# Belousov-Zhabotinsky Reaction

Τεχνίτη Νοημοσύνη και Ρομποτική

1 Δεκ 2013 (πριν από 4 χρόνια και 7 μήνες)

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Cellular Automata:

Exploring Applications

Erik Aguilar

Amelia
Yzaguirre

Amy
Femal

Goal

Explore 3 specific types of cellular automata
applications and use MATLAB to model them.

Types of Cellular Automata

Boolean

Excitable Media

Lattice Gas Automata

Excitable Media

Boolean Cellular Automata

can be seen in The Game of Life

Rules for the Game of Life

Any live cell with fewer than 2 live neighbors dies

Any live cell with more than 3 neighbors dies

Any live cell with 2 or 3 live neighbors lives on to the next
generation

Any dead cell with exactly 3 live neighbors becomes a live cell

Writing the code

%First we must create an n x n matrix of zeros.

n=75;

z=zeros(
n,n
);

%Produces random "0" and "1" throughout matrix

cells = (rand(
n,n
))<.12;

%Creates an image using the matrix

imh

= image(cat(3,cells,z,z));

%When the image is replaced, the old image is not erased first

set(
imh
, '
erasemode
', 'none')

x=(2:n
-
1);
%creates array x which is used for position

y=(2:n
-
1);
%creates array y which is used for position

%creates matrix for summations

sum(
x,y
)=zeros(n
-
2,n
-
2);

a=0;

while a==0;

%equations for the corner cells to check for infected

%neighbors

sum(1,1)=cells(1,2)+cells(2,1)+cells(2,2);

sum(1,n)=cells(2,n)+cells(2,n
-
1)+cells(1,n
-
1);

sum(n,1)=cells(n,2)+cells(n
-
1,1)+cells(n
-
1,2);

sum(
n,n
)=cells(n,n
-
1)+cells(n
-
1,n)+cells(n
-
1,n
-
1);

%equations for the edge rows/columns to check for infected

%neighbors

sum(1,y)=cells(1,y
-
1)+cells(1,y+1)+cells(2,y)+cells(2,y+1)+...

cells(2,y
-
1);

sum(
n,y
)=cells(n,y+1)+cells(n,y
-
1)+cells(n
-
1,y)+cells(n
-
1,y+1)+...

cells(n
-
1,y
-
1);

sum(x,1)=cells(x
-
1,1)+cells(x+1,1)+cells(x,2)+cells(x+1,2)+...

cells(x
-
1,2);

sum(x,n)=cells(x
-
1,n)+cells(x+1,n)+cells(x,n
-
1)+cells(x+1,n
-
1)+...

cells(x
-
1,n
-
1);

%equations for the interior cells to check for infected

%neighbors

sum(x,y)=cells(x,y+1)+cells(x,y
-
1)+cells(x+1,y)+cells(x
-
1,y)+...

cells(x+1,y+1)+cells(x+1,y
-
1)+cells(x
-
1,y+1)+cells(x
-
1,y
-
1);

%updating the cells matrix

cells = (sum(1:n,1:n)==3)|(sum(1:n,1:n)==2 & cells);

%draw the new image

set(
imh
, '
CData
', cat(3,cells,z,z))

drawnow

%displaying the updated image

waitforbuttonpress

end

Excitable Media

can be observed in the BZ reaction

Belousov
-
Zhabotinsky

reaction

What is the BZ reaction?

It is a chemical reaction caused by the mixture of

Sulfuric Acid

Sodium
Bromate

Malonic

Acid

Sodium Bromide

Phenanthroline

Ferrous Sulfate

Triton X
-
100 Surfactant

Rules for the BZ Reaction

Cells can be in 10 different states.

State 0 = resting

States 1

5 = active

States 6

9 = refractory

Like LIFE, each cell of the BZ reaction is dependent on its 8
surrounding neighbors.

If 3 or more neighbors are active, cell = 1

A cell in State 1 will change to State 2.

A cell in State 2 will change to State 3 and so on.

A cell in State 9 will change to State 0.

Code for the BZ Reaction

clear all

n=200;
%size

M=zeros(n);
%this will give us an n by n matrix

grid=M;
%the grid will be made up of the n by n matrix

grid=(rand(n))<.06;

sum=M;

bz
=image(cat(3,grid,M,M));

x=[2:n
-
1];

y=[2:n
-
1];

axis tight

t = 6; % when t is in active state

for
i
=1:1000
%duration of loop

sum(x,y) = ((grid(x,y
-
1)>0)&(grid(x,y
-
1)<t)) + ...

((grid(x,y+1)>0)&(grid(x,y+1)<t)) + ...

((grid(x
-
1, y)>0)&(grid(x
-
1, y)<t)) + ...

((grid(x+1,y)>0)&(grid(x+1,y)<t)) + ...

((grid(x
-
1,y
-
1)>0)&(grid(x
-
1,y
-
1)<t)) + ...

((grid(x
-
1,y+1)>0)&(grid(x
-
1,y+1)<t)) + ...

((grid(x+1,y
-
1)>0)&(grid(x+1,y
-
1)<t)) + ...

((grid(x+1,y+1)>0)&(grid(x+1,y+1)<t));

%the sum of each cell in active state of the 8 surrounding neighbors

grid = ((grid==0) & (sum>=3)) + 2*(grid==1) + 3*(grid==2) + ...

4*(grid==3) + 5*(grid==4) + 6*(grid==5) +...

7*(grid==6) + 8*(grid==7) + 9*(grid==8) +...

0*(grid==9);

%when state=1, next state=2...

set(
bz,'cdata
', cat(3,M,grid/10,M)
) %
bz

is the image,
cdata

contains

%data array

drawnow

%creates image

end

Lattice Gas Automata

Evolution of Gas Particles

Lattice Gas Automata

Rules

Cells have 2 states

0 = empty

1 = moving gas particle

Each cell has 3 neighbors for a given time step where a block
rule is applied to a 2 x 2 block of cells.

Odd

Odd

Odd

Cell

Even

Even

Even

Code for Lattice Gas Automata

%Cellular Automata model of gas particles in a box with a partition

%This will make use of a
Margolus

neighborhood to create motion of an HPP

%(Hardy,
Pazzis
,
Pomeau
) lattice gas... Curious if we meet Gibbs' paradox!

clear all

clf

%clears any frames being used.

%
--------------
We must first create our grid
----------

%These variables will be used to define the dimension of the matrix in our

%grid

nx
=52;
%must be divisible by 4, since each cell will be divided into groups of four

ny
=100;

z=zeros(
nx,ny
);
%Creates an
nx

by
ny

matrix of zeros called z

o=ones(
nx,ny
);
%Creates an
nx

by
ny

matrix of ones called o

%Initialize each of the matrices to be used later

cells = z ;
cellsNew

= z; ground = z ; diag1 = z; diag2 = z; and12 = z;

or12 = z; sums = z;
orsum

= z;

%create the box

ground(1:nx,ny
-
3)=1 ;
% right ground line

ground(1:nx,3)=1 ;
% left ground line

ground(
nx
/4:nx/2
-
2,ny/2)=1;
% the hole in the middle of the partition

ground(
nx
/2+2:nx,ny/2)=1;
%the hole in the middle of the partition

ground(
nx
/4, 1:ny) = 1;
%top line

ground(3*
nx
/4, 1:ny) = 1;
%bottom line

%We now want to "fill" the left side of the container with "gas particles"

r = rand(
nx,ny
);

cells(
nx
/4+1:3*
nx
/4
-
1, 4:ny/2
-
1) = r(
nx
/4+1:3*
nx
/4
-
1, 4:ny/2
-
1)<0.3;

%Define the image of the gas particles in the container!

imh

= image(cat(3,z,cells,ground));

set(
imh
, '
erasemode
', 'none')

axis equal

axis tight

%This is where we define the motion of the particles

for
i
=1:1000

p=mod(i,2);
%
Margolus

neighborhood defined

%upper left cell update

xind

= [1+p:2:nx
-
2+p];

yind

= [1+p:2:ny
-
2+p];

%See if exactly one diagonal is ones

%We can only have one of the following to hold:

diag1(
xind,yind
) = (cells(
xind,yind
)==1) & (cells(xind+1,yind+1)==1) & ...

(cells(xind+1,yind)==0) & (cells(xind,yind+1)==0);

diag2(
xind,yind
) = (cells(xind+1,yind)==1) & (cells(xind,yind+1)==1) & ...

(cells(
xind,yind
)==0) & (cells(xind+1,yind+1)==0);

%This gives the diagonals both not occupied by two particles

andboth
(
xind,yind
) = (diag1(
xind,yind
)==0) &
(diag2(
xind,yind
)==0);

%This gives one diagonal occupied by two particles

orone
(
xind,yind
) = diag1(
xind,yind
) | diag2(
xind,yind
);

%For a given gas particle, check if it is near the boundary

sums(
xind,yind
) = ground(
xind,yind
) | ground(xind+1,yind)
| ...

ground(xind,yind+1) | ground(xind+1,yind+1) ;

%Rules:

%If (no walls) and (diagonals are both empty)

%then there are no particles to swap, so the block stays
the same

cellsNew
(
xind,yind
) = ...

(
andboth
(
xind,yind
) & ~sums(
xind,yind
) &
cells(xind+1,yind+1)) + ...

(
orone
(
xind,yind
) & ~sums(
xind,yind
) &
cells(xind,yind+1)) + ...

(sums(
xind,yind
) & cells(
xind,yind
));

cellsNew
(xind+1,yind) = (
andboth
(
xind,yind
) & ~sums(
xind,yind
) &
cells(xind,yind+1)) + (
orone
(
xind,yind
) & ~sums(
xind,yind
) &
cells(
xind,yind
))+ ...

(sums(
xind,yind
) & cells(xind+1,yind));

%If (no walls) and (only one diagonal occupied)

%then this is representative of a collision
---

treat as though the

%particles hit and deflect each other at 90 degrees, i.e. one diagonal

%is converted to the other on the time step.

cellsNew
(xind,yind+1) = ...

(
andboth
(
xind,yind
) & ~sums(
xind,yind
) & cells(xind+1,yind)) + ...

(
orone
(
xind,yind
) & ~sums(
xind,yind
) & cells(xind+1,yind+1))+ ...

(sums(
xind,yind
) & cells(xind,yind+1));

%If (wall)

%then the cell stays the same in the block (causes a reflection)

cellsNew
(xind+1,yind+1) = ...

(
andboth
(
xind,yind
) & ~sums(
xind,yind
) & cells(
xind,yind
)) + ...

(
orone
(
xind,yind
) & ~sums(
xind,yind
) & cells(xind+1,yind))+ ...

(sums(
xind,yind
) & cells(xind+1,yind+1));

cells =
cellsNew
;

set(
imh
, '
cdata
',
cat(3,z,cells,ground) )

drawnow

waitforbuttonpress
;

end