Arnold tongues and devil´s staircase in a digital filter employing saturation arithmetic

The authors analyse the dynamic behaviors encountered in a digital filter employing a saturation-type adder overflow characteristic. The influence of changes of filter parameters on its dynamics is studied in detail. A two-parameter bifurcation diagram has been constructed, revealing an extremely rich and complex structure of Arnold tongues in the parameter space. These tongues are composed of a sequence of self-similar, `sausage-like´ regions. In one-parameter analyses, devil´s staircase changes of winding numbers of encountered orbits can be observed. Mathematical considerations show that the tongues do not overlap and the system cannot exhibit chaotic behavior. However, it is possible to find oscillations with any arbitrarily chosen period