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)
Hopf bifurcation calculations for scalar neutral delay differential equations. (English) Zbl 1116.34057

The paper studies Hopf bifurcations of scalar neutral delay differential equations of the form

x ˙-ax ˙(t-τ)=L(γ)x t +f(x t ,γ),

where γ is the bifurcation parameter and |a|<1. The characteristic equation has the form

λ(1-exp(-λτ))-b-cexp(-λτ)=0·

First the paper presents the stability charts depending on the coefficients of the characteristic equation. Then the paper presents a sequence of formulas leading to the criticality coefficient determining the stability of the emerging periodic solutions in the center direction. The normal form computation is done with the technique introduced by T. Faria and L. T. Magalhães [J. Differ. Equ. 122, No. 2, 181–200 (1995; Zbl 0836.34068)]. Two examples are included. Both have a delay τ=1 and a linear part of the form γx(t-1). The nonlinear parts are γx(t)x(t-1) and γx(t)x(t-1) 2 , respectively.

MSC:
34K18Bifurcation theory of functional differential equations
34K40Neutral functional-differential equations
34K17Transformation and reduction of functional-differential equations and systems; normal forms