A Fractional Delay–Stochastic Framework for Quantum-Inspired Biological Feedback: Stability, Hopf Bifurcation, and Resonance Phenomena

This study introduces a fractional dynamical framework that combines memory effects, time delay, and stochastic environmental fluctuations in a unified mathematical model. Memory is represented through the Caputo fractional derivative, while a discrete delay captures latency in feedback interactions. Random environmental variability is incorporated using an Ornstein–Uhlenbeck stochastic process. The analytical investigation focuses on equilibrium behavior and stability properties of the resulting system. In the absence of delay, stability is determined by the fractional stability sector condition. When delay is introduced, the system may experience a Hopf-type transition, leading to oscillatory dynamics once the delay exceeds a critical threshold. Numerical simulations based on a predictor–corrector scheme illustrate how the interaction between fractional memory and delay reshapes system behavior. The results show that the pair of parameters governing memory strength and delay length plays a central role in determining whether the system converges to equilibrium or develops sustained oscillations. These findings provide a general modeling framework for complex feedback systems where memory, latency, and stochastic fluctuations interact.