Journal of Mathematical Sciences

, Volume 135, Issue 1, pp 2764–2793 | Cite as

Definition and properties of characteristic frequencies of a linear equation

  • I. N. Sergeev


The definition of the characteristic frequencies of zeroes and changes of sign for solutions is given. It is equal to the upper medium (with respect to the time half-axis) of their number on the half-interval of length π. We also define the main frequencies for a linear homogeneous equation of order n. These main frequencies for an equation with constant coefficients coincide with the absolute values of the imaginary parts of the roots of the corresponding characteristic polynomial. It is proved that for the second-order equation the main frequencies are the same for all solutions and that they are stable with respect to uniformly small and infinitely small perturbations of the coefficients. For the third-order equation they can be different, and for any of the main frequencies an example of nonstability is given.


Imaginary Part Linear Equation Characteristic Frequency Small Perturbation Characteristic Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. F. Filippov, Introduction to the Theory of Differential Equations [in Russian], Editorial URSS, Moscow (2004).Google Scholar
  2. 2.
    V. I. Arnold, Ordinary Differential Equations [in Russian], Nauka, Moscow (1984).Google Scholar
  3. 3.
    B. F. Bylov, R. E. Vinograd, D. M. Grobman, and V. V. Nemytskyi, Theory of Liapunov Exponents and Its Applications to the Problems of Stability [in Russian], Nauka, Moscow (1966).Google Scholar
  4. 4.
    A. Yu. Levin, “Non-oscillation of solutions of the equation x (n)+p 1(t)x (n−1)+…+p n(t)x=0,” Usp. Mat. Nauk, 24, No. 2, 43–96 (1969).MATHGoogle Scholar
  5. 5.
    N. L. Korshikova, “On zeroes of solutions of a class of linear nth-order equations,” Differ. Uravn., 15, No. 5, 757–764 (1985).MathSciNetGoogle Scholar
  6. 6.
    I. V. Astashova, “Uniform estimates for positive solutions of quasilinear differential equations of even order,” Tr. Sem. Petrovsk., 25, 3–17 (2005).MathSciNetGoogle Scholar
  7. 7.
    V. M. Millionshchikov, “The proof of the attainability of central exponents,” Sib. Mat. Zh., 10, No. 1, 99–104 (1969).MATHGoogle Scholar
  8. 8.
    B. F. Bylov and N. A. Izobov, “Necessary and sufficient conditions for the stability of characteristic exponents for a linear system,” Differ. Uravn., 5, No. 10, 1794–1803 (1969).MathSciNetGoogle Scholar
  9. 9.
    N. A. Izobov, “On the set of lower exponents for a linear differential system,” Differ. Uravn., 1, No. 4, 469–477 (1965).MATHMathSciNetGoogle Scholar
  10. 10.
    N. A. Izobov, “Linear systems of ordinary differential equations,” in: Results in Science and Technology. Mathematical Analysis, 12, VINITI, Moscow (1974), pp. 71–146.Google Scholar
  11. 11.
    K. A. Dib, “Simultaneous attainability of central exponents,” Differ. Uravn., 10, No. 12, 2125–2136 (1974).MATHMathSciNetGoogle Scholar
  12. 12.
    E. A. Barabanov, “The construction of the set of lower exponents for linear differential systems,” Differ. Uravn., 25, No. 12, 1084–1085 (1989).Google Scholar
  13. 13.
    I. N. Sergeev, “On the theory of Liapunov exponents for linear systems of differential equations,” Tr. Sem. Petrovsk., 9, 111–166 (1983).MATHMathSciNetGoogle Scholar
  14. 14.
    I. N. Sergeev, “On lower characteristic Perron exponents for linear systems,” in: Int. Conf. Dedicated to the 103rd Anniversary of I. G. Petrovskyi: Proceedings, Izd. Mosk. Univ., Moscow (2004), pp. 199–200.Google Scholar
  15. 15.
    I. N. Sergeev, “Calculation of characteristic frequencies for a linear equation,” Differ. Uravn., 40, No. 11, 1573 (2004).Google Scholar
  16. 16.
    I. N. Sergeev, “The mobility of characteristic frequencies of a linear equation under uniformly small and infinitely small perturbations,” Differ. Uravn., 40, No. 11, 1576 (2004).Google Scholar
  17. 17.
    I. N. Sergeev, “On Baire classes of characteristic frequencies of a linear equation,” Differ. Uravn., 41, No. 6, 852 (2005).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • I. N. Sergeev

There are no affiliations available

Personalised recommendations