A perturbative algorithm for quasi-periodic linear systems close to constant coefficients

A perturbative procedure is proposed to formally construct analytic solutions for a linear differential equation with quasi-periodic but close to constant coefficients. The scheme constructs the necessary linear transformations involved in the reduction process up to an arbitrary order in the perturbation parameter. It is recursive, can be implemented in any symbolic algebra package and leads to accurate analytic approximations sharing with the exact solution important qualitative properties. This algorithm can be used, in particular, to carry out systematic stability analyses in the parameter space of a given system by considering variational equations.

[1]  V. A. I︠A︡kubovich,et al.  Linear differential equations with periodic coefficients , 1975 .

[2]  V. I. Arnol'd,et al.  PROOF OF A THEOREM OF A.?N.?KOLMOGOROV ON THE INVARIANCE OF QUASI-PERIODIC MOTIONS UNDER SMALL PERTURBATIONS OF THE HAMILTONIAN , 1963 .

[3]  Vladimir Burd Method of Averaging for Differential Equations on an Infinite Interval: Theory and Applications , 2017 .

[4]  C. Simó,et al.  Analytic families of reducible linear quasi-periodic differential equations , 2006, Ergodic Theory and Dynamical Systems.

[5]  F. Casas,et al.  A LIE{DEPRIT PERTURBATION ALGORITHM FOR LINEAR DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS , 2013 .

[6]  A. M. Samoilenko,et al.  Methods of Accelerated Convergence in Nonlinear Mechanics , 1976 .

[7]  J. Cary LIE TRANSFORM PERTURBATION THEORY FOR HAMILTONIAN SYSTEMS , 1981 .

[8]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[9]  Àngel Jorba,et al.  On the reducibility of linear differential equations with quasiperiodic coefficients , 1992 .

[10]  Ronald F. Boisvert,et al.  NIST Handbook of Mathematical Functions , 2010 .

[11]  H. Broer,et al.  On a quasi-periodic Hopf bifurcation , 1987 .

[12]  R Bellman LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS , 1985 .

[13]  J. Villanueva,et al.  Effective reducibility of quasi-periodic linear equations close to constant coefficients , 1997 .

[14]  Randolph S. Zounes,et al.  Transition Curves for the Quasi-Periodic Mathieu Equation , 1998, SIAM J. Appl. Math..

[15]  Averaging method in the asymptotic integration problem for systems with oscillatory-decreasing coefficients , 2007 .

[16]  H. Broer,et al.  Resonance and Fractal Geometry , 2012 .

[17]  New perturbation algorithms for time-dependent quantum systems , 1997 .

[18]  F. Casas,et al.  Unitary transformations depending on a small parameter , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[19]  W. Magnus On the exponential solution of differential equations for a linear operator , 1954 .

[20]  Constantin Corduneanu,et al.  Almost periodic functions , 1968 .

[21]  A. Lichtenberg,et al.  Regular and Stochastic Motion , 1982 .

[22]  S. Blanes,et al.  The Magnus expansion and some of its applications , 2008, 0810.5488.

[23]  Maurice Roseau,et al.  Vibrations non linéaires et théorie de la stabilité , 1966 .