Basins of attraction and Newtons Method.

I am working on basins of attraction for system of two equations and I need to know what is the problem of the code to get error?

the equations:

-7.716*10^-8 + 1.*10^-6*X(1)^2 +X(1)^2*(-0.01 + 0.05*X(2)^2)^2

-4.37*10^-7 + 2.14*10^-7*X(2)^2 + X(2)^2*(-0.007 + 0.21*X(1)^2 - 0.05*X(2)^2)^2

and use the code with newton method but it gives error saying'Error in BasinofAttractiontry1_3_fixedpoint (line 20)

X = NewtonRaphson(X0) ;

I don't know what is the problem exactly.

this is the code of bassin of attraction

clc ; clear all
warning('off') % To off the warning which shows "Matrix is close to singular 
               % badly scaled" when algorithm passes through a point where the Jacobian
               % matrix is singular

% The roots of the given governing equations
r1 = [-0.266794 ;-0.433493] ;
r2 = [0.0330718 ;0.0899008] ;
r3 = [0.260471 ;0.416437] ;
% Initial conditions 
x = linspace(-2,2,200) ;
y = linspace(-2,2,200) ;
% Initialize the required matrices
Xr1 = [] ; Xr2 = [] ; Xr3 = [] ; Xr4 = [] ;
tic 
for i = 1:length(x)
     for j = 1:length(y)
          X0 = [x(i);y(j)] ;
          % Solve the system of Equations using Newton's Method
          X = NewtonRaphson(X0) ;
          % Locating the initial conditions according to error
          if norm(X-r1)<1e-8 
             Xr1 = [X0 Xr1]  ;
          elseif norm(X-r2)<1e-8
             Xr2 = [X0 Xr2] ;
          elseif norm(X-r3)<1e-8
             Xr3 = [X0 Xr3] ;
          else               % if not close to any of the roots
             Xr4 = [X0 Xr4] ;
          end
          
     end 
end
toc
warning('on') % Remove the warning off constraint

% Initialize figure
figure
set(gcf,'color','w') 
hold on
plot(Xr1(1,:),Xr1(2,:),'.','color','r') ;
plot(Xr2(1,:),Xr2(2,:),'.','color','b') ;
plot(Xr3(1,:),Xr3(2,:),'.','color','g') ;
plot(Xr4(1,:),Xr4(2,:),'.','color','k') ;

the newton raphson code

function X = NewtonRaphson(X)

NoIter = 10 ;
% Run a loop for given number of iterations
for j=1:NoIter 
     % Governing equations
     f = [-7.716*1e-8 + 1e-6*X(1)^2 + X(1)^2*(-0.01 + 0.05*X(2)^2)^2; -4.37*1e-7  + 2.14*1e-7*X(2)^2 + X(2)^2*(-0.007 + 0.21*X(1)^2 - 0.05*X(2)^2)^2];  
     % Jacobian Matrix
     Jf=[2.*1e^-6*X(1) + 2*X(1)*(-0.01 + 0.05*X(2)^2)^2 0.197*X(1)^2*X(2)*(-0.01+ 0.05*X(2)^2); 0.846*X(1)*X(2)^2*(-0.007 + 0.21*X(1)^2 - 0.0506*X(2)^2) 4.287*1e^-7*X(2) - 0.202*X(2)^3*(-0.007 + 0.212*X(1)^2 - 0.0506*X(2)^2) + 2*X(2)*(-0.007 + 0.212*X(1)^2 - 0.0506*X(2)^2)^2];   
     % Updating the root 
     X=X-Jf\f; 
end

[ahmed matouk]

How can I change this code to solve ODE system?

Answer 1

[Mischa Kim]

One problem was in the governing equations for f. I cleaned up a bit and this works now:

% Governing equations
f = [-7.716*1e-8 + 1e-6*X(1)^2 + X(1)^2*(-0.01 + 0.05*X(2)^2)^2;...
     -4.37*1e-7  + 2.14*1e-7*X(2)^2 + X(2)^2*(-0.007 + 0.21*X(1)^2 - 0.05*X(2)^2)^2]; 

However, there is another problem in the plots which are probably easy to fix.

[Mohamed Abohamer]

@Mischa Kim thanks for this editing, i do it but still have problem in plot as you said.

Do you have any advices to fix this problem?