MATLAB: How to save an output to an ODE at t =infinity for various values of a parameter and plot Output Vs Parameter

MATLABode45store

I'm using ODE45 and I'm running a loop on a parameter of an ODE, and I'm trying to plot the output of the parameter against the value of the parameter at t = infinity. My code is
for b = [0.0002:0.0001:0.0006]
fprintf('', b)
g= 0.1
s= 0.3
f = @(t,x) [-b.*x(1)*x(2) + s.*x(3); b.*x(1)*x(2)-g.*x(2); g.*x(2)-s.*x(3)]
[t,xa]=ode45(f,[0 100], [650 50 0]);
plot(t,xa(:,2))
end

Best Answer

The best way would have been to find an analytical solution of ODE and then use the limit() to find value at t=infinity. But for your ODEs, such a solution does not exist. So you will need to rely on numerical methods. You can solve the equation for a long tspan such that the output converges at the end of tspan. The value at t=infinity can be approximated by the convergence value
B = 0.0002:0.0001:0.0006;
g = 0.1;
s = 0.3;
t_inf_values = zeros(size(B));
for i = 1:numel(B)
    b = B(i);
    fprintf('b = %f\n', b) 
    
    f = @(t,x) [-b.*x(1)*x(2) + s.*x(3); b.*x(1)*x(2)-g.*x(2); g.*x(2)-s.*x(3)];
    [t,xa] = ode45(f, [0 1000], [650 50 0]);
    t_inf_values(i) = xa(end,2);
end
plot(B, t_inf_values);
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