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//Example 10.3.1 Page 357
//Non-Linear Dynamics and Chaos, First Indian Edition Print 2007
//Steven H. Strogatz
clear;
clear;
clc;
close;
set(gca(),"auto_clear","off") //hold on
//Taking r=2;
r=2;
x=poly(0,"x");
f = x-2*(x^2); //Defining Polynomial--> f(x*)-x* = 2*x(1-x)-x. Let this be f(x)
disp("Fixed Points are :")
y = roots(f)
disp("The fixed point x*=1-(1/r) does not exists for r<1, Since x(n+1)<0 and population cannot be negative.")
lambda1=r-2*r*y(1) //f'(x*) = r-2rx*
lambda2=r-2*r*y(2)
disp("Since, lambda1=2>1, thus orign is Unstable.")
disp("Since, lambda2=0<1, thus x*=1-(1/r) is Stable.")
//Number of points graphically :
r1=3; //r>1
r2=1; //r=1, tangential case
r3=0.5; //r<1
for xn=0:0.05:1
xn_one=r1*xn*(1-xn);
plot2d(xn,xn_one,style=-3)
xn_one=r2*xn*(1-xn);
plot2d(xn,xn_one,style=-3)
xn_one=r3*xn*(1-xn);
plot2d(xn,xn_one,style=-3)
y=xn; // to draw y=x line
plot2d(xn,y,style=-4)
end
xtitle("Graph Showing Number of Fixed Points for differnent values of r","x-Axis ( xn )","y-Axis ( xn+1 )")
//Similarly, check for Stability by changing r.
//End of Example.
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