Almost all interesting simulations involve some sort of closed-loop feedback, where a change in A causes a change in B which in turn affects A again.

For example, consider the velocity of an object including drag forces from air resistance.

F = -k * V

A = F/m (from F = mA)

dV / dt = A

Here F is the net force on the object, V is the object's velocity, k is the object's wind resistance (newtons of resistance per meter per second of velocity), A is the object's acceleration, and m is the mass.

If drag is the only force involved, and discretizing to first order, we get:

dV = (-k*dt/m) * V

or

V = V + (-k*dt/m) * V = V + speed * V

where we've defined speed = (-k*dt/m).

That is, the change in V itself varies with V. If that speed term (-k*dt/m) is positive, then V will feed back on itself leading to exponential growth. Luckily, speed is always negative because real drag constants, timesteps, and masses are all positive. In the theoretical non-discretized equations, a negative speed term causes V to drop exponentially, though possibly with a managably small exponent. A small negative speed term causes the discrete V to also drop exponentially.

But weird stuff starts happening if the speed term gets negative enough:

- At speed=0, V doesn't change at all.
- At speed=-0.1, V shrinks by 10% per timestep.
- At speed=-0.5, V shrinks by 50% per timestep.
- At speed=-1.0, V = 0 after the first timestep.
- At speed=-1.5, V overshoots zero every timestep, ending each step 50% shorter but facing the other way.
- At speed=-2.0, V overshoots zero completely, and ends up facing exactly the opposite direction with the same magnitude.
- At speed=-2.1, V overshoots zero and gains 10% per timestep!

Note that speed = -k*dt/m, so speed is going to get out of control if:

- k, the drag coefficient, gets too big. It'd be weird that you could simulate air, but not the slower movements expected in water or tar, but it's quite possible.
- m, the object mass, gets too small. The drag coefficient for small objects should probably get small as well.

- dt, the timestep, gets too big. This is common, but luckily we can adjust the simulation timestep easily, as long as our computer is fast enough.