In this chapter we will consider linear systems of the form x′ = A(t)x and discuss various procedures which may be used for transforming such a system (if possible) into an L-diagonal form, so that the theorems in Chap. 2 could be used to obtain an asymptotic representation for solutions.
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F.V. Atkinson, The asymptotic solution of second-order differential equations. Ann. Mat. Pura Appl.
37, 347–378 (1954)
Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations. Universitext (Springer, New York, 2000)
S. Bodine, D.A. Lutz, Asymptotic analysis of solutions of a radial Schrödinger equation with oscillating potential. Math. Nachr.
279, 1641–1663 (2006)
S. Bodine, D.A. Lutz, On asymptotic equivalence of perturbed linear systems of differential and difference equations. J. Math. Anal. Appl.
326, 1174–1189 (2007)
Method of Averaging for Differential Equations on an Infinite Interval: Theory and Applications. Lecture Notes in Pure and Applied Mathematics, vol. 255 (Chapman & Hall/CRC, Boca Raton, 2007)
V.S. Burd, P. Nesterov, Parametric resonance in adiabatic oscillators. Results Math.
58, 1–15 (2010)
J.S. Cassell, The asymptotic integration of some oscillatory differential equations. Q. J. Math. Oxford Ser. (2)
33, 281–296 (1982)
J.S. Cassell, The asymptotic integration of a class of linear differential systems. Q. J. Math. Oxford Ser. (2)
43, 9–22 (1992)
E.A. Coddington, N. Levinson,
Theory of Ordinary Differential Equations (McGraw-Hill Book Company Inc., New York/Toronto/London, 1955)
Stability and Asymptotic Behavior of Differential Equations (D. C. Heath and Co., Boston, 1965)
A. Devinatz, An asymptotic theorem for systems of linear differential equations. Trans. Am. Math. Soc.
160, 353–363 (1971)
M.S.P. Eastham, The asymptotic solution of linear differential systems. Mathematika
32, 131–138 (1985)
The Asymptotic Solution of Linear Differential Systems. Applications of the Levinson Theorem (Oxford University Press, Oxford, 1989)
H. Gingold, Almost diagonal systems in asymptotic integration. Proc. Edinb. Math. Soc.
28, 143–158 (1985)
W.A. Harris, D.A. Lutz, On the asymptotic integration of linear differential systems. J. Math. Anal. Appl.
48, 1–16 (1974)
W.A. Harris, D.A. Lutz, Asymptotic integration of adiabatic oscillators. J. Math. Anal. Appl.
51(1), 76–93 (1975)
W.A. Harris, D.A. Lutz, A unified theory of asymptotic integration. J. Math. Anal. Appl.
57(3), 571–586 (1977)
P. Hartman, A. Wintner, Asymptotic integrations of linear differential equations. Am. J. Math.
77, 45–86, 404, 932 (1955)
P.F. Hsieh, F. Xie, Asymptotic diagonalization of a linear ordinary differential system. Kumamoto J. Math.
7, 27–50 (1994)
P.F. Hsieh, F. Xie, Asymptotic diagonalization of a system of linear ordinary differential equations. Dyn. Continuous Discrete Impuls. Syst.
2, 51–74 (1996)
P.F. Hsieh, F. Xie, On asymptotic diagonalization of linear ordinary differential equations. Dyn. Contin. Discrete Impuls. Syst.
4, 351–377 (1998)
N. Levinson, The asymptotic nature of solutions of linear differential equations. Duke Math. J.
15, 111–126 (1948)
R. Medina, M. Pinto, Linear differential systems with conditionally integrable coefficients. J. Math. Anal. Appl.
166, 52–64 (1992)
Real and Complex Analysis (McGraw-Hill, New York, 1966)
Y. Sibuya, A block-diagonalization theorem for systems of linear ordinary differential equations and its applications. SIAM J. Appl. Math.
14, 468–475 (1966)
W. Trench, Extensions of a theorem of Wintner on systems with asymptotically constant solutions. Trans. Am. Math. Soc.
293, 477–483 (1986)
W. Trench, Asymptotic behavior of solutions of Poincaré recurrence systems. Comput. Math. Appl.
28, 317–324 (1994)
W. Trench, Asymptotic behavior of solutions of asymptotically constant coefficient systems of linear differential equations. Comput. Math. Appl.
30, 111–117 (1995)
Asymptotic Expansions for Ordinary Differential Equations (Dover Publications, New York, 1987)
A. Wintner, On a theorem of Bôcher in the theory of ordinary linear differential equations. Am. J. Math.
76, 183–190 (1954)
V.A. Yakubovich, V.M. Starzhinskii,
Linear Differential Equations with Periodic Coefficients 1, 2 (Halsted Press/Wiley, New York/Toronto, 1975). Israel Program for Scientific Translations, Jerusalem/London
Über dieses Kapitel
Conditioning Transformations for Differential Systems