Complex Dynamics of Systems Under Delta-Modulated Feedback

Periodicity in Delta-modulated feedback control

“…The results on the existence of periodic orbits of system (25) can be elegantly stated and rigorously proved: every orbit of system (25) is eventually periodic with a prime period of the form 2(2j + 1) for some integer j, 0 j k, such that 2(2j + 1) divides k − j. A byproduct of these results is: in case of multiple periods, all the smaller periods divide the maximal period.…”

“…The first extension to the higher-dimensional case is the Sigma-Delta modulation with multiple delays [25]:…”

Δ-Modulated Feedback in Discretization of Sliding Mode Control

“…Here, we will only be concerned with the case |a| ≤ 1. The existence of periodic points for the case |a| > 1 has been discussed in (Gai, Xia and Chen, 2003;Xia, Chen, Gai and Zinober, 2004).…”

“…An early implementation of ∆-modulated feedback control is the transmitting power control of a mobile unit in the Direct Sequence Code Division Multiple Access (DS-CDMA) cellular network (Ariyavisitakul and Chang, 1991), due to the requirement that only one bit of datum is allowed for the implementation of the power controller. More recent studies, motivated by the renewed interest in hybrid system with hard nonlinearities, include ∆-modulated feedback control systems and the associated complexities Gai, Xia and Chen, 2003;Xia, Chen, Gai and Zinober, 2004).…”

Improving the Security of Chaotic Synchronization With a $\Delta$-Modulated Cryptographic Technique

“…Another reason is that the result can be easily extended to high-dimensional -modulated control systems, which will make the cryptosystem more secure. The complex behavior of this simple control system due to -modulated feedback has been investigated in [10], [21]- [24]. When some parameter in this particular 1-D discrete system, the system is chaotic ( [23], [24]) but not a self-map.…”