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: 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].
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
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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)
- Mary L. Cartwright, Forced oscillations in nearly sinusoidal systems, J. Inst. Elec. Engrs. Part III 95 (1948), 88–96. MR 27395
- A. W. Gillies, On the transformations of singularities and limit cycles of the variational equations of van der Pol, Quart. J. Mech. Appl. Math. 7 (1954), 152–167. MR 63512, DOI https://doi.org/10.1093/qjmam/7.2.152
- Chihiro Hayashi, Nonlinear oscillations in physical systems, McGraw-Hill Electrical and Electronic Engineering Series, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR 0170071
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E. Hopf, Abzweigung einer periodischen Lösung eines Differentialsystems, Akad. Wiss. Leipzig 94, 1–22 (1942)
N. Krylov and N. N. Bogoliubov, Introduction to Non-linear Mechanics translated by S. Lefschetz, Annals of Math. Studies 11, Princeton. 1947
- Marilia C. Peixoto and M. M. Peixoto, Structural stability in the plane with enlarged boundary conditions, An. Acad. Brasil. Ci. 31 (1959), 135–160. MR 107067
- S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI https://doi.org/10.1090/S0002-9904-1967-11798-1
- J. Sotomayor, Generic one-parameter families of vector fields on two-dimensional manifolds, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 5–46. MR 339279
- Floris Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 47–100. MR 339292
- Floris Takens, Forced oscillations and bifurcations, Applications of global analysis, I (Sympos., Utrecht State Univ., Utrecht, 1973) Math. Inst. Rijksuniv. Utrecht, Utrecht, 1974, pp. 1–59. Comm. Math. Inst. Rijksuniv. Utrecht, No. 3-1974. MR 0478235
- V. S. Afraĭmovič and L. P. Šil′nikov, Small periodic perturbations of autonomous systems, Dokl. Akad. Nauk SSSR 214 (1974), 739–742 (Russian). MR 0338512
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
- M. L. Cartwright and J. E. Littlewood, On non-linear differential equations of the second order. I. The equation $\ddot y-k(1-y^2)y+y=b\lambda k\;{\rm cos} (\lambda t+a), k$ large, J. London Math. Soc. 20 (1945), 180–189. MR 16789, DOI https://doi.org/10.1112/jlms/s1-20.3.180
- Norman Levinson, A second order differential equation with singular solutions, Ann. of Math. (2) 50 (1949), 127–153. MR 30079, DOI https://doi.org/10.2307/1969357
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
V. I. Arnol’d, Lectures on bifurcations in versal families, Russian Math. Surveys 27, 54–123 (1972)
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)
M. L. Cartwright, Forced oscillations in nearly-sinusoidal systems, J. Inst. Elec. Eng. 95, 88 (1948)
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)
C. Hayashi, Nonlinear oscillations in physical systems, McGraw Hill, 1964
M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds, Bull. Am. Math. Soc. 76, 1015–1019 (1970)
E. Hopf, Abzweigung einer periodischen Lösung eines Differentialsystems, Akad. Wiss. Leipzig 94, 1–22 (1942)
N. Krylov and N. N. Bogoliubov, Introduction to Non-linear Mechanics translated by S. Lefschetz, Annals of Math. Studies 11, Princeton. 1947
M. C. Peixoto and M. M. Peixoto, Structural stability in the plane with enlarged boundary conditions, An. Acad. Brasil. Sci. 31, 135–160 (1959)
S. Smale, Differentiable dynamical systems, Bull. Amer. Math Soc. 73, 747–817 (1967)
J. Sotomayor, Generic one-parameter families of vector fields on two-dimensional manifolds, Publ. I.H.E.S. 43, 5–46 (1974)
F. Takens, Singularities of vector fields, Publ. I.H.E.S. 43 47–100 (1974)
F. Takens, Forced oscillations and bifurcations, in Applications of global analysis, Comm. 3 of the Math. Institute, Rijksuniversiteit Utrecht, 1974
V. S. Afraimovic and L. P. Sil’nikov, On small periodic perturbations of autonomous systems, Soviet Math. Dokl. 15, 206–211 (1974)
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
M. L. Cartwright and J. E. Littlewood, On non-linear differential equations of the second order: I. The equation $\ddot y + k\left ( {1 - {y^2}} \right )\dot y + y = b\lambda k\cos \left ( {\lambda t + a} \right )$, $k$ large, J. London Math. Soc. 20, 180–189 (1945)
N. Levinson, A second order differential equation with singular solutions, Annals of Math. 50, 127–153 (1949)
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© Copyright 1978
American Mathematical Society