Consider the following numerical phase plot of a two-dimensional system of ordinary differential equations.
This system was plotted for a relative error tolerance of $10^{-4}$. For tighter tolerances in relative error the phase plot doesn't change, so we can assume that this is a good approximation of the phase plot of the true solution orbit.
The problem, that I don't understand is: If I redo the plot for bigger tolerances, i.e. greater than $10^{-3}$, surprisingly I obtain again a spiral picture.
But for a tolerance $1.1 \cdot 10^{-3}$ I find that the system again looks similar to the one at the top, for tolerance $10^{-4}$. This figure shows that (plottet with tolerance $1.1 \cdot 10^{-3}$):
How can this happen, that for a worse relative error tolerance, my plot suddenly looks better? What qualitative causes could be for such a behavior?
But it gets even weirder: If I increase the tolerance more, to $1.2 \cdot 10^{-3}$ I still have a spiral-like picture (although the spiral gets uglier and uglier), but $1.3 \cdot 10^{-3}$, the system again looks like a tail picture! Then for $1.4 \cdot 10^{-3}$ we again have a spiral picture, for $1.5 \cdot 10^{-3}$ again a tail picture, for $1.6 \cdot 10^{-3}$ again a spiral picture and for $1.7 \cdot 10^{-3}$ onwards again a tail picture. How can this "oscillating" behavior happen ?
Here's the MATLAB code for those persistent enough and with a desire to reproduce the figures above.
function dotY = f(t,Y)
Y100 = Y.^100
H = Y100./(1+Y100)
dotY = [ H(1)+H(2)-2*H(1)*H(2)-0.6*Y(1) ; 1-H(1)*H(2)-0.9*Y(2) ]
end