Periodic orbits arising from two-level quantized feedback control

A quantized feedback gives rise to a system of the form x+ = f(x) = ax − q(x), in which q(x) is the quantized feedback. Polynomials with “quantized” coefficients are introduced, and their properties are investigated. With the help of the roots of some interesting groups of polynomials derived from the polynomials with quantized coefficients, we characterize the value for a such that a periodic point of a certain order appears. It is shown that there are lower and upper bounds on a for the existence of a periodic point of a certain order. An exact (minimal) upper bound is also found for periodic points of any order.