Coexistence of Strange Nonchaotic Attractors in a System with Multistability and Its Application to Optimizing PID Controllers Design for Time-Delay Systems

In this study, we aim to explore the birth mechanism of coexisting Strange Nonchaotic Attractors (SNAs) in a multistability system. The underlying mechanism behind the emergence of coexisting SNAs is understood by examining the interruption of coexisting torus-doubling bifurcations. We identify three different routes that lead to the creation of coexisting SNAs, namely, the intermittent route, Heagy–Hammel route, and fractalization route. To effectively characterize SNAs, we employ several metrics, including the nontrivial largest Lyapunov exponents, phase sensitivity exponents, and power spectrum analysis. Notably, SNAs exhibit a fractal (strange) nature, which has inspired us to develop a novel optimization algorithm termed the Strange Nonchaotic Optimization (SNO) algorithm. This algorithm is used to optimize the Proportional-Integral-Derivative (PID) controller parameters in time-delay systems based on a defined objective function. Simulation results validate that the proposed SNO algorithm presents better performance than the available algorithms in the literature for time-delay systems.