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Статья
2018

On Simplified Formulas for the Central Exponents of Differential Systems With Non-Uniform Scales


I. N. SergeevI. N. Sergeev
Русская математика
https://doi.org/10.3103/S1066369X18100079
Abstract / Full Text

We study the Vinograd–Millionshchikov central exponents, which represent the exact outer boundaries for the mobility of the extremal values of the Lyapunov and Perron exponents of a linear differential system under uniformly small perturbations of its coefficients. We prove the possibility of calculating those exponents using simplified formulas with expanding time scales and obtain concrete estimates for the central exponents with simplified ones, calculated in different scales: thick, expanding, slowly expanding and sparse.

Author information
  • Lomonosov Moscow State University, Leninskiye Gory 1, Moscow, 119991, RussiaI. N. Sergeev
References
  1. Bylov, B. F., Vinograd, R. E., Grobman, D. M., Nemytskii, V. V. The Theory of Lyapunov Exponents and its Applications to Problems of Stability (Nauka, Moscow, 1966) [in Russian].
  2. Izobov, N. A. Introduction to the Theory of Lyapunov Exponents (BGU,Minsk, 2006) [in Russian].
  3. Vinograd, R. E. “On a Central Characteristic Exponent of the Systems of Differential Equations”, Matem. sb. 42, No. 2, 207–222 (1957) [in Russian].
  4. Millionshchikov, V. M. “The Proof of Attainability of Central Exponents”, Sib.Math. Z. 10, No. 1, 99–104 (1969) [in Russian].
  5. Vetokhin, A. N. “The Baire Class of Maximal Lower Semicontinuous Minorants of Lyapunov Exponents”, Differ. Equations 34, No. 10, 1313–1317 (1998).
  6. Sergeev, I. N. “The Baire Classes ofMajorant of the Senior Perron Exponent and Minorant of theMinor One to Linear Systems”, Differ. Equations 41, No. 11, 1576 (2005) [in Russian].
  7. Sergeev, I. N. “The Exact Upper Bounds of Mobility of Lyapunov Exponents of a System of Differential Equations and the Behavior of the Exponents Under Perturbations Tending to Zero on Infinity”, Differ. Equations 16, No. 3, 438–448 (1980) [in Russian].
  8. Sergeev, I. N. “On the Universal Formulas for Central Exponents of Linear Systems”, Differ. Equations 52, No. 11, 1589–1590 (2016) [in Russian].
  9. Sergeev, I. N. “The Simplified Central Exponents of Linear Differential Systems in Different Scales”, Differ. Equations 53, No. 11, 1561–1563 (2017) [in Russian].
  10. Sergeev, I. N. “Some Properties of Central Exponents of the Linear Differential Systems”, Abstracts of XVII Int. Conf. On Differential Equations (Eruginskie Chtenia–2017), Minsk, May 16–20, 2017, Part 1, 36–37 (IM NAN Belarusi,Minsk, 2017) [in Russian].
  11. Izobov, N. A. “On a Set of Lower Exponents of a Linear Differential System”, Differ. Equations 1, No. 4, 469–477 (1965) [in Russian].
  12. Izobov, N. A. “The Exponential Exponents of the Linear System and Their Calculation”, Dokl. AN BSSR 26, No. 1, 5–8 (1982) [in Russian].
  13. Millionshchikov, V. M. “The Auxiliary Orbital Exponents in the Growing Time Scales”, Usp.Mat. Mauk 94 (4), 135 (1994) [in Russian].
  14. Sergeev, I. N. “The Extremal Values of the Upper Exponent in the Non-Uniform Time Scales”, Differ. Equations 31, No. 5, 913 (1995) [in Russian].
  15. Makarov, E. K. “A Representation of Partial Exponents of Solutions to Linear Differential Systems via Geometric Progressions”, Differ. Equations 32, No. 12, 1705–1706 (1996).
  16. Barabanov, E. A. “Calculation of Exponents of Solutions of Linear Differential Systems From Time Geometric Progressions”, Differ. Equations 33, No. 12, 1596–1603 (1997).
  17. Lipnitskii, A. V. “On the Question of Calculating the Lyapunov Exponents of Linear Systems by the Time Geometric Progressions”, Differ. Equations 39, No. 11, 1577 (2003) [in Russian].
  18. Dolgov, A. V. “On an Inequality for Lower Exponents of Solutions of Linear Systems”, Differ. Equations 34, No. 11, 1578 (1998) [in Russian].
  19. Perron, O. “Die Ordnungszahlen linearer Differentialgleichungssysteme”, Math. Z. 31, 748–766 (1930).
  20. Millionshchikov, V. M. “The SystemsWith Integral Separation are EverywhereDense in the Set of All Linear Systems of Differential Equations”, Differ. Equations 5, No. 7, 1167–1170 (1969) [in Russian].
  21. Sergeev, I. N. “To the Theory of Lyapunov Exponents of Linear Systems of Differential Equations”, Trudy Semin. im. I. G.Petrovskogo.Vyp.9. 111–166 (1983) [in Russian].