Bifurcation and chaos in digital filters: identification of periodic solutions

Dear editor, It is well known that second-order digital filters with two’s complement algorithm may exhibit complex dynamics, including periodic solutions and chaos. In various eminent papers [1–3], the periodic and chaotic trajectories of secondorder digital filters have been studied. Though for certain filter parameters digital filters may exhibit periodic behavior, the relationship between the filter parameters, initial point, trajectory period, and motion pattern has not been fully studied [4–6]. This study has also received widespread attention from researchers in various disciplines [7, 8]. We classify the periodic behavior of secondorder digital filters and discuss the relationship between filter parameters, periods of periodic orbits, and trajectory travel patterns. A complete classification of the periodic behavior is given. Interesting dynamic behavior has been demonstrated through meticulous simulation studies. Topological transformation of the system. We study the two-dimensional nonlinear map F that maps a point x = (x1, x2) T ∈ I = [−1, 1)×[−1, 1) into I = [−1, 1)× [−1, 1), defined as