Symbolic Computation of Quantities Associated with Time-Periodic Dynamical Systems

Abstract

Many dynamical systems can be modeled by a set of linear/nonlinear ordinary differential equations with periodic time-varying coefficients. The state transition matrix S(t,a) associated with the linear part of the equation can be expressed in terms of the periodic Lyapunov-Floquét (L-F) transformation matrix Q(t,a) and a time-invariant matrix R(a). Computation of Q(t,a) and R(a) in symbolic form as a function of system parameters 'a' is of paramount importance in stability, bifurcation analysis, and control system design. In the past, a methodology has been presented for computing S(t,a) in a symbolic form, however Q(t,a) and R(a) have never been calculated in a symbolic form. Since Q(t,a) and R(a) were available only in numerical forms, general results for parameter unfolding and control system design could not be obtained in the entire parameter space. In this work a technique for symbolic computation of Q(t,a), and R(a) matrices is presented. First, S(t,a) is computed symbolically using the shifted Chebyshev polynomials and Picard iteration method suggested in the literature. Then R(a) is computed using an integral quadrature formula. Finally Q(t,a) is computed using the matrix exponential summation method. Using Mathematica, the symbolic computation of Q(t,a) and R(a), associated with the damped Mathieu equation, is presented for stable, unstable, and critical cases. Bifurcation and parameter unfolding is investigated for the critical case, and compared to the results in the literature. The stable case of a linearized inverted double pendulum is presented to illustrate the application to a moderately large system.