x1(k+1) = x2(k) x2(k+1) = b x1(k) + a x2(k)In practical implementations however, the term b x1(k) + a x2(k) can become too big, resulting in overflow errors. A common overflow in digital circuitry is the 2's complement overflow nonlinearity. Ignoring quantization errors, a second digital filter with 2's complement overflow nonlinearity can be written as
x1(k+1) = x2(k) x2(k+1) = f(b x1(k) + a x2(k))where f(x) = x-2n for -1+2n <= x < 1+2n, for n an integer.
Consider the case when b = -1, a = 1/2. In the Java applet below we plot a aperiodic trajectory, i.e., the points in the trajectory never repeats itself. We see that the trajectory traverses a region which is fractal in appearance.