RK4 is about as unstable. The backwards Euler method is baby's first stable ODE solver. If you want to make RK4 stable, you have to change it into the implicit

RK4 is about as unstable. The backwards Euler method is baby's first stable ODE solver. If you want to make RK4 stable, you have to change it into the implicit RK4 method. However, the Lorenz system isn't stiff, which is usually what you use stable solvers for, nor any of the other systems here, I believe. RK4 should be fine, or even normal Euler with a reasonable step size. The chaos you see in these systems is not due to numerical inaccuracy. They're inherently chaotic. That's what makes chaos a subject worth studying.