A simple proof of the uniqueness of periodic orbits in the resonance problem

Authors:
Shui-Nee Chow, Cheng Zhi Li and Duo Wang

Journal:
Proc. Amer. Math. Soc. **105** (1989), 1025-1032

MSC:
Primary 58F14; Secondary 58F21

DOI:
https://doi.org/10.1090/S0002-9939-1989-0938908-6

MathSciNet review:
938908

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In 1979, E. Horozov considered the versal deformation of a planar vector field which is invariant under a rotation through an angle (with resonance of order 3). In his study, the most difficult part of the proof is on the uniqueness of limit cycles. In this note we give a simple and elementary (without the theory of algebraic geometry proof of the uniqueness of periodic orbits in the 1:3 resonance problem.

**[1]**V. I. Arnold,*Geometrical methods in the theory of ordinary differential equations*, Springer-Verlag, New York, 1983. MR**695786 (84d:58023)****[2]**J. Carr, S.-N. Chow, and J. K. Hale,*Abelian integrals and bifurcation theory*, J. Differential Equations**59**(1985), 413-436. MR**807855 (88m:58133)****[3]**S.-N. Chow, C. Li, and D. Wang,*Normal forms and bifurcation of vector fields*(to appear).**[4]**R. H. Cushman and J. A. Sanders,*A codimension two bifurcation with a third order Picard-Fuchs equation*, J. Differential Equations**59**(1985), 243-256. MR**804890 (87h:58139)****[5]**B. Drachman, S. van Gils, and Zhang Zhifen,*Abelian integrals for quadratic vector fields*, J. Reine Angew. Math.**382**(1987), 165-180. MR**921170 (89h:34028)****[6]**E. I Horozov,*Versal deformation of equivariant vector fields in the case of symmetry of order*2*and*3, Trans. of Petrovski Seminar,**5**(1979), 163-192, (In Russian). MR**549627 (80k:58079)****[7]**C. Rousseau,*Codimension*1*and*2*bifurcations of fixed points of diffeomorphisms and of periodic solutions of vector fields*, 1987 (preprint). MR**1038368 (90m:58155)****[8]**R. H. Cushman and J. A. Sanders,*Abelian integrals and global Hopf bifurcations*, Dynamical Systems and Bifurcations (B. L. J. Braaksma, H. W. Broer, and F. Takens, editors) Lecture Notes in Mathematics 1125, Springer-Verlag, 1985, pp. 87-98. MR**798083 (86m:58133)****[9]**Yu. S., Il' yashenko,*Zeros of special Abelian integrals in a real domain*, Functional Anal. Appl.**11**(1977), 309-311.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
58F14,
58F21

Retrieve articles in all journals with MSC: 58F14, 58F21

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1989-0938908-6

Keywords:
Bifurcation diagram,
phase portrait,
periodic orbit,
Picard-Fuchs equation

Article copyright:
© Copyright 1989
American Mathematical Society