Symbolic Bifurcation Boundaries and Time-Dependent Resonance Sets of Time-Periodic Nonlinear Systems

Abstract

Abstract A recent computational technique is utilized for symbolic computation of local stability boundaries and bifurcation surfaces for nonlinear multidimensional time-periodic dynamical systems as an explicit function of the system parameters. This is made possible by the recent development of a symbolic computational algorithm for approximating the parameter-dependent fundamental solution matrix of linear time-periodic systems. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet Transition Matrix (FTM), or the linear part of the Poincaré map, is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation surfaces in the parameter space as polynomials of the system parameters. Because this method is not based on expansion in terms of a small parameter, it can successfully be applied to periodic systems whose internal excitation is strong. In addition, the time-dependent normal forms and resonance sets for one and two degree-of-freedom time-periodic nonlinear systems are analyzed. For this purpose, the Liapunov-Floquet (L-F) transformation is employed which transforms the periodic variational equations into an equivalent form in which the linear system matrix is constant. Both quadratic and cubic nonlinearities are investigated, and all possible cases for the single degree-of-freedom case are studied. The above algorithm for computing stability boundaries may also be employed to compute the time-dependent resonance sets of zero measure in the parameter space. Two illustrative example problems, viz., a parametrically excited simple pendulum and a double inverted pendulum subjected to a periodic follower force, are included.