On the analysis of time-periodic nonlinear dynamical systems

Abstract

In this study, a general technique for the analysis of time-period nonlinear dynamical systems is presented. The method is based on the fact that all quasilinear periodic systems can be replaced by similar systems whose linear parts are time invariant via the well-known Liapunov-Floquet (L-F) transformarion. A general procedure for the computation of L-F transformation in terms of Chebyshev polynomials is outlined. Once the transformation has been applied, a periodic orbit in original coordinates has a fixed point representation in the transformed coordinates. The stability and bifurcation analysis of the transformed equations are studied by employing the time-dependent normal form theory and time-dependent centre manifold reduction. For the two examples considered, the three generic codimension-one bifurcations, viz, Hopf, flip and tangent, are analysed. The methodology is semi-analytic in nature and provides a quantitative measure of stability even under critical conditions. Unlike the perturbation or averaging techniques, this method is applicable even to those systems where the periodic term in the linear part does not contain a small parameter or a generating solution does not exist due to the absence of the time-invariant term in the linear part.

Analysis and Control of Nonlinear Attitude Motion of Gravity-Gradient Stabilized Spacecraft via Lyapunov-Floquet Transformation and Normal Forms

“…Further, the the Lyapunov-Floquet (L-F) transformation x t ðÞ¼Q t ðÞ z t ðÞ reduces the original nonautonomous linear differential system to an autonomous one with the form _ z t ðÞ¼Rz t ðÞwhere R is an n  n constant matrix. In the LFT matrix, Q t ðÞcan be approximated via the methodology described by [25,30] using shifted Chebyshev polynomials of the first kind. Chebyshev polynomials are chosen because they produce better approximation and convergence than other special orthogonal polynomials [31].…”