Pattern Generation and Spatial Entropy in Multi-Dimensional Lattice Models (The Third International Congress of Chinese Mathematician, (ICCM 2004) Hong-Kong, Dec. 17-22, 2004.)

Patterns Generation and Spatial Entropy in Multi-Dimensional Lattice Models

Song-Sun Lin,

Deaprtment of Applied Mathematics, National Chiao Tung University

Joint works with

Dr. Jung-Chao Ban, NCTS, and Ms. Yin-Heng Lin, NCTU.

Contents:

§1. Introduction

§2. Two dimensional lattices

§3. Transition matrices for admissible patterns

§4. Reduction operators – lower bounds of entropy

§5. Trace operators – upper bounds of entropy

§6. Summary

§1. Introduction

§§1.1 Motivations

(I) Lattice Dynamical System (LDS)

1. L. O. CHUA and L. YANG, Cellular neural networks: theory and applications, IEEE Trans. Circuits Systems, 35(1988), pp. 1257-1290.

2. L. O. CHUA, CNN: A paradigm for complexity. World Scientific Series on Nonlinear Science, Series A, 31. World Scientific, Singapore. (1998).

3. S. N. CHOW, J. MALLET-PARET and E. S. VAN VLECK, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Comput.

Dynam., 4(1996), pp. 109-178.

(II) 2D Patterns Generation & Spatial Entropy

4. J. JUANG and S. S. LIN, Cellular neural networks: mosaic pattern and spatial

chaos, SIAM J. Appl. Math., 60(2000), pp.891-915.

5. C. H. HSU, J. JUANG, S. S. LIN, and W. W. LIN, Cellular neural networks:

local patterns for general template, International J. of Bifurcation and Chaos, 10(2000), pp.1645-1659.

6. J. JUANG, S. S. LIN, W. W. LIN and S. F. SHIEH, The spatial entropy of two-dimensional subshifts of finite type, International J. of Bifurcation and Chaos, 10(2000), pp.2845-2852.

7. S. S. LIN and T. S. YANG, On the spatial entropy and patterns of

two-dimensional cellular neural network, International J. of Bifurcation and Chaos, 12 (2002), 115-128.

8. J. C. BAN and S. S. LIN, Patterns generation and transition matrices in multi-dimensional lattice models. Discrete and Conti. Dyn. Sys., 2005.

9. J. C. Ban, S. S. Lin and Y. H. Lin, Patterns generation and spatial entropy in

§§1.2 1-D case

1-D Lattice Z¹

Symbols (colors, alphabets) S={0, …, p-1}.

Two symbols S={0,1}; two colors S={ , }.

Local Admissible Conditions (Local Interaction Property) The state at each lattice point only influenced by its finitely many neighborhood states.

Example 1.1.

On Z2  ( Basic lattice )



₂ { , , , }, the set of all local patterns defined on Z2 .

  2 : Basic (admissible) set, the set of all admissible local patterns.

Example 1.2.

B={ , , }; basic set.

Transition matrix : ^{   ( )}





 





0 1

1 1

.

T

Questions

n( )

  : all admissible patterns on ^Zn which can be generated by ^ . (a) How to generate ^n(B) from ^B ? ^{n(B) #n(B) } ?

(b) Spatial Entropy : ^{h(B)  lim}_n ^log_n^{n(B) } ?

Answers

(a) ^{  n( ) Tn1} and ^{n( )B  n1} =

2 2

1

1 1

( ⁿ )_ij

i j



 



(b) ^{h B( )}  ^{log ( )} ^T , where ^{( )T} is the maximum eigenvalue of

)

(B



 .

§2 Two dimensional lattices

H 2 : transition matrix for Z2

H ³ : transition matrix for Z^3

H ⁿ : transition matrix for Z^n

H2

H3

Hn

H ^

: transition matrix for

:

1 1

{0,1}

S^Z S^Z 

S^Z1

: The set of symbols (vertical strips) is uncountable.

1 admissible,

( , )

0 forbidden.

S S_{ }



  H 

S_

H

Questions

(i) What is the relation between H ⁿ & H ^n1 ?

i.e. can we obtain a recursive formula for H ^n1 in terms of H ⁿ , …, H₂ ?

(ii) How to compute





2

log ( )

( ) lim

h

n





H

H

(iii) What is the relation between





,



, …_,



(iv) If ^{* lim}_n^n1/n, ^{h  log}, any “limiting” equation satisfied by *?

§§2.1 Ordering Matrices

Ref [8]：(Ban - L.)

On ^{Z2 2} , 22 can be arranged by

X2[x_{i i}1 2, ] Y2[y_{j j}1, 2]

Orderings of ^{Z2 2} by & respectively. ²

1 3

4 3 4

1 2

Observations

11 12 21 22

13 14 23 24 1 2

2

31 32 41 42 3 4

33 34 43 44

y y y y

y y y y X X

y y y y X X

y y y y

 

   

 

     

 

 

Χ





















‧For ^23, = .

ˆ



2 1j

yj

3 2j

yj x

1

3 4 2

















4 3 4 3

2 1 2 1

4 3 4 3

2 1 2 1

§3 Transition Matrices

On Z2 2 , given an admissible set B  22 , define (horizontal) transition matrix H2  H2( )B [h_{j j}1 2]4 4_ , where h_{j j}1 2 {0,1} and h_{j j}1 2  1 x_{j j}1 2 B .

11 12 13 14 11 12 21 22

2;1 2;2

21 22 23 24 13 14 23 24 1 2

2

2;3 2;4

31 32 33 34 31 32 41 42 3 4

41 42 43 44 33 34 43 44

h h h h v v v v

H H

h h h h v v v v H H

H H

h h h h v v v v H H

h h h h v v v v

   

        

   

              

   

H

,

11 12 13 14 11 12 21 22

2 ;1 2 ;2

21 22 23 24 13 14 23 24 1 2

2 ;3 2 ;4

31 32 33 34 31 32 41 42 3 4

41 42 43 44 33 34 43 44

2

v v v v h h h h

V V

v v v v h h h h V V

V V

v v v v h h h h V V

v v v v h h h h

   

        

   

           

   

   

V

Denoted by vj j j1 2 3  vj j1 2vj j2 3,

111 112 121 122 211 212 221 222

113 114 123 124 213 214 223 224

131 132 141 142 231 232 241 242

133 134 143 144 233 234 243 244

3

311 312 321 322 411 412 421 422

313 314 323 324 413 414 423 424

331

v v v v v v v v

v v v v v v v v

v v v v v v v v

v v v v v v v v

v v v v v v v v

v v v v v v v v

v v

 H

11 1 12 2 21 1 22 2

3;1 3;2

13 3 14 4 23 3 24 4

3;3 3;4

31 1 32 2 41 1 42 2

33 3 34 4 43 3 44 4

332 341 342 431 432 441 442

333 334 343 344 433 434 443 444

v H v H v H v H

H H

v H v H v H v H

H H

v H v H v H v H v H v H v H v H

v v v v v v

v v v v v v v v

 

 

 

   

     

     

     

   

 

 

 

 

 

 



Theorem 3.1.(Ban-L.) Let H2 be a transition matrix, write

; 1 ; 2

; 3 ; 4

n n

n

n n

H H

H H

 

  

 

H and

1;1 1;2

1

1;3 1;4

n n

n

n n

H H

H H

 

  

 

  

 

H .

Then

1 ; 1 2 ; 2

1;

3 ;3 4 ;4

k n k n

n k

k n k n

v H v H H

v H v H

 

  

 

1 1 2

2

1 2 1 2

1 2 2 2

3 4 3 4

( ) n n n

n

n n

H H H H

E H H H H

  



 



    

      

   

 

H H

(3.2) , where ^E is the ^{2k 2k} full matrix, i.e., all entries are 1. ^■

Example 3.2. (Golden - Mean):

1 1 1 0

 

     

H

V

2

4 4

1 1 1 0

1 0 1 0

1 1 0 0

0 0 0 0

A B C D



 

   

 

     

 

 

H ₃

8 8

0

0 0

0 0

0 0 0 0

A B A

C C

A B



 

 

 

 

 

 

 

H

Remark 3.3.

A B B A

 

  

H , ²

3

a a

A a a

 

  

 

 

and ²

3

b b B b b

 

  

 

 ^, ( )



1

lim ( ) exp( ( 2))

n

n

n h

  _  H  H can also be found：

2 2 2 2 2

2

2 3

3 2

2 3 2 3 3 2

.

4 ( ) ( 4 )( ) 2 (2 )

(2 ) 1,

( )

0 & 1,

a a b a

b a if a a

Q

a a b if a a a b a b

where b b and b b

         

 

   

 

       

   



        

   



§4. Reduction Operators for

in n

m

Hn

:

2 ,

log | | ( ) lim

1 log | | 1

lim lim lim log ( ),

m n m n

m n

n m n n

h mn

n m n 



  



 

   

 

H H

H H

or

1 log | | lim lim

m n

m^ m n^ n

 

  

 

H

(4.2)

1  log (tr ^m) 

 H

To use (4.2) or (4.3) to compute spatial entropy h H( 2), need to answer Questions Fixed m  2_{, for any} n  2

(i) Find recursive formulas from H^mn _to H^m_n1.

(ii) Find recursive formulas from tr H( ^m_n ) _to tr(H^m_n1)_.

Notations

. (4.4)

for extension to H^n1 for matrix multiplication

;1 ;2 ;11 ;12

;3 ;4 ;21 ;22

n n or n n

n

n n n n

H H H H

H H H H

   

     

   

H

When m  2_,

2

;11 ;12 ;21 ;11 ;12 ;12 ;22

2

2

;21 ;11 ;22 ;21 ;21 ;12 ;22

n n n n n n n

n

n n n n n n n

H H H H H H H

H H H H H H H

   

    

H .

Denote by

2, ;1 2, ;2 2,

2, ;3 2, ;4

n n

n

n n

X X

X X

 

  

 

X and

2

;11 ;12

;11

2, ;1 2, ;2

;12 ;22

;12 ;21

, ^{n n}

n

n n

n n

n n

H H

X H X

H H H H

   

     

 

  ,

;21 ;12

;21 ;11

2, ;3 2, ;4 2

;22 ;21 ;22

, ^{n n}

n n

n n

n n n

H H H H

X X

H H H

   

     

 

  .

Similarly, for any m  3_{, denote}

, ;1 , ;2

,

, ;3 , ;4

m n m n

m n

m n m n

X X

X X

 

  

 

X ,

which represents all “elementary patterns” in H^mn . (3.1)



, 1;1 , 1;2

, 1

, 1;3 , 1;4

m n m n

m n

m n m n

X X

X X

 



 

 

  

 

X and

, 1; ,1 , 1; ,2 , 1;

, 1; ,3 , 1; ,4

m n i m n i

m n i

m n i m n i

X X

X X X

 



 

 

  

  . (4.5)

where

X

v H

Then a recursive relation from ^{Xm n,} to ^{Xm n, 1} (or between ^{Xm n. 1; ,i j} and ^{Xm n. ;,j} ) are given as follows:

Theorem 4.1.

2,n 1; ,i j 2;ij 2, ;n j

X _  S X

,

2;11 2;12 2;13 2;14 2;11 2;12 2;21 2;22

2;21 2;22 2;23 2;24 2;13 3;14 2;23 2;24

2

2;31 2;32 2;33 2;34 2;31 2;32 2;41 2;42

2;41 2;42 2;43 2;44 2;33 2;34 2;43 2;44

R R R R S S S S

R R R R S S S S

R R R R S S S S

R R R R S S S S

   

   

   

 

   

   

   

   

R

(4.6) and

1 2

1 2

2;

j j

i i

ij

h h h h

R h h h h

 

 

    

   . (4.7)

Furthermore, for

m  3

2 2

1 1

1 1

2

1 2

1 2 1 2

; 2 2

3 4

3 4 3 4

2 2

2 2 2 2

ˆ m m

m m

m m

m

j j

i i

m ij

j j

i i

h h

h h V V

R E

h h

h h V V ^{ } _{ }

 





  

         

     

           (4.8) and

;11 ;12 ;13 ;14 ;11 ;12 ;21 ;22

;21 ;22 ;23 ;24 ;13 ;14 ;23 ;24

;31 ;32 ;33 ;34 ;31 ;32 ;41 ;42

;41 ;42 ;43 ;44 ;33 ;34 ;43 ;44

m m m m m m m m

m m m m m m m m

m

m m m m m m m m

m m m m m m m m

R R R R S S S S

R R R R S S S S

R R R R S S S S

R R R R S S S S

   

   

   

 

   

   

   

   

R

, then,

, 1; , ; , ;

m n i j m ij m n j

X _  S X . (4.9) ^■

Theorem 4.2. (Lower-bound of entropy)

For any

m  2

K  1

{1,4}, 1

j

j K



  

2 ; 1 2 ; 2 3 ; 1

( ) 1 log ( )

m m m K

h S S S

mK 

H 

Example 4.3. (Golden - Mean)

2

1 1 1 0 1 0 1 0 1 1 0 0 0 0 0 0

 

 

 

  

 

 

H _

;11 1

m m

R  H 



1 1

( )^m 1 exp( (h )) ( )^m

 H ^  H   H , for m  2.

Example 4.4.

2 2;11 2;22 2;33 2;44

0 1 1 0

1 0 1 0 0 1 1 0 1 1 0 0

, , ,

1 1 0 0 1 0 1 0 0 0 0 0

0 0 0 0

R R R R

 

         

 

               

 

 

H

.

2;14 2;41 2;22 2;33 2

1 1 log 2

, ( )

1 1 4

S S R R   h

     

  H

.

§5. Trace Operators for T

(4,3) can be used for computing and finding upper bounds of spatial entropy h H( 2)_.

Theorem 5.1.

^{(Hn)2  tr(H2n)} . (5.1)

Trace operator ^{m m,11 m,22}

R R

 

  

T .

Theorem 5.2.

For any ^{m  2,} and ^{n  2, tr(Hmn ) | Tmn1 |}

.

)

Furthermore,

l o g

l i m s u p l o g

( )

( )

tr m

n

n_ nH  ^{ T}m

(5.3) and

lim sup 2

log ( )

( ) ^m

m

h m



  ^T

H . (5.4)

When

H2 is symmetric, for any m 1,

log 2

2 2

( )

( )

h



H .

§§5.1. Simplified trace Operator J

Using ^{tr AB( )  tr BA( )}, the 2^m 2^m trace operator

T

* 1

[ ][ ] 2

2 2

m  m m  . (5.5)

For each

m  2

.

Ordering H_n^l1;11(H_n^;12H_n^;21)^l2 H_n^l3;22 by the anti-lexicographic order in

Denote by

3

1 2 2

1 2 3

, ; , ,

(

)

m n l l l n n n n

t  tr H H H H

and

1, 2 3

,

(

)

m n m n l l l

t  t

Theorem 5.3.

Jm such that for any ^{n  2}

, 1 ,

m n m m n

t _  J t (5.6) and

 (J

)   ( T

)

Example 5.4.

2 2

11 12 21 22

2 13 31 14 41 23 32 24 42

2 2

33 34 43 44

2 2

h h h h

h h h h h h h h

h h h h

 

 

   

 

 

J ,

3 3

11 11 12 21 12 22 21 22

11 13 31 11 14 41 12 23 31 21 13 32 12 24 41 21 14 42 22 23 32 22 24 42

3

13 33 31 13 34 41 14 43 31 23 33 32 14 44 41 23 34 42 24 43 32 24 42 44

3

33 33 34 43 3

3 3

3 3

h h h h h h h h

h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h

h h h h h

   

    

J

3

4 44 43h h h44

 

 

 

 

 

 

 

.

Example 5.5.

1

2 1

1

1 1 1 0

1 0 0 0 ˆ

, ˆ

1 0 0 0 0

0 0 0 0

n n

n

n







 

   

 

   

   

 

 

H H

H H

H where

1 1

ˆ 0

0 0

n n





 

  

 

H H

.

2 4 6 8

1 1 1 1 1 1 1 1 1

1 1 1 8 5 3 1 0

1 1 6 3 1 0

, 4 1 0 , , 20 6 1 0 0

2 0 9 1 0 0

2 0 0 16 1 0 0 0

2 0 0 0

2 0 0 0 0

 

   

     

     

            

J J J J

.

m

1

( _m)^m 1

 H ^

1

( _m)^m

 J

1

( _m)^m

 H

2 1.25992 1.41421 1.41421 3 1.29514 1.32054 1.41174 4 1.29841 1.35019 1.38601 5 1.300843 1.33977 1.3711 6 1.31204 1.34688 1.37279 7 1.31639 1.36987 1.36911 8 1.31902 1.33328 1.36547 9 1.32149 1.36306

§6. Summary

1. Higher order transition matrices ^H_n, n  3, can be recursively derived from ^H₂. (Ref. 8)

2. Lower-bound of entropy can be found by introducing reduction

operator R , m m 2. A powerful method to verify the positivity of entropy. (Ref. 9)

3. Trace (and simplified trace) operator T^m (and ^Jm) have been introduced to compute and give a upper-bound of entropy. (Ref. 9)