Mathematical Problems in Engineering
Volume 2 (1996), Issue 2, Pages 165-190

Solution and stability of a set of PTH order linear differential equations with periodic coefficients via Chebyshev polynomials

S. C. Sinha and Eric A. Butcher

Nonlinear Systems Research Laboratory, Department of Mechanical Engineering, Auburn University, 36849, Alabama, USA

Received 5 April 1995; Revised 11 May 1995

Copyright © 1996 S. C. Sinha and Eric A. Butcher. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Chebyshev polynomials are utilized to obtain solutions of a set of pth order linear differential equations with periodic coefficients. For this purpose, the operational matrix of differentiation associated with the shifted Chebyshev polynomials of the first kind is derived. Utilizing the properties of this matrix, the solution of a system of differential equations can be found by solving a set of linear algebraic equations without constructing the equivalent integral equations. The Floquet Transition Matrix (FTM) can then be computed and its eigenvalues (Floquet multipliers) subsequently analyzed for stability. Two straightforward methods, the ‘differential state space formulation’ and the ‘differential direct formulation’, are presented and the results are compared with those obtained from other available techniques. The well-known Mathieu equation and a higher order system are used as illustrative examples.