Uncertainty quantification in stability analysis of chaotic systems with discrete delays

Time delay is ubiquitous in many real-world physical and biological systems. It typically gives rise to rich dynamic behaviors, from aperiodic to chaotic. The stability of such dynamic behaviors is of considerable interest for process control purposes. While stability analysis under deterministic conditions has been extensively studied, not too many works addressed the issue of stability under uncertainty. Nonetheless, uncertainty, in either modeling or parameter estimation, is inevitable in complex system studies. Even for high-fidelity models, the uncertainty of input parameters could lead to divergent behaviors compared to the deterministic study. This is especially true when the system is at or near the bifurcation point. To this end, we investigated generalized polynomial chaos (GPC) to quantify the impact of uncertain parameters on the stability of delay systems. Our studies suggested that uncertainty quantification in delay systems provides richer information for system stability compared to deterministic analysis. In contrast to the robust yet time-consuming Monte Carlo or Latin hypercube sampling method, GPC approach achieves the same accuracy but only with a fraction of the computational overhead.