Analysis of quasilinear dynamical systems with periodic coefficients via Liapunov—Floquet transformation

Abstract

In this paper, a new analysis technique in the study of general quasilinear systems with periodically varying parameters 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) transformation. A general technique for the computation of the L-F transformation matrices is outlined. In this technique, the state vector and the periodic matrix of the linear system equations are expanded in terms of the shifted Chebyshev polynomials over the principal period. Such an expansion reduces the original problem to a set of linear algebraic equations from which the state transition matrix can be constructed over the period as an explicit function of time. Application of Floquet theory and use of symbolic 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 application of the time-dependent normal form theory. The method is suitable for both numerical and symbolic computations and in some cases approximate closed form solutions can be obtained. Two simple examples of quasilinear periodic systems—namely, a commutative system with quadratic nonlinearity and a Mathieu equation with cubic non-linearity—are used to demonstrate the effectiveness of the method. For verification, results 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 periodic matrix does not contain a small parameter, which is not the case with averaging and perturbation procedures. It can also be used even for those systems for which the generating solutions do not exist in the classical sense.