Asynchronous resonance caused by phase delay on adaptive networks

In this paper, we study three states of stochastic resonance of the field-driven Ising model on adaptive networks: normal resonance, transition state and asynchronous resonance. We studied the changes in the position and height of the resonance peak upon the structure evolution probability, and also studied the changes in the size and number of state clusters with the evolution of structure. To make a comparison in the phase between the magnetization and driving field, we plot the time series of the dynamic order parameter in three types of responses. It is found that negative feedback will be generated in the external field if the coevolution of the topological structure and state of the system is strong enough. Further study showed that the asynchronous resonance comes from the $\pi$-shift-like difference of phase between the system and the driving field.