For the DE $$\frac{dy}{dx} = y^2-x,$$ the long term behavior of the Euler approximations with step sizes $h = 0.5, 0.25, 0.125$ with initial condition $(-1,0)$ all tend to $\infty$. However, the long term behavior of the actual solution with the same initial condition actually tends to $-\infty$.
What would I call this failure——failure of self-consistency, failure of convergence, failure of structural stability, or failure of stability?
I had thought it would be failure of structural stability since in structural stability qualitative behavior of the trajectories is unaffected by small perturbations, and here, even tiny deviations using $h$ values resulted in huge effects. However, apparently that's not the case, and I'm not sure which failure this falls under...