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Hyperbolicity criterion for periodic solutions of functional-differential equations with several delays

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Abstract

In this paper, a hyperbolicity criterion for periodic solutions of nonlinear functional-differential equations is constructed in terms of zeros of the characteristic function. In the earlier papers in this area, necessary and sufficient conditions were different from each other. Moreover, it was assumed that if the period of the investigated solution is irrational, then that solution admits a rational approximation. In this paper, we obtain necessary and sufficient conditions of the hyperbolicity. It is proved (and the proof is constructive) that a rational approximation exists for any irrational period. All the results are obtained for the case of several rational delays.

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References

  1. O. Arino and A. Chérif, “More on ordinary differential equations which yield periodic solutions of differential delay equations,” J. Math. Anal. Appl., 180, No. 2, 361–385 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  2. S. N. Chow, O. Diekmann, and J. Mallet-Paret, “Stability, multiplicity and global continuation of symmetric periodic solutions of a nonlinear Volterra integral equation,” Japan J. Indust. Appl. Math., 2, 433–469 (1985).

    MATH  MathSciNet  Google Scholar 

  3. S. N. Chow and H.-O. Walther, “Characteristic multipliers and stability of symmetric periodic solutions of {ie708-01},” Trans. Amer. Math. Soc., 307, No. 1, 127–142 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  4. O. Diekmann, S. van Gils, S. M. Verduyn Lunel, and H.-O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York (1995).

    MATH  Google Scholar 

  5. J. Dieudonné, Foundations of Modern Analysis [Russian translation], Mir, Moscow (1964).

    MATH  Google Scholar 

  6. N. Dunford and J. T. Schwartz, Linear Operators, John Wiley & Sons, New York (1988).

    Google Scholar 

  7. I. C. Gohberg and E. I. Sigal, “An operator generalization of the logarithmic residue theorem and the theorem of Rouche,” Sb. Math., 13, 603–625 (1971).

    Article  MATH  Google Scholar 

  8. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York (1993).

    MATH  Google Scholar 

  9. J. L. Kaplan and J. A. Yorke, “Ordinary differential equations which yield periodic solutions of delay differential equations,” J. Math. Anal. Appl., 48, No. 1, 317–324 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  10. T. Kato, Perturbation Theory for Linear Operators [Russian translation], Mir, Moscow (1972).

    MATH  Google Scholar 

  11. S. G. Krein, Linear Equations in Banach Spaces, Birkhäuser, Boston (1982).

    MATH  Google Scholar 

  12. G. Sell and J. Mallet-Paret, “Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions,” J. Dynam. Differential Equations, 125, 385–440 (1996).

    MATH  MathSciNet  Google Scholar 

  13. A. L. Skubachevskii and H.-O. Walther, “On the Floquet multipliers of periodic solutions to nonlinear functional differential equations,” J. Dynam. Differential Equations, 18, No. 2, 257–355 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  14. H.-O. Walther, “Hyperbolic periodic solutions, heteroclinic connections and transversal homoclinic points in autonomous differential delay equations,” Mem. Amer. Math. Soc., 79, No. 402 (1989).

    Google Scholar 

  15. H.-O. Walther and A. L. Skubachevskii, “On the spectrum of the monodromy operator for slowly oscillating periodic solutions to functional differential equations,” Dokl. Math., 65, No. 3, 442–445 (2002).

    MathSciNet  Google Scholar 

  16. H.-O. Walther and A. L. Skubachevskii, “On Floquet multipliers for slowly oscillating periodic solutions of nonlinear functional-differential equations,” Trans. Moscow Math. Soc., 1–44 (2003).

  17. H.-O. Walther and A. L. Skubachevskii, “On the hyperbolicity of rapidly oscillating periodic solutions of functional-differential equations,” Funct. Anal. Appl., 39, No. 1, 68–70 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  18. H.-O. Walther and A. L. Skubachevskii, “On the hyperbolicity of solutions with irrational periods of some functional-differential equations,” Dokl. Akad. Nauk, 402, No. 2, 151–154 (2005).

    MathSciNet  Google Scholar 

  19. X. Xie, “Uniqueness and stability of slowly oscillating periodic solutions of delay equations with unbounded nonlinearity,” J. Differential Equations, 103, 350–374 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  20. N. B. Zhuravlev, “On the spectrum of the monodromy operator for slowly oscillating periodic solutions of functional differential equations with several delays,” Funct. Differ. Equ., 13, No. 2, 323–344 (2006).

    MATH  MathSciNet  Google Scholar 

  21. N. B. Zhuravlev and A. L. Skubachevskii, “On hyperbolicity of periodic solutions of functional-differential equations with several delays,” Proc. Steklov Inst. Math., To appear.

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Authors and Affiliations

  1. Department of Differential Equations and Mathematical Physics, Peoples’ Friendship University of Russia, Ordzhonikidze str., 3, 117198, Moscow, Russia

    N. B. Zhuravlev

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  1. N. B. Zhuravlev
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__________

Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 21, Proceedings of the Seminar on Differential and Functional Differential Equations Supervised by A. Skubachevskii (Peoples’ Friendship University of Russia), 2007.

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Zhuravlev, N.B. Hyperbolicity criterion for periodic solutions of functional-differential equations with several delays. J Math Sci 153, 683–709 (2008). https://doi.org/10.1007/s10958-008-9142-z

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