Effect of time delay in a bistable synthetic gene network

The essence of logical stochastic resonance is the dynamic manipulation of potential wells. The effect of time delay on the depth of potential wells and the width of a bistable region can be inferred by logic operations in the bistable system with time delay. In a time-delayed synthetic gene network, time delay in the synthesis process can increase the depth of the potential wells, while that in the degradation process, it can reduce the depth of the potential wells, which will result in a decrease in the width of the bistable region (the reason for time delay to induce logic operations without external driving force) and the instability of the system (oscillation). These two opposite effects imply stretching and folding, leading to complex dynamical behaviors of the system, including period, chaos, bubble, chaotic bubble, forward and reverse period doubling bifurcation, intermittency, and coexisting attractors.