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)
Bifurcation analysis in a neutral differential equation. (English) Zbl 1259.34061
Summary: The dynamics of a neural network model in neutral form is investigated. We prove that a sequence of Hopf bifurcations occurs at the origin as the delay increases. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using the normal form method and center manifold theory. Global existence of periodic solutions is established using a global Hopf bifurcation result of Krawcewicz et al. and a Bendixson’s criterion for higher-dimensional ordinary differential equations due to Li and Muldowney.
MSC:
34K18Bifurcation theory of functional differential equations
34K40Neutral functional-differential equations
34K13Periodic solutions of functional differential equations
34K20Stability theory of functional-differential equations
92B20General theory of neural networks (mathematical biology)
34K17Transformation and reduction of functional-differential equations and systems; normal forms
34K19Invariant manifolds (functional-differential equations)
References:
[1]A. Ardjouni, A. Djoudi, Fixed points and stability in linear neutral differential equations with variable delays, Nonlinear Anal., in press, doi:10.1016/j.na.2010.10.050.
[2]Bellen, A.; Maset, S.; Zennaro, M.; Guglielmi, N.: Recent trends in the numerical solution of retarded functional differential equations, Acta numer. 18, 1-110 (2009) · Zbl 1178.65078 · doi:10.1017/S0962492906390010
[3]Carr, J.: Applications of centre manifold theory, (1981)
[4]Coppel, W. A.: Stability and asymptotic behavior of differential equations, (1965) · Zbl 0154.09301
[5]El-Morshedy, H. A.; Gopalsamy, K.: Nonoscillation, oscillation and convergence of a class of neutral equations, Nonlinear anal. 40, 173-183 (2000) · Zbl 0954.34066 · doi:10.1016/S0362-546X(00)85010-5
[6]Fan, D.; Wei, J.: Hopf bifurcation analysis in a tri-neuron network with time delay, Nonlinear anal. Real world appl. 9, 9-25 (2008) · Zbl 1149.34044 · doi:10.1016/j.nonrwa.2006.08.008
[7]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
[8]Gopalsamy, K.; Leung, I.; Liu, P.: Global Hopf-bifurcation in a neural netlet, Appl. math. Comput. 94, 171-192 (1998) · Zbl 0946.34065 · doi:10.1016/S0096-3003(97)10087-X
[9]Guo, S.: Equivariant normal forms for neutral functional differential equations, Nonlinear dynam. 61, 311-329 (2010) · Zbl 1204.34093 · doi:10.1007/s11071-009-9651-4
[10]Guo, S.; Lamb, J.: Equivariant Hopf bifurcation for neutral functional differential equations, Proc. amer. Math. soc. 136, 2031-2041 (2008) · Zbl 1149.34045 · doi:10.1090/S0002-9939-08-09280-0
[11]Hadd, S.; Nounou, H.; Nounou, M.: Eventual norm continuity for neutral semigroups on Banach spaces, J. math. Anal. appl. 375, 543-552 (2011)
[12]Hale, J. K.; Lunel, M. V.: Introduction to functional to differential equations, (1993)
[13]Hartung, F.: Differentiability of solutions with respect to parameters in neutral differential equations with state-dependent delays, J. math. Anal. appl. 324, 504-524 (2006)
[14]Hassard, B.; Kazarinoff, N.; Wan, Y.: Theory and applications of Hopf bifurcation, (1981)
[15]Kamenskii, M. I.; Lysakova, Yu.V.; Nistri, P.: On bifurcation of periodic solutions for functional differential equations of the neutral type with small delay, Autom. remote control 69, 2027-2032 (2008) · Zbl 1155.93027 · doi:10.1134/S0005117908120023
[16]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
[17]Li, W.; Chang, Y.; Nieto, Juan J.: Solvability of impulsive neutral evolution differential inclusions with state-dependent delay, Math. comput. Modelling 49, 1920-1927 (2009) · Zbl 1171.34304 · doi:10.1016/j.mcm.2008.12.010
[18]Li, M. Y.; Muldowney, J. S.: On Bendixson’s criterion, J. differential equations 106, 27-39 (1993) · Zbl 0786.34033 · doi:10.1006/jdeq.1993.1097
[19]Liz, E.; Röst, G.: Dichotomy results for delay differential equations with negative Schwarzian derivative, Nonlinear anal. Real world appl. 11, 1422-1430 (2010) · Zbl 1207.34093 · doi:10.1016/j.nonrwa.2009.02.030
[20]Liz, E.; Röst, G.: On the global attractor of delay differential equations with unimodal feedback, Discrete contin. Dyn. syst. 24, 1215-1224 (2009) · Zbl 1188.34102 · doi:10.3934/dcds.2009.24.1215
[21]Nam, P. T.; Phat, V. N.: An improved stability criterion for a class of neutral differential equations, Appl. math. Lett. 22, 31-35 (2009) · Zbl 1163.34392 · doi:10.1016/j.aml.2007.11.006
[22]Park, J. H.: Delay-dependent criterion for asymptotic stability of a class of neutral equations, Appl. math. Lett. 17, 1203-1206 (2004) · Zbl 1122.34339 · doi:10.1016/j.aml.2003.05.013
[23]Sun, Y. G.; Wang, L.: Note on asymptotic stability of a class of neutral differential equations, Appl. math. Lett. 19, 949-953 (2006) · Zbl 1122.34340 · doi:10.1016/j.aml.2005.10.015
[24]Walther, H. O.: The 2-dimensional a tractor of x’(t)=-μx(t)+f(x(t-1)), Mem. amer. Math. soc. 113 (1995)
[25]Wang, C.; Wei, J.: Hopf bifurcation for neutral functional differential equations, Nonlinear anal. Real world appl. 11, 1269-1277 (2010) · Zbl 1194.34137 · doi:10.1016/j.nonrwa.2009.02.017
[26]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
[27]Weedermann, M.: Hopf bifurcation calculations for scalar neutral delay differential equations, Nonlinearity 19, 2091-2102 (2006) · Zbl 1116.34057 · doi:10.1088/0951-7715/19/9/005
[28]Wei, J.: Bifurcation analysis in a scalar delay differential equation, Nonlinearity 20, 2483-2498 (2007) · Zbl 1141.34045 · doi:10.1088/0951-7715/20/11/002
[29]Wei, J.; Fan, D.: Hopf bifurcation analysis in a MacKey-Glass system, Internat. J. Bifur. chaos 17, 2149-2157 (2007) · Zbl 1159.34056 · doi:10.1142/S0218127407018282
[30]Wei, J.; Li, Michael Y.: Hopf bifurcation analysis in a delayed Nicholson blowflies equation, Nonlinear anal. 60, 1351-1367 (2005) · Zbl 1144.34373 · doi:10.1016/j.na.2003.04.002
[31]Wei, J.; Ruan, S.: Stability and global Hopf bifurcation for neutral differential equations, Acta math. Sinica 45, 94-104 (2002) · Zbl 1018.34068
[32]Wei, J.; Yu, C.: Hopf bifurcation analysis in a model of oscillatory gene expression with delay, Proc. roy. Soc. Edinburgh sect. A 139, 879-895 (2009) · Zbl 1185.34124 · doi:10.1017/S0308210507000091
[33]Wiggins, S.: Introduction to applied nonlinear dynamical systems and chaos, (1990)
[34]Wu, J.: Global continua of periodic solutions to some difference-differential equations of neutral type, J. tohoku math. 45, 67-88 (1993) · Zbl 0778.34054 · doi:10.2748/tmj/1178225955
[35]Wu, J.: Symmetric functional-differential equations and neural networks with memory, Trans. amer. Math. soc. 350, 4799-4838 (1998) · Zbl 0905.34034 · doi:10.1090/S0002-9947-98-02083-2
[36]Wu, J.; Xia, H.: Self-sustained oscillations in a ring array of coupled lossless transmission lines, J. differential equations 124, 247-278 (1996) · Zbl 0840.34080 · doi:10.1006/jdeq.1996.0009