Abstract
For families of n-dimensional linear differential systems (n ≥ 2) whose dependence on a parameter ranging in a metric space is continuous in the sense of the uniform topology on the half-line, we obtain a complete description of the ith Lyapunov exponent as a function of the parameter for each i = 1,..., n. As a corollary, we give a complete description of the Lebesgue sets and (in the case of a complete separable parameter space) the range of an individual Lyapunov exponent of such a family.
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Nemytskii, V.V. and Stepanov, V.V., Kachestvennaya teoriya differentsial’nykh uravnenii (Qualitative Theory of Differential Equations), Moscow: Gos. Izd. Tekh. Teor. Lit., 1947.
Millionshchikov, V.M., Formulas for the Lyapunov exponents of linear systems of differential equations, Tr. Inst. Prikl. Mat. im. I.N. Vekua, 1987, vol. 22, pp. 150–179.
Bykov, V.V., Some properties of majorants of Lyapunov exponents for systems with unbounded coefficients, Differ. Equations, 2014, vol. 50, no. 10, pp. 1279–1289.
Millionshchikov, V.M., Baire function classes and Lyapunov indices: I, Differ. Uravn., 1980, vol. 16, no. 8, pp. 1408–1416.
Hausdorff, F., Set Theory, New York: Chelsea Publishing Company, 1962.
Millionshchikov, V.M., Lyapunov exponents as functions of a parameter, Math. USSR Sb., 1990, vol. 65, no. 2, pp. 369–384.
Karpuk, M.V., Lyapunov exponents of families of morphisms of metrized vector bundles as functions on the base of the bundle, Differ. Equations, 2014, vol. 50, no. 10, pp. 1322–1128.
Karpuk, M.V., Structure of the semicontinuity sets of the Lyapunov exponents of linear differential systems continuously depending on a parameter, Differ. Equations, 2015, vol. 51, no. 10, pp. 1397–1401.
Perron, O., Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen, Math. Z., 1928, vol. 29, pp. 129–160.
Izobov, N.A., Lyapunov Exponents and Stability, Cambridge: Cambridge Scientific Publ., 2013.
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 Applications to Stability Problems), Moscow: Nauka, 1966.
Millionshchikov, V.M., Systems with integral division which are everywhere dense in the set of all linear systems of differential equations, Differ. Uravn., 1969, vol. 5, no. 7, pp. 1167–1170.
Rakhimberdiev, M.I., Baire class of the Lyapunov indices, Math. Notes, 1982, vol. 31, no. 6, pp. 467–470.
Vetokhin, A.N., Sharp Baire class of some Lyapunov exponents on the space of linear systems with the compact-open and uniform topologies, Sovrem. Probl. Mat. Mekh., 2015, vol. 9, no. 3, pp. 54–71.
Bykov, V.V., Structure of the sets of points of semicontinuity for the Lyapunov exponents of linear systems continuously depending on a parameter in the uniform norm on the half-line, Differ. Equations, 2017, vol. 53, no. 4, pp. 433–438.
Bykov, V.V. and Salov, E.E., The Baire class of minorants of Lyapunov’s exponents, Moscow Univ. Math. Bull., 2003, vol. 58, no. 1, pp. 36–43.
Vetokhin, A.N., Emptiness of set of points of lower semicontinuity of Lyapunov exponents, Differ. Equations, 2016, vol. 52, no. 3, pp. 272–281.
Hörmander, L., The Analysis of Linear Partial Differential Operators, Berlin: Springer-Verlag, 1983, Vol. 1.
Zorich, V.A., Mathematical Analysis II, Springer-Verlag, 2016.
Kuratowski, K., Topology, New York: Academic, 1966, Vol. 1.
Karpuk, M.V., Lyapunov exponents of generalized Millionshchikov bundles as functions on the base of the bundle, Differ. Uravn., 2016, vol. 52, no. 8, pp. 1140–1141.
Karpuk, M.V., Lyapunov exponents of families of morphisms of generalized Millionshchikov bundles as functions on the base of the bundle, Tr. Inst. Mat. Nats. Akad. Nauk Belarusi, 2016, vol. 24, no. 2, pp. 55–71.
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Original Russian Text © V.V. Bykov, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 12, pp. 1579–1592.
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Bykov, V.V. Functions Determined by the Lyapunov Exponents of Families of Linear Differential Systems Continuously Depending on the Parameter Uniformly on the Half-Line. Diff Equat 53, 1529–1542 (2017). https://doi.org/10.1134/S0012266117120011
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DOI: https://doi.org/10.1134/S0012266117120011