Hamiltonian and dissipative second-order polynomial flows on spheres S^2

The dynamics of nondissipative and dissipative autonomous Bose-Hubbard dimers is considered in second-order polynomial approximation as flow dynamics on the Bloch sphere. Special emphasis is laid on the stationary-point and singularity structure of the flows, related underlying algebraic stability features encoded in 10th-order homogeneous polynomials describing algebraic discriminant varieties over 3-dimensional projective parameter spaces. Reduced resolvent techniques, hidden Jordan block structures and relations to singularity theory provide further insights into the dynamics and possibly existing limit cycles.