International Journal of Modern Nonlinear Theory and Application, 2014, 3, 6-14 Published Online March 2014 in SciRes. http://www.scirp.org/journal/ijmnta http://dx.doi.org/10.4236/ijmnta.2014.31002 How to cite this paper: Ezekiel, E. and Redkar, S. (2014) Reducibility of Periodic Quasi-Periodic Systems. International Journal of Modern Nonlinear Theory and Application, 3, 6-14. http://dx.doi.org/10.4236/ijmnta.2014.31002 Reducibility of Periodic Quasi-Periodic Systems Evi Ezekiel, Sangram Redkar Department of Engineering, Arizona State University at Polytechnic Campus, Mesa, USA Email: ssredkar@gmail.com Received 28 September 2013; revised 28 October 2013; accepted 5 November 2013 Copyright © 2014 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract In this work, the reducibility of quasi-periodic systems with strong parametric excitation is stu- died. We first applied a special case of Lyapunov-Perron (L-P) transformation for time periodic system known as the Lyapunov-Floquet (L-F) transformation to generate a dynamically equivalent system. Then, we used the quasi-periodicnear-identity transformation to reduce this dynamically equivalent system to a constant coefficient system by solving homological equations via harmonic balance. In this process, we obtained the reducibility/resonance conditions that needed to be sa- tisfied to convert a quasi-periodic system in to a constant one. Assuming the reducibility is possi- ble, we obtain the L-P transformation that can transform original quasi-periodic system into a system with constant coefficients. Two examples are presented that show the application of this approach. Keywords L-P Transformation; Quasi-Periodic System; Reducibility 1. Introduction A matrix function () At with a square matrix of dimension n is termed quasi-periodic with k incommensura- ble frequencies 1 , , k ω ω  [1] [2]. A quasi-periodic function () f t can be showed in the form () ( ) 1 , , n f t F t t ω ω =  (1) where a continuous function is ( ) 1 , , n F x x  of period 2π in 1 , , n x x  . In addition, we can always assume that 1 , , n ω ω  are independent [2]. As Moser [3] stated, the class of all almost periodic functions is not separa- ble while ( ) 0 1 , , n C ω ω  is. The integral of a quasi-periodic function is not quasi-periodic even if the mean value of () f t is zero [3].