Numerical analysis of some characteristics of the limit cycle of the free van der Pol equation

  • S. P. Suetin


Singular Point Power Series STEKLOV Institute Analytic Continuation Frequency Function 
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  1. 1.
    E. F. Mishchenko and N. Kh. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations, Math. Concepts Methods Sci. Engrg. (Plenum, New York, 1980), Vol. 13.Google Scholar
  2. 2.
    E. F. Mishchenko, Yu. S. Kolesov, A. Yu. Kolesov, and N. Kh. Rozov, Periodic Motions and Bifurcation Processes in Singularly Perturbed Systems (Fizmatlit, Moscow, 1995) [in Russian].zbMATHGoogle Scholar
  3. 3.
    C. M. Andersen and J. F. Geer, “Power Series Expansions for the Frequency and Period of the Limit Cycle of the van der Pol Equation,” SIAM J. Appl. Math. 42(3), 678–693 (1982).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    M. B. Dadfar, J. F. Geer, and C. M. Andersen, “Perturbation Analysis of the Limit Cycle of the Free van der Pol Equation,” SIAM J. Appl. Math. 44(5), 881–895 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    G. A. Baker, Jr., and P. Graves-Morris, Padé Approximants, Part 1: Basic Theory, Encyclopedia Math. Appl. (Addison-Wesley, Reading, Mass., 1981), Vol. 13; G. A. Baker Jr. and P. Graves-Morris, Padé Approximants, Part 2: Extensions and Applications, Encyclopedia Math. Appl. (Addison-Wesley, Reading, Mass., 1981), Vol. 14.Google Scholar
  6. 6.
    I. V. Andrianov, “Application of Padé-Approximants in Perturbation Methods,” Adv. in Mech. 14(2), 3–25 (1991).MathSciNetGoogle Scholar
  7. 7.
    A. A. Gončar, “Poles of Rows of the Padé Table and Meromorphic Continuation of Functions,” Math. USSR-Sb. 43(4), 527–546 (1982).CrossRefGoogle Scholar
  8. 8.
    S. P. Suetin, “On an Inverse Problem for themth Row of the Padé Table,” Math. USSR-Sb. 52(1), 231–244 (1985).zbMATHCrossRefGoogle Scholar
  9. 9.
    S. P. Suetin, “Padé Approximants and Efficient Analytic Continuation of a Power Series,” Russ. Math. Surv. 57(1), 43–141 (2002).MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    R. de Montessus, “Sur les fractions continues algébriques,” Bull. Soc. Math. France 30, 28–36 (1902).MathSciNetzbMATHGoogle Scholar
  11. 11.
    A. A. Gončar, “On the Convergence of Generalized Padé Approximants of Meromorphic Functions,” Math. USSR-Sb. 27(4), 503–514 (1975).CrossRefGoogle Scholar
  12. 12.
    J. Hadamard, “Essai sur l’étude des fonctions données par leur développement de Taylor,” J. Math. Pures Appl. 8, 101–186 (1892); Thèse (Gauthier-Villars et Fils, Paris, 1892).Google Scholar
  13. 13.
    G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Transl. Math. Monogr. (Amer. Math. Soc., Providence, R.I., 1969), Vol. 26.zbMATHGoogle Scholar
  14. 14.
    S. P. Suetin, “On Poles of the mth Row of a Pade Table,” Math. USSR-Sb. 48(2), 493–497 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    E. Fabry, “Sur les points singuliers d’une fonction donnée par son développement en série et l’impossibilité du prolongement analytique dans des cas très génèraux,” Ann. Sci.École Norm. (3) 13, 367–399 (1896).MathSciNetzbMATHGoogle Scholar
  16. 16.
    L. Bieberbach, Analytische Fortsetzung, Ergeb. Math. Grenzgeb. (Springer-Verlag, Berlin, 1955), Vol. 3.zbMATHGoogle Scholar
  17. 17.
    P. L. Chebyshev, “On Continued Fractions,” Uchen. Zap. Imper. Akad. Nauk III, 636–664 (1855); French transl.: P. Tchébycheff, “Sur les fractions continues,” J. Math. Pures Appl. Sér. 2 3, 289–323 (1858); in Complete Collection of Works (Izd. Akad. Nauk SSSR, Moscow, 1948), Vol. 2, pp. 103–126 [in Russian].Google Scholar
  18. 18.
    A. Markoff, “Deux démonstrations de la convergence de certaines fractions continues,” Acta Math. 19(1), 93–104 (1895); A. A. Markov, “Two Proofs of Convergence of Some Continued Fractions,” in Selected Works on the Theory of Continued Fractions and the Theory of Functions Least Deviating from Zero (Gostekhizdat, Moscow, 1948), pp. 106–119 [in Russian].MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    G. Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. (Amer. Math. Soc., Providence, R.I., 1959), Vol. 23.Google Scholar
  20. 20.
    A. A. Gončar, “On Convergence of Padé Approximants for Some Classes of Meromorphic Functions,” Math. USSR-Sb. 26(4), 555–575 (1975).CrossRefGoogle Scholar
  21. 21.
    E. A. Rahmanov, “Convergence of Diagonal Padé Approximants,” Math. USSR-Sb. 33(2), 243–260 (1977).CrossRefGoogle Scholar
  22. 22.
    S. Dumas, Sur le développement des fonctions elliptiques en fractions continues, Thèse (Zürich, 1908).Google Scholar
  23. 23.
    V. I. Buslaev, “On the Baker-Gammel-Wills Conjecture in the Theory of Padé Approximants,” Sb. Math. 193(6), 811–823 (2002).MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    S. P. Suetin, “Approximation Properties of the Poles of Diagonal Padé Approximants for Certain Generalizations of Markov Functions,” Sb. Math. 193, 1837–1866 (2002).MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    S. P. Suetin, “On Interpolation Properties of Diagonal Padé Approximants of Elliptic Functions,” Russ. Math. Surv. 59, 800–802 (2004).MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    J. Nuttall, “Asymptotics of Diagonal Hermite-Padé Polynomials,” J. Approx. Theory 42(4), 299–386 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    H. Stahl, “Orthogonal Polynomials with Complex Valued Weight Function, I,” Constr. Approx. 2(1), 225–240 (1986); “Orthogonal Polynomials with Complex Valued Weight Function, II,” Constr. Approx. 2 (1), 241–251 (1986).MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    A. A. Gonchar, “On Uniform Convergence of Diagonal Padé Approximants,” Math. USSR-Sb. 46(4), 539–559 (1983).MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    A. A. Gonchar, “Convergence of Diagonal Padé Approximants in a Spherical Metric,” in Mathematical Structure-Computational Mathematics-Mathematical Modeling (Publ. House Bulgar. Acad. Sci., Sofia, 1984), Vol. 2, pp. 29–35 [in Russian].Google Scholar
  30. 30.
    S. P. Suetin, “Convergence of Chebyshëv Continued Fractions for Elliptic Functions,” Sb. Math. 194(12), 1807–1835 (2003).MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    A. A. Gonchar, “Analytic Function,” in Mathematical Encyclopaedia (Sovetskaya Entsiklopediya, Moscow, 1977), pp. 261–268 [in Russian].Google Scholar
  32. 32.
    E. M. Chirka, “Analytic Continuation,” in Mathematical Encyclopaedia (Sovetskaya Entsiklopediya, Moscow, 1977), pp. 283–285 [in Russian].Google Scholar
  33. 33.
    J. Nuttall, “Sets of Minimal Capacity, Padé Approximations, and the Bubble Problem,” in Bifurcation Phenomena in Mathematical Physics and Related Topics, NATO Adv. Stud. Inst. Ser. Ser. C: Math. Phys. Sci. (D. Reidel, Dordrecht, 1980), Vol. 54, pp. 185–201.CrossRefGoogle Scholar
  34. 34.
    I. V. Andrianov and J. Awrejcewicz, “New Trends in Asymptotic Approaches: Summation and Interpolation Methods,” Appl. Mech. Rev. 54(1), 69–92 (2001).CrossRefGoogle Scholar
  35. 35.
    Z. I. Krupka, V. G. Novikov, and V. B. Uvarov, Application of the Method of Summation of Power Series by Means of Padé Approximants for Calculation of the Zone Structure of the Electron Spectrum in Matter Preprint no. 44, (Inst Prikl. Mat. im. M. V. Keldysha, Russian Acad. Sci., Moscow, 1985).Google Scholar
  36. 36.
    J. M. McNamee, “A 2002 Update of the Supplementary Bibliography on Roots of Polynomials,” J. Comput. Appl. Math. 142(2), 433–434 (2002).MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    A. A. Dorodnicyn, Asymptotic Solution of van der Pol’s Equation, Amer. Math. Soc. Translation 88 (Amer. Math. Soc., Providence, R.I., 1953).Google Scholar

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© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  • S. P. Suetin
    • 1
  1. 1.Steklov Institute of MathematicsRussian Academy of SciencesMoscowRussia

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