zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the number of limit cycles in double homoclinic bifurcations. (English) Zbl 1013.34026
The authors consider perturbations of Hamiltonian systems in $\bbfR^2$. They assume that the unperturbed system possesses a double homoclinic loop (figure-eight-configuration) at a hyperbolic fixed point. The main results concern the maximal number of limit cycles near the given homoclinic orbits. The condition permitting the existence of the limit cycles are formulated in terms of Melnikov functions.

34C05Location of integral curves, singular points, limit cycles (ODE)
Full Text: DOI
[1] Roussarir, R., On the number of limit cycles which appear by perturbation of separatnx loop of plannar fields, Bol. Soc. Brasil Mat., 1986, 17: 67. · Zbl 0628.34032 · doi:10.1007/BF02584827
[2] Han Maoan, Ye Yanqian, On the coefficients appearing in the expansion of Melnikov functions in homoclinic bifurcations, Ann. of Diff. Equs., 1998, 14(2): 156. · Zbl 0968.34028
[3] Han Maoan, Cyclicity of plannar homoclinic loops and quadratic integratable systems, Science in China, Ser. A, 1997, 40(12): 1247. · Zbl 0930.37035 · doi:10.1007/BF02876370
[4] Joyal, P., Generalized Hopf bifurcation and its dual generalized homoclinic bifurcation, SIAM J. Math., 1988, 48: 481. · Zbl 0642.34041 · doi:10.1137/0148027
[5] Han Maoan, Luo Dingjun, Zhu Deming, The uniqueness of limit cycles bifurrated fiwm a separatrix cycle (II), Acta Math. Sinica (in Chinese), 1992, 4: 541. · Zbl 0772.34027
[6] Han Maoan, Bifurcations of limit cycles from a heteroclinic cycle of Hamiltonian systems, Chin. Ann. of Math., 1998, 19B(2): 189. · Zbl 0905.34032
[7] Han Maoan, Zhu Deming, Bifurcation Theory of Differential Equations (in Chinese), Beijing: Coal Industry Publishing House, 1994. · Zbl 1016.34040
[8] Chow, S. N., Hale, J. K., Methods of Bifurcation Theory, New York: Springer-Verlag, 1982. · Zbl 0487.47039