IJMNTA  Vol.3 No.1 , March 2014
Reducibility of Periodic Quasi-Periodic Systems
ABSTRACT

In this work, the reducibility of quasi-periodic systems with strong parametric excitation is studied. 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 satisfied to convert a quasi-periodic system in to a constant one. Assuming the reducibility is possible, 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.


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. doi: 10.4236/ijmnta.2014.31002.
References
[1]   Murdock, J.A. (1978) On the Floquet Problem for Quasiperiodic Systems. Proceedings of the American Mathematical Society, 68, 179. http://dx.doi.org/10.1090/S0002-9939-1978-0481275-8

[2]   Jorba, A. and Simó, C. (1992) On the Reducibility of Linear Differential Equations with Quasi-Periodic Coefficients. Journal of Differential Equations, 98, 111-124 http://dx.doi.org/10.1016/0022-0396(92)90107-X

[3]   Moser, J. (1966) On the Theory of Quasi-Periodic Motions. Siam Review, 8, 145-172.
http://dx.doi.org/10.1137/1008035

[4]   Puig, J. (2002) Reducibility of Linear Equations with Quasi-Periodic Coefficients. A Survey.
www.maia.ub.edu/dsg/2002/0201puig.pdf

[5]   Chavaudret, C. (2010) Reducibility of Quasi-Periodic Cocycles in Linear Lie Groups. Ergodic Theory and Dynamical Systems, 31, 741-769 http://dx.doi.org/10.1017/S0143385710000076

[6]   Fink, A.M. (1974) Almost Periodic Differential Equations, Lecture Notes in Math. Springer Verlag, Berlin and New York, 377.

[7]   Richard, R., Randolph, Z. and Rachel, H. (1997) A Quasi-Periodic Mathieu Equation. Series on Stability, Vibration and Control of Systems Series B, 2, 203-221. http://dx.doi.org/10.1142/9789812831132_0009

[8]   Zounes, R. and Rand, R. (1998) Transition Curves for the Quasi-Periodic Mathieu Equation. SIAM Journal on Applied Mathematics, 58, 1094-1115. http://dx.doi.org/10.1137/S0036139996303877

[9]   Xu, J. and Jiang, S. (2011) Reducibility for a Class of Nonlinear Quasi-Periodic Differential Equations with Degenerate Equilibrium Point under Small Perturbation. Ergodic Theory and Dynamical Systems, 31, 599-611.
http://dx.doi.org/10.1017/S0143385709001114

[10]   Arnold, V.I. (1988) Geometrical Methods in the Theory of Ordinary Differential Equations. Springer-Verlag, New York.

[11]   Bogoljubov, N.M., Mitropoliskii, J.A. and Samoilenko, A.M. (1976) Methods of Accelerated Convergence in Nonlinear Mechanics. Springer-Verlag, New York.

[12]   Sinha, S.C. and Butcher, E.A. (1997) Symbolic Computation of Fundamental Solution Matrices for Linear Time Periodic Dynamical Systems. Journal of Sound and Vibration, 206, 61-85.

[13]   Sinha, S.C. and Gourdon, E. (2005) Control of Time-Periodic Systems via Symbolic Computation with Application to Chaos Control. Communications in Nonlinear Science and Numerical Simulation, 10, 835-854.

[14]   Sinha, S.C. and Joseph P. (1994) Control of General Dynamic Systems with Periodically Varying Parameters via Liapunov-Floquet Transformation. Journal of Dynamic Systems, Measurement, and Control, 116, 650-658.
http://dx.doi.org/10.1115/1.2899264

[15]   Wooden, S.M. and Sinha, S.C. (2007) Analysis of Periodic-Quasiperiodic Nonlinear Systems via Lyapunov-Floquet Transformation and Normal Forms. Journal of Nonlinear dynamics, 47, 263-273.

[16]   Belhaq, M., Guennoun, K. and Houssni, M. (2002) Asymptotic Solutions for a Damped Non-Linear Quasi-Periodic Mathieu Equation. International Journal of Non-Linear Mechanics, 37, 445-460.
http://dx.doi.org/10.1016/S0020-7462(01)00020-8

[17]   Puig, J. and Simó, C. (2011) Resonance Tongues in the Quasi-Periodic Hill-Schrodinger Equation with Three Frequencies. Regular and Chaotic Dynamics, 16, 62-79. http://dx.doi.org/10.1134/S1560354710520047

[18]   Broer, H.W. and Simó, C. (1998) Hill’s Equation with Quasi-Periodic Forcing: Resonance Tongues, Instability Pockets and Global Phenomena. Boletim da Sociedade Brasileira de Matemática, 29, 253-293.
http://dx.doi.org/10.1007/BF01237651

 
 
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