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 for neutral functional differential equations. (English) Zbl 1194.34137
Summary: We extend the computation of the properties of Hopf bifurcation, such as the direction of bifurcation and stability of bifurcating periodic solutions, of DDE to a kind of neutral functional differential equation (NFDE). As an example, a neutral delay logistic differential equation is considered, and the explicit formulas for determining the direction of bifurcation and the stability of bifurcating periodic solutions are derived. Finally, some numerical simulations are carried out to support the analytic results.
MSC:
34K18Bifurcation theory of functional differential equations
92D25Population dynamics (general)
34K28Numerical approximation of solutions of functional-differential equations
34K13Periodic solutions of functional differential equations
34K20Stability theory of functional-differential equations
References:
[1]Brayton, R. K.: Bifurcation of periodic solutions in a nonlinear difference-differential equation of neutral type, Quart. appl. Math. 24, 215-224 (1966) · Zbl 0143.30701
[2]Gopalsamy, K.; Zhang, B.: On a neutral delay-logistic equation, Dyn. stab. Syst. 2, 183-195 (1988) · Zbl 0665.34066 · doi:10.1080/02681118808806037
[3]Györi, I.; Ladas, G.: Oscillation theory of delay differential equations with applications, (1991) · Zbl 0780.34048
[4]Hale, J.; Lunel, Y. H.: Introduction to functional differential equations, (1993)
[5]Krawcewicz, W.; Ma, S.; Wu, J.: Multiple slowly oscillating periodic solutions in coupled lossless transmission lines, Nonlinear anal. 5, 309-354 (2004) · Zbl 1144.34365 · doi:10.1016/S1468-1218(03)00040-3
[6]Krawcewicz, W.; Wu, J.; Xia, H.: Global Hopf bifurcation theory for condensing fields and neutral equations with applications to lossless transmission problems, Can. appl. Math. Q. 1, 167-219 (1993) · Zbl 0801.34069
[7]Weedermann, M.: Normal forms for neutral functional differential equations, Fields inst. Commun. 29, 361-368 (2001) · Zbl 0989.34060
[8]Faria, T.; Magalhaes, L.: Normal forms for retarded functional differential equation and applications to bogdanov–Takens singularity, J. differential equations 122, 201-224 (1995) · Zbl 0836.34069 · doi:10.1006/jdeq.1995.1145
[9]Faria, T.; Magalhaes, L.: Normal forms for retarded functional differential equation with parameters and applications to Hopf bifurcation, J. differential equations 122, 181-200 (1995) · Zbl 0836.34068 · doi:10.1006/jdeq.1995.1144
[10]Weedermann, M.: Hopf bifurcation calculations for scalar delay differential equations, Nonlinearity 19, 2091-2102 (2006) · Zbl 1116.34057 · doi:10.1088/0951-7715/19/9/005
[11]Kazarinoff, N. D.; Den Driessche, P. Van; Wan, Y. H.: Hopf bifurcation and stability of periodic solutions of differential-difference and integro-differential equations, J. inst. Math. appl. 21, 461-477 (1978) · Zbl 0379.45021 · doi:10.1093/imamat/21.4.461
[12]Hassard, B.; Kazarinoff, N.; Wan, Y.: Theory and applications of Hopf bifurcation, (1981)
[13]Wei, J.; Ruan, S.: Stability and global Hopf bifurcation for neutral differential equations, Acta math. Sinica 45, 94-104 (2002) · Zbl 1018.34068
[14]Carr, J.: Applications of centre manifold theory, (1981)
[15]Chow, S. -N.; Lu, K.: Ck center unstable manifolds, Proc. roy. Soc. Edinburgh sect. A 108, 303-320 (1988) · Zbl 0707.34039 · doi:10.1017/S0308210500014682
[16]Hale, J.; Weedermann, M.: On perturbations of delay-differential equations with periodic orbits, J. differential equations 197, 219-246 (2004) · Zbl 1071.34074 · doi:10.1016/S0022-0396(02)00063-3
[17]Wiggins, S.: Introduction to applied nonlinear dynamical systems and chaos, (1990)
[18]Wang, C.; Wei, J.: Normal forms for nfdes with parameters and application to the lossless transmission line, Nonlinear dynam. 52, 199-206 (2008) · Zbl 1187.34094 · doi:10.1007/s11071-007-9271-9