Skip to main content
SpringerLink
Log in

On stability of solutions of nonlinear nonstationary systems of impulsive differential equations in a critical case

  • Published:
Nonlinear Oscillations

We study the problem of stability of critical equilibrium states for a nonlinear system of impulsive differential equations in a special case. The investigation is carried out on the basis of direct Lyapunov method with the use of two auxiliary functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations [in Russian], Vyshcha Shkola, Kiev (1987).

    Google Scholar 

  2. N. A. Perestyuk, “Stability of solutions of impulsive linear systems,” Vestn. Kiev. Univ., Ser. Mat. Mekh., No. 19, 71–76 (1977).

  3. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore (1989).

    MATH  Google Scholar 

  4. N. A. Perestyuk, “On stability of the equilibrium position of impulsive systems,” GOD na VUZ, Prilozh. Mat., Sofia, 11, Issue 1, 145–150 (1976).

    MATH  Google Scholar 

  5. A. O. Ignat’ev, O. A. Ignat’ev, and A. A. Soliman, “Asymptotic stability and instability of the solutions of systems with impulse action,” Math. Notes, 80, No. 4, 491–499 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  6. A. O. Ignat’ev, “On the stability of invariant sets of systems with impulse effect,” Nonlin. Anal., 69, 53–72 (2008).

    Article  MathSciNet  Google Scholar 

  7. A. A. Martynyuk and V. I. Slyn’ko, “On stability of motion of a nonlinear impulsive system,” Prikl. Mekh., 40, No. 2, 112–122 (2004).

    MathSciNet  Google Scholar 

  8. M. O. Perestyuk and O. S. Chernikova, “Some modern aspects of the theory of impulsive differential equations,” Ukr. Mat. Zh., 60, No. 1, 81–94 (2008); English translation: Ukr. Math. J., 60, No. 1, 91–107 (2008).

    Article  MathSciNet  Google Scholar 

  9. O. S. Chernikova, “A reduction principle for systems of differential equations that have impulses,” Ukr. Mat. Zh., 34, No. 5, 601–607 (1982); English translation: Ukr. Math. J., 34, No. 5, 487–492 (1982).

    Article  MathSciNet  Google Scholar 

  10. V. I. Arnol’d and Yu. S. Il’yashenko, “Ordinary differential equations,” in: Dynamical Systems–1, VINITI Series in Contemporary Problems of Mathematics, Fundamental Trends [in Russian], Vol. 1, VINITI, Moscow (1985), pp. 7–140.

    Google Scholar 

  11. S. V. Babenko and V. I. Slyn’ko, “Stability of motion of nonlinear impulsive systems in critical cases,” Dopov. Nats. Akad. Nauk Ukr., No. 6, 46–52 (2008).

    Google Scholar 

  12. V. I. Slyn’ko, “Construction of Poincaré mappings for a holonomic mechanical system with two degrees of freedom with impacts,” Prikl. Mekh., 44, No. 5, 115–122 (2008).

    MathSciNet  MATH  Google Scholar 

  13. R. I. Gladilina and A. O. Ignat’ev, “On the stability of periodic impulsive systems,” Mat. Notes, 76, No. 1, 41–47 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  14. A. I. Dvirnyi and V. I. Slyn’ko, “Stability of solutions to impulsive differential equations in critical cases,” Sib. Math. J., 52, No. 1, 54–62.

Download references

Author information

Authors and Affiliations

  1. Institute of Mechanics, Ukrainian National Academy of Sciences, Kiev, Ukraine

    A. I. Dvirnyi & V. I. Slyn’ko

Authors
  1. A. I. Dvirnyi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. I. Slyn’ko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. I. Dvirnyi.

Additional information

Translated from Neliniini Kolyvannya, Vol. 14, No. 4, pp. 445–467, October–December, 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dvirnyi, A.I., Slyn’ko, V.I. On stability of solutions of nonlinear nonstationary systems of impulsive differential equations in a critical case. Nonlinear Oscill 14, 472–496 (2012). https://doi.org/10.1007/s11072-012-0171-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11072-012-0171-7

Keywords

Search

Navigation