ANALYSIS OF QUASI-LINEAR DYNAMICAL-SYSTEMS WITH PERIODIC COEFFICIENTSVIA LIAPUNOV-FLOQUET TRANSFORMATION

In this paper, a new analysis technique in the study of general quasil inear systems with periodically varying parameters is presented. The m ethod is based on the fact that all quasilinear periodic systems can b e replaced by similar systems whose linear parts are time-invariant, v ia the well-known Liapunov-Floquet (L-F) transformation. A general tec hnique for the computation of the L-F transformation matrices is outli ned. In this technique, the state vector and the periodic matrix of th e linear system equations are expanded in terms of the shifted Chebysh ev polynomials over the principal period. Such an expansion reduces th e original problem to a set of linear algebraic equations from which t he state transition matrix can be constructed over the period as an ex plicit function of time. Application of Floquet theory and use of symb olic software yields the L-F transformation matrix in a form suitable for algebraic manipulations. Once the transformation has been applied, the solution of the resulting system is obtained through an applicati on of the time-dependent normal form theory. The method is suitable fo r both numerical and symbolic computations and in some cases approxima te closed form solutions can be obtained. Two simple examples of quasi linear periodic systems-namely, a commutative system with quadratic no nlinearity and a Mathieu equation with cubic non-linearity-are used to demonstrate the effectiveness of the method. For verification, result s obtained from the proposed technique are compared with the numerical solutions computed using a standard Runge-Kutta type algorithm. It is shown that the present technique is applicable to systems where the p eriodic matrix does not contain a small parameter, which is not the ca se with averaging and perturbation procedures. It can also be used eve n for those systems for which the generating solutions do not exist in the classical sense.