Coupled-microresonator systems, enabled by recent breakthroughs in the nanofabrication of low-loss integrated photonic microresonators, deliver significant performance benefits for coherent frequency comb generation beyond single resonators. They also display emergent nonlinear phenomena including soliton hopping—a dynamic process in which solitons periodically transit between coupled resonators. We employ a nonlinear dynamical systems approach, leveraging numerical techniques developed in the context of hydrodynamics, to identify exact periodic orbit solutions of the coupled Lugiato-Lefever equations underlying soliton hopping in photonic dimers and trimers. A bifurcation-theoretic origin of hopping reveals a fundamental difference in dimers and trimers: In dimers, hopping emerges from a branch of stable soliton solutions, while in trimers it originates from an unstable branch. This distinction lowers pump power requirements for trimers, rendering them promising for experimental demonstrations of hopping. We further show that subcritical Hopf bifurcations of unstable equilibria explain hysteresis, coexistence of multiple attractors, and path-dependent access to different dynamical regimes as observed in simulated laser scans mimicking typical experimental investigations. Our findings provide insights into the nonlinear dynamics of coupled multimode microresonators, offering a foundation for the controlled implementation of soliton-based technologies in future integrated photonic systems.