Some stability analysis of a non-linear time-delayed feedback circuit

A key focus of this paper is on understanding stability in a non-linear circuit design subject to delayed feedback. The system we consider is an autonomous first order RC circuit, coupled with a non-linear element, with a fixed feedback delay. With time delays, stability is a key concern. So for the non-linear circuit model we first establish a sufficient condition for local stability. We then establish necessary and sufficient conditions for local stability. Additionally we also exhibit that when local stability is just lost, a Hopf type bifurcation would occur. For the non-linear system, this bifurcation signifies the emergence of limit cycles which can degrade performance. Practically, it is useful to obtain an estimate of the amplitude and frequency of the bifurcating periodic solutions. Such estimates are calculated using the method of harmonic balance. In addition to the local stability analysis, we also establish some global stability results and some conditions to ensure that the solutions of the governing non-linear system are bounded. Some of the theoretical insights obtained are complemented with numerical simulations and also with experiments which are carried out using off-the-shelf components. The combination of theory, both local and global, simulations and experiments serve to enhance our understanding of the underlying model.