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On the Lack of Topologically Essential Values of the Perron Exponent on Solutions to a Linear Differential System with Bounded Coefficients

  • ORDINARY DIFFERENTIAL EQUATIONS
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Abstract

It is proved that for each positive integer \(n\ge 2\) there exist \(n\)-dimensional linear systems of first-order ordinary differential equations with bounded infinitely differentiable coefficients on whose solutions the Perron exponent does not have a single topologically essential value (i.e., the difference between any open set in the space of solutions with compact-open topology and the complete preimage of each Perron exponent value is a set of the second Baire category).

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REFERENCES

  1. Perron, O., Die Ordnungszahlen linearer Differentialgleichungssysteme, Math. Z., 1930, vol. 31, no. 4, pp. 748–766.

    Article  MathSciNet  Google Scholar 

  2. Izobov, N.A., Vvedenie v teoriyu pokazatelei Lyapunova (Introduction to the Theory of Lyapunov Exponents), Ìèíñê: Belarus. Gos. Univ., 2006.

    Google Scholar 

  3. Izobov, N.A., On the measure of the set of solutions to a linear system with the greatest lower exponent, Differ. Uravn., 1988, vol. 24, no. 12, pp. 2168–2170.

    MATH  Google Scholar 

  4. Sergeev, I.N., Topologically typical and essential values of the exponents of linear systems, Differ. Uravn., 2012, vol. 48, no. 11, pp. 1567–1568.

    Google Scholar 

  5. Gargyants, A.G., On the existence of a linear differential system with given values of the Perron exponent, Izv. Math., 2019, vol. 83, no. 2, pp. 214–231.

    Article  MathSciNet  Google Scholar 

  6. 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 Applciations to Problems of Stability), Moscow: Nauka, 1966.

    Google Scholar 

  7. Mazanik, S.A., Preobrazovaniya Lyapunova lineinykh differentsial’nykh sistem (Lyapunov Transformations of Linear Differential Systems), Minsk: Belarus. Gos. Univ., 2008.

    Google Scholar 

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    A. G. Gargyants

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  1. A. G. Gargyants
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Correspondence to A. G. Gargyants.

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Translated by V. Potapchouck

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Gargyants, A.G. On the Lack of Topologically Essential Values of the Perron Exponent on Solutions to a Linear Differential System with Bounded Coefficients. Diff Equat 56, 51–59 (2020). https://doi.org/10.1134/S0012266120010061

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  • DOI: https://doi.org/10.1134/S0012266120010061

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