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On the improperness sets of families of linear differential systems

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Abstract

We consider families of linear differential systems continuously depending on a real parameter with continuous (or piecewise continuous) coefficients on the half-line. The improperness set of such a family is defined as the set of all parameter values for which the corresponding systems in the family are Lyapunov improper. We show that a subset of the real axis is the improperness set of some family if and only if it is a G δσ -set. The result remains valid for families in which the matrices of the systems are bounded on the half-line. Almost the same result holds for families in which the parameter occurs only as a factor multiplying the system matrix: their improperness sets are the G δσ -sets not containing zero. For families of the last kind with bounded coefficient matrix, we show that their improperness set is an arbitrary open subset of the real line.

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References

  1. Lyapunov, A.M., Sobranie sochinenii. T. 2 (Collected Works, Vol. II), Moscow: Izdat. Akad. Nauk SSSR, 1956.

    Google Scholar 

  2. Bylov, B.F., Vinograd, R.E., Grobman, D.M., and Nemytskii, V.V., Teoriya pokazatelei Lyapunova i ee prilozheniya k voprosam ustoichivosti (Theory of Lyapunov Exponents and Its Application to Problems of Stability), Moscow: Nauka, 1966.

    Google Scholar 

  3. Adrianova, L.Ya., Vvedenie v teoriyu lineinykh sistem differentsial’nykh uravnenii (Introduction to the Theory of Linear Systems of Differential Equations), St. Petersburg: Izdat. St. Petersburg. State Univ., 1992.

    Google Scholar 

  4. Izobov, N.A. and Makarov, E.K., Lyapunov Improper Linear Systems with a Parameter Multiplying the Derivative, Differ. Uravn., 1988, vol. 24, no. 11, pp. 1870–1880.

    MathSciNet  Google Scholar 

  5. Izobov, N.A., Investigations in Belarus in the Theory of Characteristic Lyapunov Exponents and Its Applications, Differ. Uravn., 1993, vol. 29, no. 12, pp. 2034–2055.

    MathSciNet  Google Scholar 

  6. Makarov, E.K., Improperness Sets of Linear Systems with a Parameter Multiplying the Derivative, Differ. Uravn., 1988, vol. 24, no. 12, pp. 2091–2098.

    Google Scholar 

  7. Makarov, E.K., Linear Systems with Improperness Sets of Full Measure, Differ. Uravn., 1989, vol. 25, no. 2, pp. 209–212.

    Google Scholar 

  8. Leonov, G.A., Khaoticheskaya dinamika i klassicheskaya teoriya ustoichivosti dvizheniya (Chaotic Dynamics and Classical Theory of Motion Stability), Moscow, 2006.

  9. Makarov, E.K., The Measure of the Improperness Set of a Linear System with a Parameter Multiplying the Derivative, Dokl. Akad. Nauk BSSR, 1989, vol. 33, no. 4, pp. 302–305.

    MATH  MathSciNet  Google Scholar 

  10. Daletskii, Yu.L. and Krein, M.G., Ustoichivost’ reshenii differentsial’nykh uravnenii v banakhovom prostranstve (Stability of Solutions of Differential Equations in Banach Space), Moscow: Nauka, 1970.

    Google Scholar 

  11. Perron, O., Über eine Matrixtransformation, Math. Z., 1930, vol. 32, no. 3, pp. 465–473.

    Article  MATH  MathSciNet  Google Scholar 

  12. Diliberto, S.P., On Systems of Ordinary Differential Equations, Contr. to the Theory of Nonlinear Oscillations. Ann. of Math. Stud., 1950, no. 20, pp. 1–38.

  13. Vinograd, R.E., A New Proof of Perron’s Theorem and Certain Properties of Proper Systems, Uspekhi Mat. Nauk, 1954, vol. 9, no. 2 (60), pp. 129–136.

    MATH  MathSciNet  Google Scholar 

  14. Petrovskii, I.G., Lektsii po teorii obyknovennykh differentsial’nykh uravnenii (Lectures on the Theory of Ordinary Differential Equations), Moscow: Nauka, 1970.

    Google Scholar 

  15. Hausdorff, F., Set Theory, Berlin: Walter de Gruyter, 1935. Translated under the title Teoriya mnozhestv, Moscow: ONTI, 1937.

    MATH  Google Scholar 

  16. Millionshchikov, V.M., Baire Classes of Functions and Lyapunov Exponents. I, Differ. Uravn., 1980, vol. 16, no. 8, pp. 1408–1416.

    MathSciNet  Google Scholar 

  17. Barabanov, E.A., Singular Exponents and Properness Criteria for Linear Differential Systems, Differ. Uravn., 2005, vol. 41, no. 2, pp. 147–157.

    MathSciNet  Google Scholar 

  18. Rakhimberdiev, M.I., Baire Class of Lyapunov Exponents, Mat. Zametki, 1982, vol. 31, no. 6, pp. 925–931.

    MathSciNet  Google Scholar 

  19. Basov, V.P., On the Structure of a Solution of a Proper System, Vestnik Leningr. Univ., 1952, no. 12, pp. 3–8.

  20. Bogdanov, Yu.S., Lyapunov’s Norms in Linear Spaces, Dokl. Akad. Nauk SSSR, 1957, vol. 113, no. 2, pp. 255–257.

    MathSciNet  Google Scholar 

  21. Husemoller, D., Fibre Bundles, New York, 1966. Translated under the title Rassloennye prostranstva, Moscow: Mir, 1970.

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences, Minsk, Belarus

    E. A. Barabanov

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  1. E. A. Barabanov
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Original Russian Text © E.A. Barabanov, 2009, published in Differentsial’nye Uravneniya, 2009, Vol. 45, No. 8, pp. 1067–1084.

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Barabanov, E.A. On the improperness sets of families of linear differential systems. Diff Equat 45, 1087–1104 (2009). https://doi.org/10.1134/S0012266109080011

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  • DOI: https://doi.org/10.1134/S0012266109080011

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