Emergent rhythmicity and control in a stochastic trirhythmic system

In the trirhythmic system, the coexistence of three stable limit cycles (LCs) renders the system highly sensitive to subtle input variations. This study meticulously examines how system parameters, filtered feedback control, and Gaussian perturbations impact the system’s bifurcation behaviors. Using the harmonic decomposition method, the deterministic bifurcation behaviors of the trirhythmic system under filtered feedback control are explored. By applying the stochastic averaging method, the stationary probability density function(SPDF) of the system’s amplitude is derived, further facilitating the construction of one-parameter and two-parameter bifurcation diagrams. These diagrams contribute to a thorough understanding of the multirhythmic oscillation phenomenon within the system. It should be noted that the presence of noise with higher intensity significantly disrupts the rhythmicity of the system, while the introduction of filtered feedback control ensures multirhythmic oscillation to a large extent. Furthermore, the magnitudes of parameters α and β profoundly influence the amplitude distribution range: larger values correspond to narrower amplitude distribution ranges, whereas smaller values result in broader distributions. The proposed method in this paper exhibits high precision, and numerical results validate the accuracy of the theoretical analysis.