zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Bifurcations of heteroclinic loops. (English) Zbl 0993.34040
Summary: By generalizing the Floquet method from periodic systems to systems with exponential dichotomy, a local coordinate system is established in a neighborhood of a heteroclinic loop Γ to study the bifurcation problems of homoclinic and periodic orbits. Asymptotic expressions for the bifurcation surfaces and their relative positions are given. Results obtained in the literature concerning the 1-hom bifurcation surfaces are improved and extended to the nontransversal case. Existence regions of the 1-per orbits bifurcated from Γ are described, and the uniqueness and noncoexistence of 1-hom and 1-per orbits and nonexistence of 2-hom and 2-per orbits are also obtained.

MSC:
34C23Bifurcation (ODE)
34C37Homoclinic and heteroclinic solutions of ODE
34C25Periodic solutions of ODE
34C28Complex behavior, chaotic systems (ODE)
34D09Dichotomy, trichotomy
References:
[1]Chow, S.N., Deng, B., Terman, D., The bifurcation of a homoclinic orbit from two heteroclinic orbits,SIAM J. Math. Anal., 1990, 21: 179. · Zbl 0693.34055 · doi:10.1137/0521010
[2]Chow, S.N., Deng, B., Terman, D., The bifurcation of a homoclinic orbit from two heteroclinic orbits–a topological approach,Appl. Anal., 1991, 42: 275.
[3]Kokubu, H., Homoclinic and heteroclinic bifurcations of vector fields,Japan J. Appl. Math., 1988, 5: 455. · Zbl 0668.34039 · doi:10.1007/BF03167912
[4]Zhu, D.M., Melnikov vector and heteroclinic manifolds,Science in China, Ser. A, 1994, 37: 673.
[5]Zhu, D.M., Xu, M., Exponential trichotomy, orthogonality condition and their application,Chin. Ann. Math. B, 1997, 18:1.
[6]Zhu, D.M., Exponential trichotomy and heteroclinic bifurcation,Nonlinear Analysis, 1997, 28: 547. · Zbl 0877.34037 · doi:10.1016/0362-546X(95)00164-Q
[7]Abraham, R., Marsden, J.E.,Manifolds, Tensor Analysis and Applications, London: Addison Wesley, 1983.