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Centers and Isochronous Centers of Liénard Systems

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Abstract

For holomorphic Liénard equations, necessary and sufficient conditions for the existence of a center and an isochronous center are obtained without calculating the focus quantities and the isochronicity constants.

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References

  1. Amel’kin, V.V., Lukashevich, N.A., and Sadovskii, A.P., Nelineinye kolebaniya v sistemakh vtorogo poryadka (Nonlinear Oscillations in Systems of Second Order), Minsk: Belarus. Gos. Univ., 1982.

    MATH  Google Scholar 

  2. Kukles, I.S., On centers and foci, Dokl. Akad. Nauk SSSR, 1938, vol. 19, no. 6–7, pp. 459–464.

    Google Scholar 

  3. Kukles, I.S., Several criteria for the difference between a focus and a center, Tr. Navoi Uzbek. Gos. Univ., 1951, no. 47, pp. 29–98.

  4. Cherkas, L.A., On an indication distinguishing center from focus for the Liénard equation, Dokl. Akad. Nauk BSSR, 1978, vol. 22, no. 11, pp. 969–970.

    MathSciNet  MATH  Google Scholar 

  5. Cherkas, L.A., Limit Cycles of Two-Dimensional Autonomous Systems with Polynomial Right-Hand Sides, Doctoral (Phys.-Math.) Dissertation, Tbilisi, 1986.

    Google Scholar 

  6. Christopher, C., An algebraic approach to the classification of centers in polynomial Liénard systems, J. Math. Anal. Appl., 1999, vol. 229, pp. 319–329.

    Article  MathSciNet  MATH  Google Scholar 

  7. Cherkas, L.A., On conditions for some equations of the form yy′ = P(x) + Q(x)y + R(x)y 2 to have a center, Differ. Uravn., 1972, vol. 8, no. 8, pp. 1435–1439.

    Google Scholar 

  8. Cherkas, L.A., Conditions for the equation \(yy\prime = \sum\nolimits_{i = 0}^3 {{p_i}(x){y^i}}\) to have a center, Differ. Uravn., 1978, vol. 14, no. 9, pp. 1594–1600.

    Google Scholar 

  9. Cherkas, L. and Romanovski, V.G., Trivial centers of a cubic system of Liénard type, Nonlinear Phenom. Complex Systems, 2005, vol. 8, no. 1, pp. 89–93.

    MathSciNet  Google Scholar 

  10. Sadovskii, A.P., Solution of the center-focus problem for the Liénard system with polynomial coefficients, Differ. Uravn., 1975, vol. 11, no. 11, pp. 2102–2104.

    Google Scholar 

  11. Amel’kin, V.V., On one conjecture in the theory of isochronous Liénard systems, Differ. Equations, 2017, vol. 53, no. 10, pp. 1247–1253.

    Article  MathSciNet  MATH  Google Scholar 

  12. Rudenok, A.E., Generalized symmetry of the Liénard system, Differ. Equations, 2019, vol. 55, no. 2, pp. 181–193.

    Article  MathSciNet  Google Scholar 

  13. Abdullaev, N., On isochronicity in nonlinear oscillations, Tr. Aini Tadzh. Uchit. Inst., 1954, no. 2, pp. 71–78.

  14. Amel’kin, V.V. and Chin’ Zan’ Dang, On strong isochronicity of the Cauchy-Riemann differential systems, Vestsi Nats. Akad. Navuk Belarusi Ser. Fiz.-Mat. Navuk, 1993, no. 2, pp. 26–30.

  15. Kukles, I.S. and Piskunov, N.S., On isochronicity of oscillations for conservative and nonconservative systems, Dokl. Akad. Nauk SSSR, 1937, vol. 17, no. 9, pp. 467–470.

    Google Scholar 

  16. Amel’kin, V.V., On the order of strong isochronicity of plane dynamical systems, Differ. Equations, 2007, vol. 43, no. 10, pp. 1464–1467.

    Article  MathSciNet  MATH  Google Scholar 

  17. Chavarriga, J. and Sabatini, M., A survey of isochronous centers, Qual. Theory Dyn. Syst., 1999, vol. 1, Art. no. 1, pp. 1–70.

    Article  MathSciNet  Google Scholar 

  18. Rudenok, A.E., Strong isochronicity of center. On periods of limit cycles of the Liénard system, Differ. Uravn., 1975, vol. 11, no. 5, pp. 811–819.

    MathSciNet  Google Scholar 

  19. Algaba, A., Freire, E., and Gamero, E., Isochronicity via Normal Form, Preprint. Universidad de Sevilla, 1998.

    MATH  Google Scholar 

  20. Algaba, A., Freire, E., and Gamero, E., Isochronicity via normal form, Qual. Theory Dyn. Syst., 2000, vol. 1, no. 2, Art. no. 4, pp. 133–156.

    Article  MathSciNet  MATH  Google Scholar 

  21. Sabatini, M., On the period function of Liénard systems, J. Differential Equations, 1999, vol. 152, pp. 467–487.

    Article  MathSciNet  MATH  Google Scholar 

  22. Christopher C. and Devlin, J., On the classification of Liénard systems with amplitude-independent periods, J. Differential Equations, 2004, vol. 200, no. 1, pp. 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  23. Amel’kin, V.V., Strong isonchronicity of the Liénard system, Differ. Equations, 2006, vol. 42, no. 5, pp. 615–618.

    Article  MathSciNet  MATH  Google Scholar 

  24. Erugin, N.P., Kniga dlya chteniya po obshchemu kursu differentsial’nykh uravnenii (Reading Book in the General Course of Differential Equations), Minsk: Nauka i Tekhnika, 1979.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Belarusian State University, Minsk, 220030, Belarus

    V. V. Amel’kin & A. E. Rudenok

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  1. V. V. Amel’kin
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  2. A. E. Rudenok
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Correspondence to V. V. Amel’kin or A. E. Rudenok.

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Amel’kin, V.V., Rudenok, A.E. Centers and Isochronous Centers of Liénard Systems. Diff Equat 55, 283–293 (2019). https://doi.org/10.1134/S0012266119030017

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