Skip to main content

Conditioning Transformations for Differential Systems

  • Chapter
  • 1102 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 2129)

Abstract

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.

Keywords

  • Trigonometric Polynomial
  • Approximate Equation
  • High Order Derivative
  • Fundamental Matrix
  • Dichotomy Condition

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   59.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   79.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. F.V. Atkinson, The asymptotic solution of second-order differential equations. Ann. Mat. Pura Appl. 37, 347–378 (1954)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. W. Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations. Universitext (Springer, New York, 2000)

    MATH  Google Scholar 

  3. S. Bodine, D.A. Lutz, Asymptotic analysis of solutions of a radial Schrödinger equation with oscillating potential. Math. Nachr. 279, 1641–1663 (2006)

    CrossRef  MATH  MathSciNet  Google Scholar 

  4. 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)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. V.S. Burd, 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)

    Google Scholar 

  6. V.S. Burd, P. Nesterov, Parametric resonance in adiabatic oscillators. Results Math. 58, 1–15 (2010)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. J.S. Cassell, The asymptotic integration of some oscillatory differential equations. Q. J. Math. Oxford Ser. (2) 33, 281–296 (1982)

    Google Scholar 

  8. J.S. Cassell, The asymptotic integration of a class of linear differential systems. Q. J. Math. Oxford Ser. (2) 43, 9–22 (1992)

    Google Scholar 

  9. E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill Book Company Inc., New York/Toronto/London, 1955)

    MATH  Google Scholar 

  10. W.A. Coppel, Stability and Asymptotic Behavior of Differential Equations (D. C. Heath and Co., Boston, 1965)

    MATH  Google Scholar 

  11. A. Devinatz, An asymptotic theorem for systems of linear differential equations. Trans. Am. Math. Soc. 160, 353–363 (1971)

    MATH  MathSciNet  Google Scholar 

  12. M.S.P. Eastham, The asymptotic solution of linear differential systems. Mathematika 32, 131–138 (1985)

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. M.S.P. Eastham, The Asymptotic Solution of Linear Differential Systems. Applications of the Levinson Theorem (Oxford University Press, Oxford, 1989)

    Google Scholar 

  14. H. Gingold, Almost diagonal systems in asymptotic integration. Proc. Edinb. Math. Soc. 28, 143–158 (1985)

    CrossRef  MATH  MathSciNet  Google Scholar 

  15. W.A. Harris, D.A. Lutz, On the asymptotic integration of linear differential systems. J. Math. Anal. Appl. 48, 1–16 (1974)

    CrossRef  MATH  MathSciNet  Google Scholar 

  16. W.A. Harris, D.A. Lutz, Asymptotic integration of adiabatic oscillators. J. Math. Anal. Appl. 51(1), 76–93 (1975)

    CrossRef  MATH  MathSciNet  Google Scholar 

  17. W.A. Harris, D.A. Lutz, A unified theory of asymptotic integration. J. Math. Anal. Appl. 57(3), 571–586 (1977)

    CrossRef  MATH  MathSciNet  Google Scholar 

  18. P. Hartman, A. Wintner, Asymptotic integrations of linear differential equations. Am. J. Math. 77, 45–86, 404, 932 (1955)

    Google Scholar 

  19. P.F. Hsieh, F. Xie, Asymptotic diagonalization of a linear ordinary differential system. Kumamoto J. Math. 7, 27–50 (1994)

    MATH  MathSciNet  Google Scholar 

  20. P.F. Hsieh, F. Xie, Asymptotic diagonalization of a system of linear ordinary differential equations. Dyn. Continuous Discrete Impuls. Syst. 2, 51–74 (1996)

    MATH  MathSciNet  Google Scholar 

  21. P.F. Hsieh, F. Xie, On asymptotic diagonalization of linear ordinary differential equations. Dyn. Contin. Discrete Impuls. Syst. 4, 351–377 (1998)

    MATH  MathSciNet  Google Scholar 

  22. N. Levinson, The asymptotic nature of solutions of linear differential equations. Duke Math. J. 15, 111–126 (1948)

    CrossRef  MATH  MathSciNet  Google Scholar 

  23. R. Medina, M. Pinto, Linear differential systems with conditionally integrable coefficients. J. Math. Anal. Appl. 166, 52–64 (1992)

    CrossRef  MATH  MathSciNet  Google Scholar 

  24. W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1966)

    MATH  Google Scholar 

  25. Y. Sibuya, A block-diagonalization theorem for systems of linear ordinary differential equations and its applications. SIAM J. Appl. Math. 14, 468–475 (1966)

    CrossRef  MATH  MathSciNet  Google Scholar 

  26. W. Trench, Extensions of a theorem of Wintner on systems with asymptotically constant solutions. Trans. Am. Math. Soc. 293, 477–483 (1986)

    CrossRef  MATH  MathSciNet  Google Scholar 

  27. W. Trench, Asymptotic behavior of solutions of Poincaré recurrence systems. Comput. Math. Appl. 28, 317–324 (1994)

    CrossRef  MATH  MathSciNet  Google Scholar 

  28. W. Trench, Asymptotic behavior of solutions of asymptotically constant coefficient systems of linear differential equations. Comput. Math. Appl. 30, 111–117 (1995)

    CrossRef  MATH  MathSciNet  Google Scholar 

  29. W. Wasow, Asymptotic Expansions for Ordinary Differential Equations (Dover Publications, New York, 1987)

    MATH  Google Scholar 

  30. A. Wintner, On a theorem of Bôcher in the theory of ordinary linear differential equations. Am. J. Math. 76, 183–190 (1954)

    CrossRef  MATH  MathSciNet  Google Scholar 

  31. 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

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Bodine, S., Lutz, D.A. (2015). Conditioning Transformations for Differential Systems. In: Asymptotic Integration of Differential and Difference Equations. Lecture Notes in Mathematics, vol 2129. Springer, Cham. https://doi.org/10.1007/978-3-319-18248-3_4

Download citation