| A technique is suggested for the symbolic computation of local stability and bifurcation surfaces for nonlinear time-periodic dynamical systems of arbitrary dimension. This is accomplished via a new symbolic computational technique which allows the state transition matrix (STM) associated with the linearized system to be obtained in closed form as an explicit function of the system parameters and time. By evaluating the STM at the end of the principal internal excitation period, the Floquet transition matrix (FTM) is obtained from which the parameter-dependent local stability/bifurcation relations are derived by employing a variation of the Routh-Hurwitz criteria. In this way, closed form expressions are achieved for primary and secondary codimension 1 bifurcations as well as for the time-dependent resonance sets associated with the method of time-dependent normal forms. In addition, it is suggested how one may apply time-dependent canonical perturbation theory to nonlinear time-periodic Hamiltonian systems in an effort to obtain accurate solutions and bifurcation conditions. For this purpose, the Liapunov-Floquet (L-F) transformation matrix associated with the linearized system is computed and used to transform the time-periodic quasilinear system into an equivalent one in which the linear system matrix is time-invariant. In order to demonstrate the usefulness of these methods, six illustrative examples, viz., a commutative system, a parametrically excited simple pendulum, a double inverted pendulum subjected to a periodic follower force, a gyroscopic system, and two different perturbed Hamiltonian systems are considered. The proposed techniques and results are compared and contrasted with those obtained from the classical asymptotic techniques as well as the method of truncated point mapping. |
