Bifurcations of the forced van der Pol oscillator

Authors:
P. J. Holmes and D. A. Rand

Journal:
Quart. Appl. Math. **35** (1978), 495-509

MSC:
Primary 34C15

DOI:
https://doi.org/10.1090/qam/492551

MathSciNet review:
492551

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Abstract | References | Similar Articles | Additional Information

Abstract: We discuss the bifurcations of the variational equation of the forced van der Pol oscillator and prove the existence of bifurcations of saddle connection type as postulated by M. L. Cartwright [4] and A. W. Gillies [5].

**[1]**A. A. Andronov, E. A. Leontovich. I. I. Gordon and A. G. Maier,*Theory of bifurcations of dynamic systems in the plane*(translated from the original Russian), Israel Program of Scientific Translations, Jerusalem, 1971**[2]**V. I. Arnol'd,*Lectures on bifurcations in versal families*, Russian Math. Surveys**27**, 54-123 (1972) MR**0413191****[3]**M. L. Cartwright*Forced oscillations in non-linear systems: contributions to the theory of nonlinear oscillations*, Annals Math. Studies No. 20, 1, 149-241 (1950)**[4]**M. L. Cartwright,*Forced oscillations in nearly-sinusoidal systems*, J. Inst. Elec. Eng.**95**, 88 (1948) MR**0027395****[5]**A. W. Gillies,*On the transformation of singularities and limit cycles of the variational equations of van der Pol*, Quart. J. Mech. Appl. Math.**7**, 152-167 (1954) MR**0063512****[6]**C. Hayashi,*Nonlinear oscillations in physical systems*, McGraw Hill, 1964 MR**0170071****[7]**M. W. Hirsch, C. C. Pugh and M. Shub,*Invariant manifolds*, Bull. Am. Math. Soc.**76**, 1015-1019 (1970) MR**0292101****[8]**E. Hopf,*Abzweigung einer periodischen Lösung eines Differentialsystems*, Akad. Wiss. Leipzig**94**, 1-22 (1942)**[9]**N. Krylov and N. N. Bogoliubov,*Introduction to Non-linear Mechanics*translated by S. Lefschetz,*Annals of Math. Studies***11**, Princeton. 1947**[10]**M. C. Peixoto and M. M. Peixoto,*Structural stability in the plane with enlarged boundary conditions*, An. Acad. Brasil. Sci.**31**, 135-160 (1959) MR**0107067****[11]**S. Smale,*Differentiable dynamical systems*, Bull. Amer. Math Soc.**73**, 747-817 (1967) MR**0228014****[12]**J. Sotomayor,*Generic one-parameter families of vector fields on two-dimensional manifolds*, Publ. I.H.E.S.**43**, 5-46 (1974) MR**0339279****[13]**F. Takens,*Singularities of vector fields*, Publ. I.H.E.S.**43**47-100 (1974) MR**0339292****[14]**F. Takens,*Forced oscillations and bifurcations*, in*Applications of global analysis*, Comm. 3 of the Math. Institute, Rijksuniversiteit Utrecht, 1974 MR**0478235****[15]**V. S. Afraimovic and L. P. Sil'nikov,*On small periodic perturbations of autonomous systems*, Soviet Math. Dokl.**15**, 206-211 (1974) MR**0338512****[16]**E. C. Zeeman,*Morse inequalities for diffeomorphisms with shoes and flows with solenoids*, in*Dynamical systems-Warwick 1974*, ed. A. Manning, Lecture Notes in Mathematics No. 468, Springer-Verlag, 1975**[17]**M. L. Cartwright and J. E. Littlewood,*On non-linear differential equations of the second order: I. The equation*, large, J. London Math. Soc.**20**, 180-189 (1945) MR**0016789****[18]**N. Levinson,*A second order differential equation with singular solutions*, Annals of Math.**50**, 127-153 (1949) MR**0030079**

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DOI:
https://doi.org/10.1090/qam/492551

Article copyright:
© Copyright 1978
American Mathematical Society