how to draw a trajectory

Hi

i have a differential equation like this:

x'' + x + x^2 = 0

its a chaos problem and i want to display the trajectories.it has two fix point, a saddle at (-1,0) and a repellor at (0,0).

we can transform the equation to ODE by this:

x'= y

y'= -x -x^2

[mostafa]

if true
  x0 = input('Enter the initial position [x y] - ');
  tspan=[0,10*pi];
  [t,x]=ode45(@simple,tspan,x0,[]);
  plot(x(:,1),x(:,2));
  title('Lorenz Model Trajectory XZ')
  xlabel('X')
  ylabel('Z')
end

but it give me error. i write equations in simple.m .

Answer 1

[Mischa Kim]

Mostafa, try something like

 function my_phase()
 IC = rand(10,2);
 hold on
 for ii = 1:length(IC(:,1))
    [~,X] = ode45(@EOM,[0 2],IC(ii,:));
    x = X(:,1);
    y = X(:,2);
    plot(x,y,'r')
 end
 xlabel('x')
 ylabel('y')
 grid
 end
 function dZ = EOM(t, z)
 dZ = zeros(2,1);
 x  = z(1);
 y  = z(2);
 dZ = [  y;...
       - x - x^2];
 end

You can use the IC vector to place the initial conditions to specifically show the dynamics around the fixed points.

[mostafa]

it shows me only the repellor one around the zero. how can i see the saddle one??

[Mischa Kim]

As I pointed, use the IC vector, e.g.,

 IC = bsxfun(@plus, [-1 0], rand(100,2)-0.5*ones(100,2));

You probably need to zoom to display the interesting part of the plot.

[mostafa]

thanks a lot. i got it