On a Discrete-Time Nonlinear System Associated with the Second-Order Digital Filter

This paper presents analysis of extremely simple second-order digital lter. In our analysis we take into account the nonlinearity caused by the overrow eeect. This digital lter corresponds to a nonlinear semidynamical system on the plane depending on two parameters. The symbolic theory for analysing this system is introduced. Several theorems on existence and stability of periodic orbits are given. With these theorems all periodic solutions with low periods are found. The properties of limit sets of the system for diierent values of parameters are discussed. The limit sets are classiied. We single out the regions in the parameter space with various kind of asymptotic trajectory behaviour. In particular we prove the existence of periodic and quasi-periodic limits sets only. In the last section the layout of Arnold tongues on the parameter plane is discussed. Several properties of Arnold tongues are stated.