Extended Fundamental Model of Resonance

Fundamental models are the simplest, one degree of freedom Hamiltonians that serve as a tool to understand the qualitative effects of various resonances. A new, extended fundamental model (EFM) is proposed in order to improve the classical, Andoyer type, second fundamental model (SFM). The EFM Hamiltonian differs from the SFM by the addition of a term with the third power of momentum; it depends on two free parameters. The new model is studied for the case of a first-order resonance, where up to five critical points can be present. Similarly, to the respective SFM, it admits only the saddle-node bifurcations of critical points, but its advantage lies in the capability of generating the separatrix bifurcations, known also as saddle connections. The reduction of parameters for the EFM has been performed in a way that allows the use of the model in the case of the so-called abnormal resonance.