Complex behavior in digital filters with overflow nonlinearity: analytical results

In this paper we present more analytical results about the complex behavior of a second order digital filter with overflow nonlinearity. We explore the parameter space to obtain a taxonomy of the different behaviors that occurs. In particular, we give a complete description of the chaotic behavior of the map F (which models the second order digital filter) in the parameter space (a,b). We prove that in the region R¯₅ (the closure of R₅, where R₅={(a,b):b<-a+1, b<a+1, b>-1}) F is not chaotic; in the region |b|<1 and (a,b)∉R¯₅, F has a generalized hyperbolic attractor; and in the region |b|>1, if (a,b) are integers and b=-2(a-1), then F is an exact map. In addition, we obtain some results concerning the fractal behavior of the map F. We find an estimate of the Hausdorff dimension of the generalized hyperbolic attractor. We obtain results regarding the symbolic dynamics of F. For example, we prove that the set of points with aperiodic admissible sequences in the case |a|<2 and b=-1 is uncountable