The document provides an introduction to applied symbolic dynamics of unimodal maps. It discusses patterns, maximal shift sequences (MSS), and Gray codes. For patterns, it defines order and legal inverse paths. For MSS, it discusses observations and theorems about sequences generated by unimodal maps. It then introduces Gray ordering numbers (GON) and their application to visualize sequences generated by maps like the logistic map and Mandelbrot map.

1. Contents Introduction Pattern MSS Gray References
Applied symbolic dynamics of
unimodal maps
˜
David Arroyo Guardeno
Instituto de F´sica Aplicada, Madrid
ı
CSIC
LIT FernUni Hagen
September, 9th 2007
David Arroyo (IFA,CSIC) -1/41- Applied Symbolic Dynamics

2. Contents Introduction Pattern MSS Gray References
Contents
1 Introduction
2 Pattern
Definition
Order
LIP
3 MSS
Observations
Theorems
4 Gray
GON
Application
David Arroyo (IFA,CSIC) -2/41- Applied Symbolic Dynamics

3. Contents Introduction Pattern MSS Gray References CV Scenario
Curriculum Vitae
Wu, Hu and
Zhang 2004
GRAY Metropoli, Stein Beyer, Mauldin
CODES and Stein, 1973 and Stein, 1986
Cusick
1999
Alvarez
1998
Hao and
REFORMULATION
Zheng, 1998
Wang and
Kazarinoff, 1987
David Arroyo (IFA,CSIC) -3/41- Applied Symbolic Dynamics

4. Contents Introduction Pattern MSS Gray References CV Scenario
What are we going to do?
Unimodal maps as generators of bit sequences
Bit sequence
Control parameter?
Initial condition?
David Arroyo (IFA,CSIC) -4/41- Applied Symbolic Dynamics

5. Contents Introduction Pattern MSS Gray References CV Scenario
Scenario
f (x) is defined in I = [a, b]
xc is the point where f (x) reaches its maximum
(minimum) value
f (x) is an increasing (decreasing) function in
[a, xc ) and a decreasing (increasing) function
(xc , b]
David Arroyo (IFA,CSIC) -5/41- Applied Symbolic Dynamics

6. Contents Introduction Pattern MSS Gray References CV Scenario
Tent map
1
0.9
0.8
0.7
0.6
xi+1
0.5
2xi + 1, xi ∈ [−1, 0] 0.4
xi+1 =
−2xi + 1, xi ∈ (0, 1] 0.3
0.2
0.1
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x
i
David Arroyo (IFA,CSIC) -6/41- Applied Symbolic Dynamics

7. Contents Introduction Pattern MSS Gray References CV Scenario
Logistic map
xi+1 = λxi (1 − xi )
David Arroyo (IFA,CSIC) -7/41- Applied Symbolic Dynamics

8. Contents Introduction Pattern MSS Gray References CV Scenario
Mandelbrot map
xi+1 = λxi2 + c
David Arroyo (IFA,CSIC) -8/41- Applied Symbolic Dynamics

9. Contents Introduction Pattern MSS Gray References CV Scenario
Class of functions F
Definition
F is the class of functions defined over the interval
I = [a, b] so each f ∈ F satisfies:
1 f is a continuous function in I
2 f reaches its maximum value fmax = f (xc ) in a
subinterval [am , bm ] so that am ≤ bm
3 f is an strictly increasing function in [a, am ] and
strictly decreasing function in [bm , b]
David Arroyo (IFA,CSIC) -9/41- Applied Symbolic Dynamics

10. Contents Introduction Pattern MSS Gray References Definition Order LIP
What is a pattern?
P = A1 A2 · · · Ak = L i(1) RL i(2) R · · · L i(m−1) RL i(m)
Ak ∈ {L, R}
L ≡ [a, xc )
R ≡ (xc , b]
f L (x) = f −1 (x) ([a, xc ) (xc )), ∀x ∈ f (I)
f R (x) = f −1 (x) ((xc , b] (xc )), ∀x ∈ f (I)
f P (x) = f A1 f A2 · · · f Ak (x)
Power Sequence of P relative to L
{i(1), i(2), . . . , i(m)}
David Arroyo (IFA,CSIC) -10/41- Applied Symbolic Dynamics

11. Contents Introduction Pattern MSS Gray References Definition Order LIP
Patterns order definition
x0 = xc
x1 = f Ak (xc )
x2 = f Ak−1 (x1 )
.
.
.
Final point: L(WP,f ) = xk = f A1 (xk−1 )
David Arroyo (IFA,CSIC) -11/41- Applied Symbolic Dynamics

12. Contents Introduction Pattern MSS Gray References Definition Order LIP
Patterns order definition
x0 = xc
x1 = f Ak (xc )
x2 = f Ak−1 (x1 )
.
.
.
Final point: L(WP,f ) = xk = f A1 (xk−1 )
Definition
P <P Q ⇔ L(WP,f ) < L(WQ,f )
David Arroyo (IFA,CSIC) -12/41- Applied Symbolic Dynamics

13. Contents Introduction Pattern MSS Gray References Definition Order LIP
Universality
Theorem
Let it be f , g ∈ F and P, Q ∈ P, so P = A1 A2 · · · Ak
and Q = B1 B2 · · · Bn . If f P (y0 ), f Q (xc ), gP (xc ), gQ (xc )
are well defined, it is satisfied
L(WP,f ) < L(WQ,f )
if and only if
L(WP,g ) < L(WQ,g ).
David Arroyo (IFA,CSIC) -13/41- Applied Symbolic Dynamics

14. Contents Introduction Pattern MSS Gray References Definition Order LIP
Power sequences order
Definition
{i(1), i(2), . . . , i(m)} <l {j(1), j(2), . . . , j(n)} if and
only if one of the next three conditions is satisfied:
1 It exists r so that 1 ≤ r ≤ min(m, n) and
i(β) = j(β) for β = 1, . . . , r − 1 and
(−1)r i(r) < (−1)r j(r).
2 m < n, i(β) = j(β) for β = 1, . . . , m, being m an
odd value.
3 m > n, i(β) = j(β) for β = 1, . . . , n, and n even.
David Arroyo (IFA,CSIC) -14/41- Applied Symbolic Dynamics

15. Contents Introduction Pattern MSS Gray References Definition Order LIP
Orders equivalence
Theorem
The next statements are equivalent:
1 P <P Q,
2 LP < LQ ,
3 {i(1), . . . , i(m)} <l {j(1), . . . , j(n)},
m β β
β=1 (−1) [i(β) + 1] /x <
4
n β β
β=1 (−1) [j(β) + 1] /x .
David Arroyo (IFA,CSIC) -15/41- Applied Symbolic Dynamics

16. Contents Introduction Pattern MSS Gray References Definition Order LIP
Legal Inverse Path (Shift Maximal Sequence)
S = S0 S1 . . . SN −1
i = 1
LIP no i <N
yes
T = Si . . . SN −1 i = i+1
S>T yes
no
NO LIP
David Arroyo (IFA,CSIC) -16/41- Applied Symbolic Dynamics

17. Contents Introduction Pattern MSS Gray References Observations Theorems
What happens if f (x) = fλ(x)?
I fλ (x) = I0 I1 I2 · · ·
(i)
Ii = R ⇔ fλ (x) > xc
(i)
Ii = L ⇔ fλ (x) < xc
(i)
Ii = C ⇔ fλ (x) = xc
I fλ (x) finishes when the first C appears
Definition (MSS sequence)
(k)
Pλi = I fλi (xc )| fλi (xc ) = xc
David Arroyo (IFA,CSIC) -17/41- Applied Symbolic Dynamics

18. Contents Introduction Pattern MSS Gray References Observations Theorems
A new order?
L <S C <S R
S = {Si }, T = {Ti }, S <S T
1 S0 < T0
2 S0 S1 · · · Si−1 = T0 T1 · · · Ti−1 has an even number of
R’s and Si <S Ti
3 S0 S1 · · · Si−1 = T0 T1 · · · Ti−1 has an odd number of
R’s and Si >S Ti
Proposition
The orders <S and <P are equivalent
David Arroyo (IFA,CSIC) -18/41- Applied Symbolic Dynamics

19. Contents Introduction Pattern MSS Gray References Observations Theorems
Some important observations
Proposition
Any MSS sequence is a superstable orbit
Lema
If I fλ (x) < I fλ (y) then x < y
Theorem
For each value of λ, I fλ (fλ (xc )) is a shift maximal
sequence. Any MSS sequence is a shift maximal
sequence
David Arroyo (IFA,CSIC) -19/41- Applied Symbolic Dynamics

20. Contents Introduction Pattern MSS Gray References Observations Theorems
Some important results
Theorem
Let it F an unimodal, Lipschitz, continuous function
and with continuos derivative in a neighborhood of
x = xc . Assuming 0 ≤ λ1 < λ2 ≤ 1 and A is a shift
maximal sequence. A is any sequence different from
L ∞ , C, R ∞ o RL ∞ . It is also satisfied
I λ1 F (λ1 ) < A < I λ2 F (λ2 ).
Then it exists λ ∈ (λ1 , λ2 ) so that
I λF (λ) = A.
David Arroyo (IFA,CSIC) -20/41- Applied Symbolic Dynamics

21. Contents Introduction Pattern MSS Gray References Observations Theorems
Some important results
Theorem
Let it be F an unimodal, continuous, concave and
Lipstchitz function whose derivative is continuous in
a neighborhood of x = xc . For a sequence A which
is shift maximal there exists a value of λ such
I λF (λ) = A. Particularly, it exists a value λ for each
MSS sequence.
David Arroyo (IFA,CSIC) -21/41- Applied Symbolic Dynamics

22. Contents Introduction Pattern MSS Gray References GON Application
David Arroyo (IFA,CSIC) -22/41- Applied Symbolic Dynamics

23. Contents Introduction Pattern MSS Gray References GON Application
f (0) (x)
L R
x
a xc b
David Arroyo (IFA,CSIC) -23/41- Applied Symbolic Dynamics

24. Contents Introduction Pattern MSS Gray References GON Application
f (x) LL LR RR RL
xc
x
a xc b
David Arroyo (IFA,CSIC) -24/41- Applied Symbolic Dynamics

25. Contents Introduction Pattern MSS Gray References GON Application
f (2) (x) L L L L RR RR L RL R
LLR L RL RRR RL L
xc
x
a xc b
David Arroyo (IFA,CSIC) -25/41- Applied Symbolic Dynamics

26. Contents Introduction Pattern MSS Gray References GON Application
f (3) (x) L L LR L LRL LRRR LRL L RRLR RRRL RLRR RL L L
LLLL L LRR LRRL LRLR RRL L RRRR RLRL RL LR
xc
x
a xc b
David Arroyo (IFA,CSIC) -26/41- Applied Symbolic Dynamics

27. Contents Introduction Pattern MSS Gray References GON Application
Gray Ordering Number
P = p1 p2 . . . pn , pi ∈ R, L
1 G(P) = g1 g2 . . . gn
1 if pi = R
gi =
0 if pi = L
2 U (P) = u1 u2 . . . un
u1 = g1
ui+1 = gi ⊕ ui+1
Gray Ordering Number
GON (P) = 2−1 · u1 + 2−2 · u2 + . . . + 2−n · un
David Arroyo (IFA,CSIC) -27/41- Applied Symbolic Dynamics

28. Contents Introduction Pattern MSS Gray References GON Application
Extended GON
David Arroyo (IFA,CSIC) -28/41- Applied Symbolic Dynamics

29. Contents Introduction Pattern MSS Gray References GON Application
1 1
0.8 0.8
GON(Pf (x))
GON(Pf (x))
0.6 0.6
λ
λ
n
n
0.4 0.4
Logistic map
0.2 0.2
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
x x
(a) λ = 3.4 (b) λ = 3.6
1 1
0.8 0.8
GON(Pf (x))
GON(Pf (x))
0.6 0.6
λ
λ
n
n
0.4 0.4
0.2 0.2
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
x x
(c) λ = 3.8 (d) λ = 4
David Arroyo (IFA,CSIC) -29/41- Applied Symbolic Dynamics

30. Contents Introduction Pattern MSS Gray References GON Application
1 1
0.8 0.8
GON(Pn (x))
GON(Pf (x))
0.6 0.6
c
c
n
f
Mandelbrot map
0.4 0.4
0.2 0.2
0 0
−2 −1 0 1 2 −2 −1 0 1 2
x x
(e) c = −1.5 (f) c = −1.7
1 1
0.8 0.8
GON(Pn (x))
GON(Pf (x))
0.6 0.6
c
c
n
f
0.4 0.4
0.2 0.2
0 0
−2 −1 0 1 2 −2 −1 0 1 2
x x
(g) c = −1.8 (h) c = −2
David Arroyo (IFA,CSIC) -30/41- Applied Symbolic Dynamics

31. Contents Introduction Pattern MSS Gray References GON Application
1
0.95
0.9
GON(ζf (0.5))
0.85
μ
n 0.8
0.75
Logistic map
0.7
0.65
3 3.2 3.4 3.6 3.8 4
μ
(i) GON of I fµ (fµ (xc )) (j) Asynthotic iteration values
0.7
0.6
0.5
GON(ζf (0.5))
0.4
n,2
μ
0.3
0.2
0.1
0
3 3.2 3.4 3.6 3.8 4
μ
(2)
(k) GON of I fµ (fµ (xc )) (l) Asynthotic GON values
David Arroyo (IFA,CSIC) -31/41- Applied Symbolic Dynamics

32. Contents Introduction Pattern MSS Gray References GON Application
Parameter estimation
The symbolic sequence generated from fλ (xc ) is
shift maximal
The symbolic sequences generated from fλ (xc )
are ordered according to λ
David Arroyo (IFA,CSIC) -32/41- Applied Symbolic Dynamics

33. Contents Introduction Pattern MSS Gray References GON Application
Looking for the shift maximal sequence
Input: S = I fλ (x0 ) = S0 S1 . . . SM+n−1
Smax = S0 S1 . . . Sn−1 , i = 1
Output: Smax no i<M
yes
T = Si Si+1 . . . Si+n−1
T > Smax no i = i +1
yes
Smax = T
David Arroyo (IFA,CSIC) -33/41- Applied Symbolic Dynamics

34. Contents Introduction Pattern MSS Gray References GON Application
Parameter estimation: logistic map
Input: Smax
λR +λL
λL = 3.5, λR = 4, λ = 2
S = I fλ (fλ (0.5))
Output: λ yes S = Smax S = I fλ (fλ (0.5))
no
S < Smax yes λR = λ λ= λR +λL
2
no
λL = λ
David Arroyo (IFA,CSIC) -34/41- Applied Symbolic Dynamics

35. Contents Introduction Pattern MSS Gray References GON Application
Initial condition estimation
Input: S = I fλ (x0 ) = S0 S1 . . . SN , λ
x0L = 0, x0R = 1,
x +x
x0 = 0R 2 0L
T = I fλ (x0 )
Output: x0 yes T =S S = I fλ (x0 )
no
yes x0R = x0R +x0L
T <S x0 x0 = 2
no
x0L = x0
David Arroyo (IFA,CSIC) -35/41- Applied Symbolic Dynamics

36. Contents Introduction Pattern MSS Gray References GON Application
Parameter error estimation
−4
c estimation error (Logarithmic scale) 10
−6
10
−8
10
−10
10
−12
10
0 2 4 6 8 10
M 5
x 10
David Arroyo (IFA,CSIC) -36/41- Applied Symbolic Dynamics

37. Contents Introduction Pattern MSS Gray References GON Application
Initial condition estimation error
0
x0 estimation error (Logarithmic scale) 10
−5
10
−10
10
−15
10
−20
10
10 20 30 40 50 60
N
David Arroyo (IFA,CSIC) -37/41- Applied Symbolic Dynamics

38. Contents Introduction Pattern MSS Gray References GON Application
GON comparing vs. Gray code comparing
−12
Error(logarithmic scale) 10
−13
10
−14
10
−15 Gray
10
GON
−16
10
50 52 54 56 58 60 62
N
David Arroyo (IFA,CSIC) -38/41- Applied Symbolic Dynamics

39. Contents Introduction Pattern MSS Gray References GON Application
Future work
1 Look for a new way to get the shift maximal
sequence
M has to be too big to get a good estimation
2 Non-unimodal maps
David Arroyo (IFA,CSIC) -39/41- Applied Symbolic Dynamics

40. Contents Introduction Pattern MSS Gray References
N. Metropolis, M.L. Stein, and P.R. Stein.
On the limit sets for transformations on the unit
interval.
Journal of Combinatorial Theory (A), 15:25–44,
1973.
W.A. Beyer, R.D. Mauldin, and P.R. Stein.
Shift-maximal sequences in function iteration:
Existence, uniqueness and multiplicity.
J. Math. Anal. Appl., 115:305–362, 1986.
David Arroyo (IFA,CSIC) -40/41- Applied Symbolic Dynamics

41. Contents Introduction Pattern MSS Gray References
Li Wang and Nicholas D. Kazarinoff.
On the universal sequence generated by a class
of unimodal functions.
Journal of Combinatorial Theory, Series A,
46:39–49, 1987.
Bai-Lin Hao and Wei-Mou Zheng.
Applied symbolic dynamics and chaos, volume 7.
Directions in Chaos, 1998.
David Arroyo (IFA,CSIC) -41/41- Applied Symbolic Dynamics

42. Contents Introduction Pattern MSS Gray References
Gonzalo Alvarez, Miguel Romera, Gerardo
Pastor, and Fausto Montoya.
Gray codes and 1d quadratic maps.
Electronic Letters, 34(13):1304–1306, 1998.
T.W. Cusick.
Gray codes and the symbolic dynamics of
quadratic maps.
Electronic Letters, 35(6):468–469, 1999.
Xiaogang Wu, Hanping Hu, and Baoliang Zhang.
Parameter estimation only from the symbolic
sequences generated by chaos system.
Chaos, solitons and Fractals, 22:359–366, 2004.
David Arroyo (IFA,CSIC) -42/41- Applied Symbolic Dynamics

43. Contents Introduction Pattern MSS Gray References
Gonzalo Alvarez, Fausto Montoya, Miguel
Romera, and Gerardo Pastor.
Cryptanalysis of an ergodic chaotic cipher.
Physics Letters A, 311:172–179, 2003.
M. S. Baptista.
Cryptography with chaos.
Physics Letters A, 240(1-2):50–54, 1998.
David Arroyo (IFA,CSIC) -43/41- Applied Symbolic Dynamics