Directions: From Flagstaff Take State Highway 89 North, then turn right on 160 toward Tuba City. The location of the villiage of Moenkopi, and the Dinosaur Tracks site are about a mile on the left. This is 5 miles West of Tuba City, not East, on 160. Caution, some maps have not correctly located this villiage. It is on the north side of 160.

Dinosau Eggs

Biggest Foot print found in Moekopi

Dinosau Trial

### Chaos, Lyapunov, and entropy increase-MATLAB CODE

## Saturday, October 10, 2009

**Chaos, Lyapunov, and entropy increase.**

Chaotic dynamical systems have sensitive dependence on initial conditions. Thisiscommonly described as the ‘butterﬂy eﬀect’ (due to Lorenz of the Lorenz attractor): the eﬀects of the ﬂap of a butterﬂy’s wings in Brazil build up with time until months later a tornado in Texas could be launched. In this exercise, we will see this sensitive dependence for a particular system (the logistic map)and measure the sensitivity by deﬁning the Lyapunov exponents. The logistic map takes the interval (0, 1) into itself:

f(x)=4μx(1 − x),

where the time evolution is given by iterating the

map:

x0,x1,x2,... = x0,f(x0),f(f(x0)),....

In particular, for μ = 1 it precisely folds the unit interval in half, and stretches it (non-uniformly)to cover the original domain. The mathematics community lumps together continuous dynamical evolution laws and discrete mappings as both being dynamical systems.

The general stretching and folding exhibited by our map is often seen in driven physical systems without conservation laws.In this exercise, we will focus on values of μ near one, where the motion is mostly chaotic. Chaos is sometimes deﬁned as motion where the ﬁnal position depends sensitively on the initial conditions. Two trajectories, starting a distance "delta" apart, will typically drift apart in time as "delta*eλt", where "λ" is the Lyapunov exponent for the chaotic dynamics. Start with μ =0.9 and two nearby points x0 and "y0 = x0 + delta" somewhere between zero and one. Investigate the two trajectories x0,f(x0),f(f(x0)),...,f[n] (x0) and

y0,f(y0),... . How fast do they separate? Why do they stop separating? Estimate the Lyapunov exponent. (Hint: "delta" can be a few times the precision of the machine. so long as you are not near the maximum value of f at x0 =0.5.) Many Hamiltonian systems are also chaotic. Two conﬁgurations of classical atoms or billiard balls, with initial positions and velocities that are almost identical, will rapidly diverge as the collisions magnify small initial deviations in angle and velocity into large ones. It is this chaotic stretching, folding, and kneading of phase space

that is at the root of our explanation that entropy increases

**MATLAB CODE**

close all, clear all, clc

avalues=0.6:0.0001:1;

N=1000;

a=avalues;

x0 = [];

for i=1:length(avalues)

x0=[x0 rand(1)];

% x0=0.1;

end

x(1,:)=x0;

for n=1:.3*N

x(n+1,:)=4*a.*x(n,:).*(1-x(n,:));

end

figure

for n=.3*N:N

x(n+1,:)=4*a.*x(n,:).*(1-x(n,:));

plot(a,x(n+1,:),'.','MarkerSize',4.5) %plot x versus a

hold on

end

zoom

title('F(x) versus a' )

xlabel('a')

ylabel('F(x)')

clear all

muvalues=0.001:0.00000005:1;

muvalues=.9;

N=120;

mu=muvalues;

x0 = [];

y0=[];

e=1e-8;

k=.8;

x(1)=k;

y(1)=k+e;

figure

for n=1:N-1

x(n+1)=4*mu.*x(n).*(1-x(n));

y(n+1)=4*mu.*y(n).*(1-y(n));

plot(x(n),x(n+1),'.') %plot x(n+1) versus x(n)

hold on

end

title('x(n+1) versus x(n) ')

xlabel('x(n)')

ylabel('x(n+1)')

figure

hist(x,100)

title('Histogram of F(x)')

figure

l=1:N;

z=x-y;

plot(l,(x),'.-r'); %plot F(x) and F(y) versus N

hold on

plot(l,y,'-xk');

zoom

title(['F(x) and F(y) versus N (No.of steps)'])

xlabel('N')

ylabel(['F(x)-Red, F(y)-Black','e-initial difference- ',num2str(e)])

figure

plot(l,(abs((z/1))),'.-k') %plot F(x)-F(y) versus N

zoom

title(['[F(x)-F(y)] versus N (No of Steps)','e-initial difference- ',num2str(e)])

xlabel('N')

ylabel('F(x)-F(y)')

figure

g=(((log(abs(z)/1)))); %plot log [F(x)-F(y)]

p=polyfit(l,g,1);

y3=p(1)*l+p(2);

lamda=p(1)

plot(l,g,'.-','MarkerSize',0.5)

hold on

plot(l,y3,'r')

title(lamda)

title(['Log(abs((F(x)-F(y)))) versus N (No of Steps), ',' Lyapunov exponent- ',num2str(lamda)])

xlabel('N')

ylabel('Log(abs( (F(x)-F(y))) ) ')

Labels:
Education

### Random walk-MatLab Code

One can eﬃciently generate and analyze random

walks on the computer.

(a)

Write a routine to generate an N-step random walk in d dimensions, with each step uniformly distributed in the range (−1/2, 1/2) in each dimension. (Generate the steps ﬁrst as an N × d array, then do a cumulative sum.) Plot xt versus t for a few 10 000-step random walks. Plot x versus y for a few two-dimensional random walks, with N = 10, 1000, and 100 000.(Try to keep the aspect ratio of the XY plot equal to one.) Does multiplying the number of steps by one hundred roughly increase the net distance by ten?

(b)

Write a routine to calculate the endpoints of W random walks with N steps each in d dimensions. Do a scatter plot of the endpoints of 10 000 random walks with N = 1 and 10, superimposed on the same plot. Notice that the longer random walks are distributed in a circularly symmetric pattern, even though the single step random walk N =1 has a square probability distribution

(c)

Calculate the RMS step-size a for onedimensional steps uniformly distributed in (−1/2, 1/2). Write a routine that plots a histogram of the endpoints of W one-dimensional random walks with N steps and 50 bins, along with the prediction of eqn 2.35 for x in (−3σ, 3σ). Do a histogram with W = 10 000 and N = 1, 2, 3, and 5. How quickly does the Gaussian distribution become a good approximation to the random walk?

N=3; % no of trials

k=100; %no of walks

t3=get(handles.dim,'string');

d=2; %dimension

s=0.5; %step size

z=(2*s).*(rand(d,N))-s;

vec=[zeros(1,d); cumsum(z')];

%x=[zeros(1,d); cumsum(z(2,N))]

colorstr=['b' 'r' 'g' 'y'];

for i=1:k

z=(2*s).*(rand(d,N))-s;

if N==1

for j=1:d

D(i,j)=cumsum(z(j,N));

%G(j)=cumsum(z(j,N));

K(i)=i;

end

else

vec=[zeros(1,d); cumsum(z')];

X(i)=vec(N+1,1);

if d>1

Y(i)=vec(N+1,2);

end

K(i)=i;

%plot(x(N+1,1),x(N+1,2),'.')

end

end

if N==1

% L=[0 ,0;1 ,D(k,1) ]

figure

% plot(L(:,1),L(:,2),'.-r')

plot(cumsum(z(1,:)'),'.-')

title(['Xt versus t, ','( Number of Steps- ',int2str(N),')'])

xlabel('t')

ylabel('Xt-X position')

figure

if d>1

plot(D(:,1),D(:,2),'.b')

else

plot(D(:,1)','.b')

end

axis([-2*s 2*s -2*s 2*s])

title(['X versus Y, ','( Number of Steps- ',int2str(N),')'])

xlabel('X position')

ylabel('Y position')

else

figure

%M=[0 ,0;K' ,X' ] ;

%plot(M(:,1),M(:,2),'.-b')

plot(cumsum(z(1,:)'),'.-')

title(['Xt versus t, ','( Number of Steps- ',int2str(N),')'])

xlabel('t-Number of walks')

ylabel('Xt-X position')

grid on

figure

plot(cumsum(z(1,:)'),cumsum(z(2,:)'),'.-b')

title(['X versus Y, ','( Number of Steps- ',int2str(N),')'])

xlabel('X position')

ylabel('Y position')

grid on;

end

% PART(B)>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

N=10;

k=10000;

for i=1:k

z=(2*s).*(rand(d,N))-s;

vec=[zeros(1,d); cumsum(z')];

X(i)=vec(N+1,1);

if d>1

Y(i)=vec(N+1,2);

end

K(i)=i;

%plot(x(N+1,1),x(N+1,2),'.')

end

figure

if d>1

plot(Y,X,'.g')

else

plot(X,'.g')

end

title(['X versus Y, ','( Number of Steps- ',int2str(N),')'])

xlabel('X position')

ylabel('Y position')

hold on

grid on

N=1;

k=10000;

for i=1:k

z=(2*s).*(rand(d,N))-s;

for j=1:d

D(i,j)=cumsum(z(j,N));

%G(j)=cumsum(z(j,N));

K(i)=i;

end

end

plot(D(:,1),D(:,2),'.b')

%%axis([-2*s 2*s -2*s 2*s])

title(['X versus Y, ','( Number of Steps- ',int2str(N),')'])

xlabel('X position')

ylabel('Y position')

%PART C>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

N=1;

k=10000;

d=1;

for i=1:k

N=1;

z=(2*s).*(rand(d,N))-s;

H1(i)=z;

N=2;

z=(2*s).*(rand(d,N))-s;

t=cumsum(z');

H2(i)=t(N);

N=3;

z=(2*s).*(rand(d,N))-s;

t=cumsum(z');

H3(i)=t(N);

N=5;

z=(2*s).*(rand(d,N))-s;

t=cumsum(z');

H4(i)=t(N);

end

figure

h1=subplot(4,1,1);

hist(H1,50);

[mu,s]=normfit(H1)

normal=normpdf(H1,mu,s);

h2=subplot(4,1,2);

hist(H2,50);

h3=subplot(4,1,3);

hist(H3,50);

h4=subplot(4,1,4);

hist(H4,50);

walks on the computer.

(a)

Write a routine to generate an N-step random walk in d dimensions, with each step uniformly distributed in the range (−1/2, 1/2) in each dimension. (Generate the steps ﬁrst as an N × d array, then do a cumulative sum.) Plot xt versus t for a few 10 000-step random walks. Plot x versus y for a few two-dimensional random walks, with N = 10, 1000, and 100 000.(Try to keep the aspect ratio of the XY plot equal to one.) Does multiplying the number of steps by one hundred roughly increase the net distance by ten?

(b)

Write a routine to calculate the endpoints of W random walks with N steps each in d dimensions. Do a scatter plot of the endpoints of 10 000 random walks with N = 1 and 10, superimposed on the same plot. Notice that the longer random walks are distributed in a circularly symmetric pattern, even though the single step random walk N =1 has a square probability distribution

(c)

Calculate the RMS step-size a for onedimensional steps uniformly distributed in (−1/2, 1/2). Write a routine that plots a histogram of the endpoints of W one-dimensional random walks with N steps and 50 bins, along with the prediction of eqn 2.35 for x in (−3σ, 3σ). Do a histogram with W = 10 000 and N = 1, 2, 3, and 5. How quickly does the Gaussian distribution become a good approximation to the random walk?

**MATLAB CODE**N=3; % no of trials

k=100; %no of walks

t3=get(handles.dim,'string');

d=2; %dimension

s=0.5; %step size

z=(2*s).*(rand(d,N))-s;

vec=[zeros(1,d); cumsum(z')];

%x=[zeros(1,d); cumsum(z(2,N))]

colorstr=['b' 'r' 'g' 'y'];

for i=1:k

z=(2*s).*(rand(d,N))-s;

if N==1

for j=1:d

D(i,j)=cumsum(z(j,N));

%G(j)=cumsum(z(j,N));

K(i)=i;

end

else

vec=[zeros(1,d); cumsum(z')];

X(i)=vec(N+1,1);

if d>1

Y(i)=vec(N+1,2);

end

K(i)=i;

%plot(x(N+1,1),x(N+1,2),'.')

end

end

if N==1

% L=[0 ,0;1 ,D(k,1) ]

figure

% plot(L(:,1),L(:,2),'.-r')

plot(cumsum(z(1,:)'),'.-')

title(['Xt versus t, ','( Number of Steps- ',int2str(N),')'])

xlabel('t')

ylabel('Xt-X position')

figure

if d>1

plot(D(:,1),D(:,2),'.b')

else

plot(D(:,1)','.b')

end

axis([-2*s 2*s -2*s 2*s])

title(['X versus Y, ','( Number of Steps- ',int2str(N),')'])

xlabel('X position')

ylabel('Y position')

else

figure

%M=[0 ,0;K' ,X' ] ;

%plot(M(:,1),M(:,2),'.-b')

plot(cumsum(z(1,:)'),'.-')

title(['Xt versus t, ','( Number of Steps- ',int2str(N),')'])

xlabel('t-Number of walks')

ylabel('Xt-X position')

grid on

figure

plot(cumsum(z(1,:)'),cumsum(z(2,:)'),'.-b')

title(['X versus Y, ','( Number of Steps- ',int2str(N),')'])

xlabel('X position')

ylabel('Y position')

grid on;

end

% PART(B)>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

N=10;

k=10000;

for i=1:k

z=(2*s).*(rand(d,N))-s;

vec=[zeros(1,d); cumsum(z')];

X(i)=vec(N+1,1);

if d>1

Y(i)=vec(N+1,2);

end

K(i)=i;

%plot(x(N+1,1),x(N+1,2),'.')

end

figure

if d>1

plot(Y,X,'.g')

else

plot(X,'.g')

end

title(['X versus Y, ','( Number of Steps- ',int2str(N),')'])

xlabel('X position')

ylabel('Y position')

hold on

grid on

N=1;

k=10000;

for i=1:k

z=(2*s).*(rand(d,N))-s;

for j=1:d

D(i,j)=cumsum(z(j,N));

%G(j)=cumsum(z(j,N));

K(i)=i;

end

end

plot(D(:,1),D(:,2),'.b')

%%axis([-2*s 2*s -2*s 2*s])

title(['X versus Y, ','( Number of Steps- ',int2str(N),')'])

xlabel('X position')

ylabel('Y position')

%PART C>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

N=1;

k=10000;

d=1;

for i=1:k

N=1;

z=(2*s).*(rand(d,N))-s;

H1(i)=z;

N=2;

z=(2*s).*(rand(d,N))-s;

t=cumsum(z');

H2(i)=t(N);

N=3;

z=(2*s).*(rand(d,N))-s;

t=cumsum(z');

H3(i)=t(N);

N=5;

z=(2*s).*(rand(d,N))-s;

t=cumsum(z');

H4(i)=t(N);

end

figure

h1=subplot(4,1,1);

hist(H1,50);

[mu,s]=normfit(H1)

normal=normpdf(H1,mu,s);

h2=subplot(4,1,2);

hist(H2,50);

h3=subplot(4,1,3);

hist(H3,50);

h4=subplot(4,1,4);

hist(H4,50);

Labels:
Education

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