I'm interested in the role of the time step for RK methods. I've attached a plot of the standard stability regime here where the x and y axis represent the real and imaginary parts of $$\Delta t \lambda$$ in the equation $$f' = \lambda f$$ and I pinched the RK45 relationship being plotted from somewhere else.
What's curious as this seems to suggest it is possible to move from a region of stability to instability by reducing the step size. In particular I'd like to know what happens when, for fixed values of the real and imaginary parts of $\lambda$, the step size is reduced.
In the plot I initially picked a point in x-y space (blue) which is inside the region of stability. I then assumed that the underlying parts of $\lambda$ were fixed and just reduced $\Delta t$ in effect scaling the first point by 0.9 in both axes, bringing to a point outside the stability region.
Is this calculation/interpretation correct?