how to plot a semi-stable solution to autonomous DE

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Will
Will on 10 Mar 2014
Commented: Will on 10 Mar 2014
I have been trying to plot this and am only able to do so for certain initial values of y (y(0)) and can't see why this is the case.
Can anyone tell me how to do this for all values of y(0)?
the equation is: dy/dt =k(1-y)^2.
First, I made an Mfile function for the RHS of this:
>>function ydot=semi(t,y) >>k=2 >>ydot=k*((1-y)^2)
Then, in the command window: >>tspan=[0 20]; >>y0=2; >>[t,y]=ode45('semi',tspan,yo); >>plot(t,y)
As I say, this seems to have worked for values of y_0<1 but when y_0>1, there is an error message reading:
Warning: Failure at t=4.999822e-01. Unable to meet integration tolerances without reducing the step size below the smallest value allowed (8.881784e-16) at time t. > In ode45 at 309

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Accepted Answer

Mischa Kim
Mischa Kim on 10 Mar 2014
Edited: Mischa Kim on 10 Mar 2014
Will, the problem is that the system is unstable for y0 > 1. E.g., for y0 = 2 the derivative
dy/dt = k(1 - y)^2
at t = 0 is dy/dt(0) = 2
is positive and remains positive for all t, therefore y is increasing without bounds. To show (plot) an unstable solution you could decrease the simuation time and initial condition to say
tspan = [0 2];
y0 = 1.1;
All other stable and neutrally stable solutions you can plot without any problems.

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