Circular periodic orbits, resonance capture and inclination excitation during type II migration

We consider planetary systems evolving under the effect of a Stokes-type dissipative force mimicking the outcome of a type II migration process. As inward migration proceeds and the planets follow the circular family (they start on circular orbits) and even though they are initially almost coplanar, resonance capture can be realized. Then, at the \textit{vertical critical orbits} (VCOs), that the circular family possesses, the inclination excitation can abruptly take place. The planets are now guided by the spatial elliptic families, which bifurcate from those critical orbits. We herein, perform a direct link of mutually inclined stable planetary systems on circular orbits trapped in \textit{mean-motion resonance} (MMR) with the existence of VCOs of high values of multiplicity. It is shown that the more the multiplicity of the periodic orbits of the circular family increases, the more VCOs (corresponding to more MMRs) appear. In this way, we can provide a justification for the existence of resonant planets on circular orbits, which could, even further to that, evolve stably if they were mutually inclined.