a 0, a 1, a 2,...) = (a 1, a 2, a 3,...). ( + m, σ) is called the one-sided symbolic dynamical system on m symbols. Page 13 of 35 It is chaotic in the sense of both Devaney and Li-Yorke.

14 First Prev Next Last 3.2 Subshift of finite type Page 14 of 35 Let A be an m m transitive matrix. + m (A) := {α = (a 0, a 1,...) + is a compact invariant set under σ. The subshift of finite type m : (A) a i a i+1 = 1, i 0} σ A := σ + m (A) : + m (A) + m (A). Q: Under what conditions the subshifts are chaotic in the sense of Li-Yorke or Devaney?

15 First Prev Next Last THEOREM [Shi, Ju and Chen, 2006] Assume that A is irreducible. Then the following statements are equivalent: (i) σ A is chaotic in the sense of Devaney on the infinite set + m (A); (ii) σ A is chaotic in the sense of Li-Yorke; Page 15 of 35 (iii) + m (A) is infinite; (iv) + m (A) is a Cantor set; (v) m j=1 (A) i 0 j 2 for some i 0 ; (vi) m i=1 (A) ij 0 2 for some j 0. * Irreducible transitive matrix: (i, j), 1 i, j m, k 1 s.t. (A k ) ij > 0.

16 First Prev Next Last 3.3. Relationships THEOREM [Shi, Ju and Chen, 2006] f : D X X is C 0. Page 16 of 35 f is topologically conjugate to σ A if and only if m disjoint compact subsets V i D, 1 i m, s.t. (i) f is SCE for A in V i, 1 i m; (ii) α = (a 0, a 1,...) + m (A), f n (V an ) is a singleton. n=0

17 First Prev Next Last THEOREM [Shi, Ju and Chen, 2006] f : D X X. Assume that m( 2) nonempty bounded and closed subsets V i D with d(v i, V j ) > 0 s.t. f is C 0 in m i=1 V i and satisfies Page 17 of 35 (i) f is SCE for some A in V i, 1 i m; (ii) α = (a 0, a 1,...) + m (A), n d( f i (V ai )) 0 as n. i=0 Then f in some invariant set V m i=1 V i is topologically conjugate to σ A.

18 First Prev Next Last 4. Chaos induced by coupled-expanding maps Page 18 of SCE maps in compact sets 4.2. SCE maps in bounded and closed sets

19 First Prev Next Last 4.1. SCE maps in compact sets THEOREM [Shi and Chen, 2004] Let V j, 1 j m, be disjoint compact sets of X, and f : m j=1 V j X be C 0. If Page 19 of 35 (i) f is SCE in V j, 1 j m; (ii) λ > 1 s.t. d(f(x), f(y)) λ d(x, y), x, y V j, 1 j m; then a Cantor set Λ m j=1 V j s.t. f : Λ Λ is topologically conjugate to σ : + m + m. = Chaotic on Λ in the sense of Devaney as well as Li-Yorke.

20 First Prev Next Last THEOREM [Shi and Yu, 2005] If f satisfies (i) and Page 20 of 35 (ii ) a j 0, and λ > 1 s.t. d(f(x), f(y)) λ d(x, y) x, y V j0 ; and f is injective in the other sets V j, j j 0 ; then f is chaotic in the sense of both Wiggins and Li-Yorke on a perfect and compact invariant set.

21 First Prev Next Last THEOREM [Shi, Ju and Chen, 2006] Assume that an m m irreducible transitive matrix A with m j=1 (A) i 0 j 2 for some i 0 ; m disjoint compact subsets V i of D, 1 i m, and f is C 0 in m i=1 V i. If f satisfies that Page 21 of 35 (i ) f is SCE for A in V i, 1 i m; (ii) λ > 1 s.t. d(f(x), f(y)) λ d(x, y) x, y V i, 1 i m. Then, f in a Cantor set V is topologically conjugate to σ A. = f is chaotic on V in the sense of Devaney as well as Li-Yorke.

22 First Prev Next Last 4.2. SCE maps in bounded and closed sets In the following, (X, d) is a complete metric space. THEOREM [Shi and Chen, 2004] Let V j X, 1 j m, be bounded and closed subsets of X with d(v i, V j ) > 0, and f : m i=1 V i X be C 0. If Page 22 of 35 (i) f is SCE in V j, 1 j m; (ii) µ λ > 1 s.t. λ d(x, y) d(f(x), f(y)) µ d(x, y) x, y V j, 1 j m; then a Cantor set Λ m i=1 V i s.t. f : Λ Λ is topologically conjugate to σ : + m + m. = f is chaotic on Λ in the sense of Devaney as well as Li-Yorke. * Similarly, these two conditions can be weakened as we have done before.

23 First Prev Next Last 5. Some applications 1. Anti-control of chaos (or chaotification) The original system x n+1 = f(x n ), n 0. Page 23 of 35 Objective: Design a control input sequence {u n } s.t. x n+1 = f(x n ) + u n, n 0 is chaotic. u n = µ g(x n ) or u n = g(µ x n ). * Shi Y and Chen G [2005] Int J Bifur Chaos, 15, (R n ) * Lu J [2005] Chinese Physics 14, ; (R n ) * Shi Y, Yu P and Chen G [2006] Int J Bifur Chaos, 16, (Banach spaces)

24 First Prev Next Last 2. Snap-back repeller theory Marotto FR [1978] J. Math. Anal. Appl. 63, f : R n R n is C 1, f(z) = z. Page 24 of 35 A snap-back repeller implies Li-Yorke chaos.

25 First Prev Next Last Recent developments: (1) The concept is extended to maps in metric spaces in 2004: Regular and singular; nondegenerate and degenerate. * In the Marotto paper, a snap-back repeller is regular and nondegenerate. Page 25 of 35 (2) C 1 maps in R n : Improvement of the Marotto theorem: A snap-back repeller in the Marotto paper implies Devaney chaos as well as Li-Yorke chaos. The assumptions of the Marotto theorem were weakened: A regular snap-back repeller implies Li-Yorke chaos. (3) Snap-back repellers in Banach spaces and in general complete metric spaces.

26 First Prev Next Last 3. Partial difference equations [Y Shi, 2006] Page 26 of Time-varying discrete systems [Y Shi and G Chen, 2005] 5. PDEs, FDEs?

27 First Prev Next Last 6. Examples with simulations Example 1. Page 27 of 35 The origin is a regular and nondegenerate snap-back repeller. = f is chaotic in the sense of both Devaney and Li-Yorke.

28 First Prev Next Last Page 28 of 35 Figure 1a: Simulation result in the (x, y) space in the rectangular box [ 8, 8] [ 8, 8].

29 First Prev Next Last Page 29 of 35 Figure 1b: Simulation result in the (x, y) space in the rectangular box [ 4, 4] [ 4, 4].

30 First Prev Next Last Page 30 of 35 Figure 1c: Simulation result in the (x, y) space in the rectangular box [ 1, 1] [ 1, 1].

31 First Prev Next Last Example 2. Page 31 of 35 The origin is a regular and degenerate snap-back repeller. = f is chaotic in the sense of both Wiggins and Li-Yorke.

32 First Prev Next Last 8 4 y(n) 0 JJ II J I -4 Page 32 of x(n) Figure 2a. Simulation result in the rectangular box [ 8, 8] [ 8, 8]

33 First Prev Next Last 4 2 y(n) 0 JJ II J I -2 Page 33 of x(n) Figure 2b. Zoom area of the rectangular box [ 4, 4] [ 4, 4]

34 First Prev Next Last Thanks for your attention! Page 34 of 35

35 First Prev Next Last DEFINITION [Shi and Chen, 2004] f : X X, f(z) = z. Expanding fixed point (EFP): λ > 1 s.t. d(f(x), f(y)) λ d(x, y), x, y B r (z). Page 35 of 35 Snap-back repeller (SBR): z is an EFP in B r (z). x 0 B r (z), x 0 z, s.t. f m (x 0 ) = z for some m 2. Nondegenerate SBR: µ > 0 s.t. d(f m (x), f m (y)) µ d(x, y), x, y B r0 (x 0 ) B r (z). Regular SBR: f(b r (z)) is open, and δ 0 > 0 s.t. z is an interior point of f m (B δ (x 0 )) for any δ δ 0. * In the Marotto paper, a SBR is regular and nondegenerate.