| We analyse periodic solutions of a family of nonlinear two-degree-of-freedom (2DOF) gyroscopic systems with weak dissipation near an equilibrium, as the family phases through a resonance where the system possesses nonsemisimple purely imaginary eigenvalues. This class of problems includes as a special case the restricted three-body problem with weak dissipation at the equilateral equilibrium point ;The discussion focuses around the Hamiltonian-Hopf bifurcation and the generalised Hopf bifurcation, using the Poincare-Birkhoff normal form for the nonsemisimple 1: 1 resonant case. The conditions in the normal form which distinguish between Hamiltonian and non-Hamiltonian systems are determined. We study such a near-Hamiltonian system as a particular degenerate or near-degenerate case of the generic 1: 1 resonant case studied in (42). Moreover, two periodic solutions at the equilibrium point are shown to exist in both Hamiltonian and near-Hamiltonian systems, using the reduced system.;A mathematical model of 2DOF gyroscopic systems with weak dissipation is investigated. Also, a computer-aided proof of the versal canonical form in a 2DOF gyroscopic system is given in this thesis. |
