Existence of at most 1, 2, or 3 zeros of a melnikov function and limit cycles

Journal of Differential Equations

Volumn 170, Issue 2, 2001, Pages 325-343

  Han, Maoan ^a  

^a SHANGHAI JIAO TONG UNIVERSITY   (China)

Author keywords

Melnikov function; limit cycle; bifurcation

Indexed keywords

EID: 0035285527     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.2000.3828     Document Type: Article

References (20)

1. Arnold V. I. Geometrical Methods in the Theory of Ordinary Differential Equations. 1983;Springer-Verlag, New York.
2. Bogdanov R. I. Versal deformations of singularity of a vector field on the plane in the case of zero eigenvalues. Proc. Petrovki Sem. 2:1976;37-65.
3. Carr J. Applications of Center Manifold Theory. 1981;Springer-Verlag, New York.
4. Carr J., Chow S. N., Hale J. K. Abelian integrals and bifurcation theory. J. Differential Equations. 59:1985;413-436.
5. Chow S. N., Li C., Wang D. Uniqueness of periodic orbits in some vector fields with codimension two singularities. J. Differential Equations. 77:1989;231-253.
6. Chow S. N., Li C., Wang D. Normal Forms and Bifurcations of Planar Vector Fields. 1994;Cambridge Univ. Press, Cambridge.
7. Cushman R., Sanders J. A. A codimension two bifurcation with a third order Picard-Fuchs equation. J. Differential Equations. 59:1985;243-256.
8. Han M., Jiang K. Uniqueness of limit cycles of certain planar vector fields with codimension two singularities. Alavi Y., Hsieh Po-Fang. Trends and Developments in Ordinary Differential Equations. 1994;179-188 World Scientific, Singapore.
9. Han M. Bifurcations of limit cycles and the cusp of order n. Acta Math. Sinica (N.S.). 13:1997;64-75.
10. Horozov E., Iliev I. D. On the number of limit cycles in perturbations of quadratic Hamiltonian systems. Proc. London Math. Soc. (3). 69:1994;198-224.
11. Horozov E., Iliev I. D. Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians. Nonlinearity. 11:1998;1521-1537.
12. Joyal P. The cusp of order n. J. Differential Equations. 88:1990;1-14.
13. Li B., Zhang Z. A note on a result of G. S. Petrov about the weakened 16th Hilbert problem. J. Math. Anal. Appl. 190:1995;489-516.
14. Li C., Rousseau C., Wang X. A simple proof for the unicity of the limit cycle in the Bogdanove-Takens system. Canad. Math. Bull. 33:1990;84-92.
15. Li C., Zhang Z. A criterion for determining the monotonicity of the ratio of two Abelian integrals. J. Differential Equations. 124:1996;407-424.
16. Luo D., Wang X., Zhu D., Han M. Bifurcation Theory and Methods of Dynamical Systems. 1997;World Scientific, Singapore.
17. Sanders J. A., Cushman R. Abelian integrals and global Hopf bifurcations. Lecture Notes in Math. 1985;Springer-Verlag, New York/Berlin. p. 87-98.
18. Petrov G. S. The Chebyshev property of elliptic integrals. Funct. Anal. Appl. 22:1988;72-73.
19. Wang M., Luo D. Global bifurcation of some cubic planar systems. Nonlinear Anal. 8:1984;711-722.
20. Zoladek H. Abelian integrals in unfoldings of codimension 3 singular planar vector fields. Lecture Notes in Math. 1991;Springer-Verlag, New York/Berlin. p. 165-224.