Time-reversal symmetric quadratic differential systems having an invariant cylinder: global dynamics and mechanical application

This paper provides a comprehensive analysis of the global dynamics and an exact mechanical application for a class of time-reversal symmetric quadratic differential systems in R³ R 3 having an elliptic cylinder as an invariant algebraic surface. A normal form for this family is first derived. When the polynomial defining the cylinder is a first integral, the phase space is foliated by a continuum of invariant cylinders. For fixed coefficients of the differential system, the initial condition selects a level of this first integral, and the vector fields restricted to the corresponding cylinders undergo pitchfork-type and degenerate topological transitions as the level crosses critical values. In the broader terminology used for systems with first integrals, these phenomena may be interpreted as initial-condition-induced bifurcations without parameters. The critical equilibrium collision is analyzed in regular coordinates and the resulting nilpotent singularities are classified. We also prove that every periodic orbit of the restricted flow on a nondegenerate invariant cylinder belongs to a local period annulus. Consequently, there are no limit cycles on the invariant cylinders, neither when they foliate the phase space nor when there is a single cylinder. Furthermore, we characterize all possible invariant parallels and meridians and describe the compactified flow at infinity, including the precise conditions under which invariant meridians form direct heteroclinic connections between the antipodal directions of the cylinder. Finally, we give an exact embedding into the classical problem of a bead sliding on a rotating hoop, showing that the same equilibrium-collision geometry is realized on the unit cylinder as the classical Hamiltonian pitchfork bifurcation when the angular velocity is varied.