How can I limit the state variables of ODE to upper and lower limits ?

Dear friends; I have the differential equations given as follows:

    x1'=x1*(1-x2^2)-x2
    x2'=x1

I have created an m-file which contains these differential equations and I have constrained the variables x1 and x2 to their upper and lower limits (-limit and +limit)

function xdot=fun(t,x)

limit1=2;
limit2=1;

if (abs(x(1))>limit1)
x(1)=sign(x(1))*limit1;
end

if (abs(x(2))>limit2)
  x(2)=sign(x(2))*limit2;
end

 xdot=zeros(2,1);
 xdot(1)=x(1)*(1-x(2)^2)-x(2);
 xdot(2)=x(1);

end

then I have simulated the differential equation defined in the function fun over the interval 0<=t<=20;

    x0=[0;0.25];
    [t,x]=ode45('fun',[0:0.01:20],x0);
    plot(t,x)

My problem consists in how to limit the state variable x because I have the condition

    -2<=x(1)<=2
      -1<=x(2)<=1

So, how can I simulate the differential equation over the interval 0<=t<=20 with satisfying this condition, I will be very grateful if someone can help me, because I have tried but it not works

Thanks

Answer 1

[Jan]

If you do this without event functions and a restart of the integration, you try to integrate a non-smooth function. This collides with the specifications of standard ODE functions such that the results has eitehr an extremely poor accuracy or the integration might even stop. See http://www.mathworks.com/matlabcentral/answers/59582#answer_72047 .

Event functions are the only reliable solution: Detect the limit, stop the integration, change the function to be integrated and restart the ODE solver.