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Specific features of oscillations in adiabatic oscillators with delay

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Abstract

We investigate the features of oscillations in adiabatic oscillators for a delay differential equation. We describe a technique for constructing asymptotic formulas for a class of linear delay differential systems in the vicinity of infinity. In addition, we analyze the dynamics of the transformation of the parametric resonance zone of one adiabatic oscillator for different values delay.

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References

  1. Bellman, R., Stability theory of differential equations, New York: McGraw-Hill, 1953.

    MATH  Google Scholar 

  2. Burd, V.Sh. and Karakulin, V.A., On the asymptotic integration of systems of linear differential equations with oscillatory decreasing coefficients, Math. Notes, 1998, vol. 64, no. 5, p. 571–578.

    Article  MathSciNet  MATH  Google Scholar 

  3. Demidovich, B.P., Lektsii po matematicheskoi teorii ustoichivosti (Lectures on Mathematical Theory of Stability), Moscow: Nauka, 1967.

    Google Scholar 

  4. Coddington, E.A. and Levinson, N., Theory of Ordinary Differential Equations, New York: McGraw-Hill, 1955.

    MATH  Google Scholar 

  5. Mayorov, V.V., Investigation of the stability of solutions of a linear differential equation with aftereffect, encountered in applications, Vestn. Yaroslav. Univ., Issled. Ustoich. Teor. Kolebanii, 1973, no. 5, p. 86–93.

    Google Scholar 

  6. Nesterov, P.N., Averaging method in the asymptotic integration problem for systems with oscillatory-decreasing coefficients, Differ. Equations, 2007, vol. 43, no. 6, p. 745–756.

    Article  MathSciNet  MATH  Google Scholar 

  7. Hale, J.K., Theory of Functional Differential Equations, New York: Springer-Verlag, 1977.

    Book  MATH  Google Scholar 

  8. Burd, V. and Nesterov, P., Parametric resonance in adiabatic oscillators, Results Math., 2010, vol. 58, nos. 1–2, pp. 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  9. Cassel, J.S. and Hou, Z., Asymptotically diagonal linear differential equations with retardation, J. Lond. Math. Soc., 1993, vol. 47, p. 473–483.

    Article  Google Scholar 

  10. Eastham, M.S.P., The asymptotic solution of linear differential systems, in London Math. Soc. Monographs, Oxford: Clarendon Press, 1989.

    Google Scholar 

  11. Harris, W.A., Jr. and Lutz, D.A., On the asymptotic integration of linear differential systems, J. Math. Anal. Appl., 1974, vol. 48, p. 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  12. Harris, W.A., Jr. and Lutz, D.A., A unified theory of asymptotic integration, J. Math. Anal. Appl., 1977, vol. 57, p. 571–586.

    Article  MathSciNet  MATH  Google Scholar 

  13. Nesterov, P., Asymptotic integration of functional differential systems with oscillatory decreasing coefficients, Monatsh. Math., 2013, vol. 171, no. 2, p. 217–240.

    Article  MathSciNet  MATH  Google Scholar 

  14. Wintner, A., The adiabatic linear oscillator, Am. J. Math., 1946, vol. 68, p. 385–397.

    Article  MathSciNet  MATH  Google Scholar 

  15. Wintner, A., Asymptotic integration of the adiabatic oscillator, Am. J. Math., 1946, vol. 69, p. 251–272.

    Article  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Demidov Yaroslavl State University, ul. Sovetskaya 14, Yaroslavl, 150000, Russia

    P. N. Nesterov & E. N. Agafonchikov

Authors
  1. P. N. Nesterov
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  2. E. N. Agafonchikov
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Correspondence to P. N. Nesterov.

Additional information

Original Russian Text © P.N. Nesterov, E.N. Agafonchikov, 2013, published in Modelirovanie i Analiz Informatsionnykh Sistem, 2013, No. 5, pp. 25–44.

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Nesterov, P.N., Agafonchikov, E.N. Specific features of oscillations in adiabatic oscillators with delay. Aut. Control Comp. Sci. 49, 582–596 (2015). https://doi.org/10.3103/S0146411615070160

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  • DOI: https://doi.org/10.3103/S0146411615070160

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