# American Institute of Mathematical Sciences

March  2014, 34(3): 959-975. doi: 10.3934/dcds.2014.34.959

## A Lie--Deprit perturbation algorithm for linear differential equations with periodic coefficients

 1 Institut de Matemàtiques i Aplicacions de Castelló (IMAC) and Departament de Matemàtiques, Universitat Jaume I, E-12071 Castellón, Spain 2 Institut de Matemàtiques i Aplicacions de Castelló (IMAC), and Departament de Matemàtiques, Universitat Jaume I, E-12071 Castellón, Spain

Received  November 2012 Revised  March 2013 Published  August 2013

A perturbative procedure based on the Lie--Deprit algorithm of classical mechanics is proposed to compute analytic approximations to the fundamental matrix of linear differential equations with periodic coefficients. These approximations reproduce the structure assured by the Floquet theorem. Alternatively, the algorithm provides explicit approximations to the Lyapunov transformation reducing the original periodic problem to an autonomous system and also to its characteristic exponents. The procedure is computationally well adapted and converges for sufficiently small values of the perturbation parameter. Moreover, when the system evolves in a Lie group, the approximations also belong to the same Lie group, thus preserving qualitative properties of the exact solution.
Citation: Fernando Casas, Cristina Chiralt. A Lie--Deprit perturbation algorithm for linear differential equations with periodic coefficients. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 959-975. doi: 10.3934/dcds.2014.34.959
##### References:
 [1] L. Ya. Adrianova, "Introduction to Linear Systems of Differential Equations," American Mathematical Society, Providence, R.I., 1995. [2] S. Blanes, F. Casas, J. A. Oteo and J. Ros, The Magnus expansion and some of its applications, Phys. Rep., 470 (2009), 151-238. doi: 10.1016/j.physrep.2008.11.001. [3] V. Burd, "Method of Averaging for Differential Equations on an Infinite Interval," Chapman & Hall / CRC, 2007. doi: 10.1201/9781584888758. [4] F. Casas, J. A. Oteo and J. Ros, Floquet theory: Exponential perturbative treatment, J. Phys. A: Math. Gen., 34 (2001), 3379-3388. doi: 10.1088/0305-4470/34/16/305. [5] F. Casas, J. A. Oteo and J. Ros, Unitary transformations depending on a small parameter, Proc. R. Soc. A, 468 (2012), 685-700. doi: 10.1098/rspa.2011.0388. [6] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill, 1955. [7] A. Deprit, Canonical transformations depending on a small parameter, Celes. Mech., 1 (1969), 12-30. doi: 10.1007/BF01230629. [8] M. G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Ann. École Norm. Sup., Ser. (2), 12 (1883), 47-89. [9] J. K. Hale, "Ordinary Differential Equations," Krieger Publishing, Florida, 1980. [10] A. Iserles and S. P. Nørsett, On the solution of linear differential equations in Lie groups, Phil. Trans. Royal Soc. A, 357 (1999), 983-1019. doi: 10.1098/rsta.1999.0362. [11] A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods, Acta Numerica, 9 (2000), 215-365. doi: 10.1017/S0962492900002154. [12] S. Klarsfeld and J. A. Oteo, Recursive generation of higher-order terms in the Magnus expansion, Phys. Rev. A, 39 (1989), 3270-3273. doi: 10.1103/PhysRevA.39.3270. [13] E. S. Mananga and T. Charpentier, Introduction of the Floquet-Magnus expansion in solid-state nuclear magnetic resonance spectroscopy, J. Chem. Phys., 135 (2011), 044109. doi: 10.1063/1.3610943. [14] W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math., 7 (1954), 649-673. doi: 10.1002/cpa.3160070404. [15] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, "NIST Handbook of Mathematical Functions," Cambridge University Press, 2010. [16] V. A. Yakubovich and V. M. Starzhinskii, "Linear Differential Equations with Periodic Coefficients," John Wiley & Sons, 1975.

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##### References:
 [1] L. Ya. Adrianova, "Introduction to Linear Systems of Differential Equations," American Mathematical Society, Providence, R.I., 1995. [2] S. Blanes, F. Casas, J. A. Oteo and J. Ros, The Magnus expansion and some of its applications, Phys. Rep., 470 (2009), 151-238. doi: 10.1016/j.physrep.2008.11.001. [3] V. Burd, "Method of Averaging for Differential Equations on an Infinite Interval," Chapman & Hall / CRC, 2007. doi: 10.1201/9781584888758. [4] F. Casas, J. A. Oteo and J. Ros, Floquet theory: Exponential perturbative treatment, J. Phys. A: Math. Gen., 34 (2001), 3379-3388. doi: 10.1088/0305-4470/34/16/305. [5] F. Casas, J. A. Oteo and J. Ros, Unitary transformations depending on a small parameter, Proc. R. Soc. A, 468 (2012), 685-700. doi: 10.1098/rspa.2011.0388. [6] E. A. Coddington and N. Levinson, "Theory of Ordinary Differential Equations," McGraw-Hill, 1955. [7] A. Deprit, Canonical transformations depending on a small parameter, Celes. Mech., 1 (1969), 12-30. doi: 10.1007/BF01230629. [8] M. G. Floquet, Sur les équations différentielles linéaires à coefficients périodiques, Ann. École Norm. Sup., Ser. (2), 12 (1883), 47-89. [9] J. K. Hale, "Ordinary Differential Equations," Krieger Publishing, Florida, 1980. [10] A. Iserles and S. P. Nørsett, On the solution of linear differential equations in Lie groups, Phil. Trans. Royal Soc. A, 357 (1999), 983-1019. doi: 10.1098/rsta.1999.0362. [11] A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods, Acta Numerica, 9 (2000), 215-365. doi: 10.1017/S0962492900002154. [12] S. Klarsfeld and J. A. Oteo, Recursive generation of higher-order terms in the Magnus expansion, Phys. Rev. A, 39 (1989), 3270-3273. doi: 10.1103/PhysRevA.39.3270. [13] E. S. Mananga and T. Charpentier, Introduction of the Floquet-Magnus expansion in solid-state nuclear magnetic resonance spectroscopy, J. Chem. Phys., 135 (2011), 044109. doi: 10.1063/1.3610943. [14] W. Magnus, On the exponential solution of differential equations for a linear operator, Commun. Pure Appl. Math., 7 (1954), 649-673. doi: 10.1002/cpa.3160070404. [15] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, "NIST Handbook of Mathematical Functions," Cambridge University Press, 2010. [16] V. A. Yakubovich and V. M. Starzhinskii, "Linear Differential Equations with Periodic Coefficients," John Wiley & Sons, 1975.
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