Bifurcation of resonances in a ring model

We consider a particle in a ring system around a primary body, which is modeled as an ellipsoid. We study the occurrence of bifurcations associated to resonances, which arise at different values of the orbital eccentricity of the particle. The analysis of the bifurcations is first conducted by establishing theoretical conditions ensuring the existence of equilibria. Then, each specific resonance is examined through the study of the associated linearized system; their stability is obtained by computing the eigenvalues of the corresponding Jacobian matrix. Our results reveal a variety of dynamical behaviours characterizing the resonances, including pitchfork, saddle-node and transcritical bifurcations.