Chaos in digital filters: identification of all periodic symbolic sequences admissible adjacent to zero

We study the symbolic sequences associated with a two-dimensional nonlinear map describing a digital filter with 2's complement arithmetic, and the role of the filter parameter a = 2 cos theta in determining admissibility of these sequences. We solve an appropriately specialized form of Chua and Lin's inequalities governing admissibility, and obtain a complete identification of the infinite class of periodic symbolic sequences admissible in an open interval having parameter value theta = 0 as left end-point. A key step in the argument, easily established by parity for sequences having odd period, requires a more penetrating analysis in the even case. Computer searches for sequences with periods up to 200 have revealed that this class accounts for the greater number of admissible periodic sequences associated with the map. Our solution provides an efficient construction for the periodic symbolic sequences admissible adjacent to zero, parametrized by the period and the number of +- digit pairs present. We also show that the precise interval of admissibility is (0, pi/N), where N is the least period, which permits us to relate the sequences to the points of state space that give rise to them. We thereby obtain a bound on the measure of the subset of state space consisting of the points associated with periodic symbolic sequences, improving previously published bounds. A similar identification of the periodic sequences admissible at the far end of the parameter range, adjacent to. = p, is deduced