E.A. Barabanov, V.V. Bykov. Description of the linear Perron effect under parametric perturbations exponentially vanishing at infinity ... P. 31-43

Let ${\cal M}_n$ be the set of linear differential systems of order $n\geqslant 2$ whose coefficients are continuous and bounded on the time semiaxis $\mathbb{R}_+$. Denote by $\lambda_1(A)\leqslant\ldots\leqslant \lambda_n(A)$ the Lyapunov exponents of a system $A\in {\cal M}_n,$ by $\Lambda(A)=(\lambda_1(A),\ldots,\lambda_n(A))$ their spectrum, and by ${\rm es}(A)$ the exponential stability index of $A$ (the dimension of the linear subspace of solutions with negative characteristic exponents). For a system $A\in {\cal M}_n$ and a metric space $M,$ we consider the class ${\cal E}_n[A](M)$ of continuous $(n\times n)$ matrix-valued functions $Q\colon \mathbb{R}_+\times M\to \mathbb{R}^{n\times n}$ satisfying the bound $\|Q(t,\mu)\|\leqslant C_Q\exp(-\sigma_Qt)$ for all $(t,\mu)\in\mathbb{R}_+\times M,$ where $C_Q$ and $\sigma_Q$ are positive constants (possibly different for each function $Q$), and such that the Lyapunov exponents of the system $A+Q,$ which are functions of $\mu\in M$ and are denoted by $\lambda_1(\mu;A+Q)\leqslant\ldots\leqslant \lambda_n(\mu;A+Q),$ are not less than the corresponding Lyapunov exponents of the system $A$; i.e., $\lambda_k(\mu;A+Q)\geqslant \lambda_k(A),$ $k=\overline{1,n},$ for all $\mu\in M$. The problem is to obtain a complete description for each $n\in\mathbb{N}$ and each metric space $M$ of the class of pairs $\bigl(\Lambda(A),\Lambda(\cdot\,;A+Q)\bigr)$ composed of the spectrum $\Lambda(A)\in\mathbb{R}^n$ of a system $A\in {\cal M}_n$ and the spectrum $\Lambda(\cdot\,;A+Q)\colon M\to \mathbb{R}^n$ of a family $A+Q,$ where $A$ ranges over ${\cal M}_n$ and the matrix-valued function $Q$ ranges over the class ${\cal E}_n[A](M)$ for each $A,$ i.e., of the class $\Pi {\cal E}_n(M)=\{\bigl(\Lambda(A),\Lambda(\cdot\,;A+Q)\bigr)\,\vert\, A\in {\cal M}_{n},\,Q\in {\cal E}_n[A](M)\}$. The solution of this problem is provided by the following statement: for each integer $n\geqslant 2$ and every metric space $M$, a pair $\bigl(l,F(\cdot)\bigr),$ where $l=(l_1,\ldots,l_n)\in\mathbb{R}^n$ and $F(\cdot)=(f_1(\cdot),\ldots,f_n(\cdot))\colon M\to \mathbb{R}^n,$ belongs to the class $\Pi {\cal E}_n(M)$ if and only if the following conditions are met: (1) $l_1\leqslant \ldots \leqslant l_n,$ (2) $f_1(\mu)\leqslant \ldots \leqslant f_n(\mu)$ for all $\mu\in M,$ (3) $f_i(\mu)\geqslant l_i$ for all $i=\overline{1,n}$ and $\mu\in M,$ (4) for each $i=\overline{1,n}$, the function $f_i(\cdot)\colon M\to \mathbb{R}$ is bounded and, for any $r\in\mathbb{R}$, the preimage $f_i^{-1}([r,+\infty))$ of the half-interval $[r,+\infty)$ is a $G_{\delta}$-set. The solution of the similar problem of describing the pairs composed of the exponential stability index ${\rm es}(A)\in \{0,\ldots,n\}$ of a system $A$ and the exponential stability index ${\rm es}(\cdot\,;A+Q)\colon M\to \{0,\ldots,n\}$ of a family $A+Q,$ i.e., the class ${\cal I}{\cal E}_n(M)=\{\bigl({\rm es}(A),{\rm es}(\cdot\,;A+Q)\bigr)\,\vert\, A\in {\cal M}_{n},\,Q\in {\cal E}_n[A](M)\}$, is contained in the following statement: for any positive integer $n\geqslant 2$ and every metric space $M$, a pair $\bigl(d,f(\cdot)\bigr),$ where $d\in\{0,\ldots,n\}$ and $f\colon M\to\{0,\ldots,n\},$ belongs to the class ${\cal I}{\cal E}_n(M)$ if and only if $f(\mu)\leqslant d$ for all $\mu\in M$ and, for any $r\in\mathbb{R}$, the preimage $f^{-1}((-\infty,r])$ of the half-interval $(-\infty,r]$ is a $G_{\delta}$-set.

Keywords: linear differential system, Lyapunov exponents, perturbations vanishing at infinity, Baire classes

Received September 30, 2019

Revised November 8, 2019

Accepted November 11, 2019

Evgenii Aleksandrovich Barabanov, Cand. Sci. (Phys.-Math.), Institute of Mathematics of National Academy of Sciences, Minsk, 119991 Belarus, e-mail: bar@im.bas-net.by

Vladimir Vladislavovich Bykov, Cand. Sci. (Phys.-Math.), Lomonosov Moscow State University, Moscow, 119991 Russia, e-mail: vvbykov@gmail.com


1.   Lyapunov A.M. Collected works: in 6 vol. Vol. 2: Obshchaya zadacha ob ustoichivosti dvizheniya [General problem of motion stability)]. Moscow; Leningrad: USSR Academy of Sciences Publ., 1956, 476 p.

2.   Demidovich B.P. Lektsii po matematicheskoi teorii ustoichivosti [Lectures on mathematical stability theory]. Moscow: Nauka Publ., 1967, 472 p.

3.   Perron O. Die Ordnungszahlen linearer Differentialgleichungssysteme. Math Z., 1930, vol. 31, no. 1, pp. 748–766. doi: 10.1007/BF01246445 .

4.   Perron O. Die StabilitЈatsfrage bei Differentialgleichungen. Math Z., vol. 32, no. 1, pp. 703–728. doi: 10.1007/BF01194662 .

5.   Leonov G.A. Khaoticheskaya dinamika i klassicheskaya teoriya ustoichivosti dvizheniya [Chaotic dynamics and classical theory of motion stability]. Izhevsk: RKhD Publ., 2006, 168 p. ISBN: 5-93972-470-1 .

6.   Korovin S.K., Izobov N.A. Realization of the Perron effect whereby the characteristic exponents of solutions of differential systems change their values. Diff. Equat., vol. 46, no. 11, pp. 1537–1551. doi: 10.1134/S0012266110110029 

7.   Barabanov E.A., Bykov V.V., Karpuk M.V. Complete description of the Lyapunov spectra of families of linear differential systems whose dependence on the parameter is continuous uniformly on the time semiaxis. Diff. Equat., 2018, vol. 54, no. 12, pp. 1535–1544. doi: 10.1134/S0012266118120017 

8.   Millionshchikov V.M. Baire classes of functions and Lyapunov exponents. I. Diff. Uravneniya, 1980, vol. 16, no. 8, pp. 1408–1416 (in Russian).

9.   Baire R. Teoriya razryvnykh funktsii [Theory of discontinuous functions]. Moscow; Leningrad: GTTI Publ., 1932, 134 p. ISBN: 978-5-4460-6297-3 . Original French text published in Baire R. Lecons sur les Fonctions Discontinues, Paris: Gauthier-Villars, 1905, 127 p.

10.   Hausdorff F. Set theory. N Y: Chelsea Publ. Company, 1962, 352 p. Translated to Russian under the title Teoriya mnozhestv. Moscow; Leningrad: ONTI Publ., 1937, 306 p.

11.   Barabanov E.A., Bykov V.V. Description of a linear Perron effect under parametric perturbations vanishing at infinity. In: T. F. Filippova, V. I. Maksimov, A. M. Tarasyev (eds.), Stability, Control, Differential Games (SCDG2019): Proc. Internat. Conf. devoted to the 95th anniversary of Academician N.N. Krasovskii (Yekaterinburg, Russia, 16–20 September 2019). Yekaterinburg, 2019, pp. 48–53 (in Russian).

12.   Millionshchikov V.M. Lyapunov exponents as functions of a parameter. Math. USSR-Sb., 1990, vol. 65, no. 2, pp. 369–384. doi: 10.1070/SM1990v065n02ABEH001147 

13.   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. Eq., 2017, vol. 53, no. 12, pp. 1529–1542. doi: 10.1134/S0012266117120011 

14.   Stepanoff W. Sur les suites des fonctions continues. Fund. Math., 1928, vol. 11, no. 1, pp. 264–274.

15.   Karpuk M.V. Lyapunov exponents of families of morphisms of metrized vector bundles as functions on the base of the bundle. Diff. Eq., 2014, vol. 50, no. 10, pp. 1322–1328. doi: 10.1134/S001226611410005X 

16.   Izobov N.A. Lyapunov exponents and stability. Cambridge: Cambridge Scientific Publ., 2012, 353 p. ISBN: 978-1-908106-25-4 . Original Russian text published in Izobov N.A. Vvedenie v teoriyu pokazatelei Lyapunova. Minsk: BGU, 2006, 319 p.

17.   Bylov B.F., Vinograd R.E., Grobman D.M., Nemytskii V.V. Teoriya pokazatelei Lyapunova i ee prilozheniya k voprosam ustoichivosti [Theory of Lyapunov exponents and its application to stability problems]. Moscow: Nauka Publ., 1966, 576 p.

Cite this article as: E.A.Barabanov, V.V.Bykov. Description of the linear Perron effect under parametric perturbations exponentially vanishing at infinity, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 4, pp. 31–43.