Periodic orbits arising from Delta-modulated feedback control

A Delta-modulated feedback gives rise to a system of the form x+=f(x)=ax−Δ sgn(ax). In this paper, we will determine the a values, 1<|a|<2, for which periodic orbits of each order exist. Polynomials with “sign” coefficients are introduced, and their properties are investigated. With the help of the roots of these polynomials, we characterize the minimal value for |a| such that a periodic point of a certain order first appears. Our results show that even though the topological properties of the tent map and the map f are different, the mechanisms of giving rise to periodic orbits via parameter variations are exactly the same for −2<a<−1, and only “slightly” different for 1<a<2.