Adaptive type-1 and type-2 fuzzy-PI control for chaos suppression in 2D and 4D delayed fermionic models

Abstract

Instanton solutions in the Gursey and Thirring models are renowned for their intricate nonlinear dynamics and profound ties to quantum field theory. In the delayed setting, time delay is considered as an effective representation of memory/retardation and feedback-latency effects, which can qualitatively alter stability and attractor structure. When these models incorporate delays, they lead to highly complex and often chaotic behavior, rendering traditional control techniques insufficient. Therefore, adaptive Type-1 fuzzy PI (T1FAC) and adaptive Interval Type-2 fuzzy PI (T2FAC) controllers are developed and compared for chaos suppression in the delayed Gursey and Thirring models. Both controllers tune the PI gains online using a filtered error signal, while T2FAC further introduces a Footprint of Uncertainty to represent membership-evaluation uncertainty and improve robustness. Numerical simulations confirm stabilization at selected chaotic delay settings. Benchmarking against PI, PID, non-adaptive fuzzy PI, and non-adaptive fuzzy PID under identical conditions is reported using channel-wise RMSE and 2% settling-time metrics computed on uniformly resampled trajectories. The Interval Type-2 controller generally yields faster settling and improved tracking accuracy relative to the Type-1 design. A theoretical stability discussion is also provided to support the numerical findings. © 2026 Elsevier B.V.