Normal forms and the structure of resonance sets in nonlinear time-periodic systems

The structure of time-dependent resonances arising in the method of time-dependent normal forms (TDNF) for one and two-degrees-of-freedom nonlinear systems with time-periodic coefficients is investigated. For this purpose, the Liapunov-Floquet (L-F) transformation is employed to transform the periodic variational equations into an equivalent form in which the linear system matrix is time-invariant. Both quadratic and cubic nonlinearities are investigated and the associated normal forms are presented. Also, higher-order resonances for the single-degree-of-freedom case are discussed. It is demonstrated that resonances occur when the values of the Floquet multipliers result in MT-periodic (M = 1, 2, ...) solutions. The discussion is limited to the Hamiltonian case (which encompasses all possible resonances for one-degree-of-freedom). Furthermore, it is also shown how a recent symbolic algorithm for computing stability and bifurcation boundaries for time-periodic systems may also be employed to compute the time-dependent resonance sets of zero measure in the parameter space. Unlike classical asymptotic techniques, this method is free from any small parameter restriction on the time-periodic term in the computation of the resonance sets. Two illustrative examples (one and two-degrees-of-freedom) are included.