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)
Two-heteroclinic orbits emerging in the reversible homoclinic pitchfork bifurcation. (English) Zbl 1067.37023
Summary: We consider reversible and 2 -symmetric systems of ordinary differential equations (ODEs) that possess a symmetric homoclinic orbit to a degenerate equilibrium. The equilibrium is supposed to undergo a reversible pitchfork bifurcation, controlled by the system’s parameter. It has been shown in [the author, ibid. 15, No. 6, 2097–2119 (2002; Zbl 1021.37014)] that a multitude of homoclinic orbits emerges in this bifurcation. In particular, if a coefficient in the normal form of the local bifurcation has the correct sign such that this bifurcation is of eye-type, then globally a reversible homoclinic pitchfork bifurcation can be observed. This means, that similar to the local bifurcation in which two new equilibria emerge, two-homoclinic orbits to these equilibria bifurcate from the primary homoclinic orbit. In this paper, we investigate the emergence of two-homoclinic and two-heteroclinic orbits, that is, orbits making two windings in a neighbourhood of the primary orbit, in this bifurcation. Using a combination of geometrical and analytical techniques, we prove the emergence of a family of two-homoclinic orbits to periodic orbits and of a two-heteroclinic cycle between equilibria. The general analysis is illustrated by numerical results for an example system of two second-order ODEs.
MSC:
37C29Homoclinic and heteroclinic orbits
37C80Symmetries, equivariant dynamical systems
37C10Vector fields, flows, ordinary differential equations
34C23Bifurcation (ODE)
34C37Homoclinic and heteroclinic solutions of ODE