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 and stability analysis of nonlinear waves in $\Bbb {D}_n$ symmetric delay differential systems. (English) Zbl 1118.34067
The paper is concermed with the array of identical coupled oscillators with delay $$ \dot x_j(t) = -x_j(t) + \alpha f(x_j(t-\tau)) + \beta \left[ g(x_{j-1}(t-\tau))+g(x_{j+1}(t-\tau)) \right],$$ where $j$ is considered mod $n$, $\tau>0$, $f(0)=g(0)=0$, and $f'(0)=g'(0)=1$. The system is ${\Bbb D}_n$-equivariant and has an equilibrium at the origin. The author studies stability of the origin and properties of the bifurcating solutions. In particular, the analysis of the characteristic equation reveals multiple Hopf (or pitchfork) bifurcations, which give rise to a variety of symmetric periodic solutions (or equilibria) as the delay or another bifurcation parameter varies.

MSC:
34K18Bifurcation theory of functional differential equations
92B20General theory of neural networks (mathematical biology)
34C14Symmetries, invariants (ODE)
34K13Periodic solutions of functional differential equations
WorldCat.org
Full Text: DOI
References:
[1] Buono, P. -L.; BĂ©lair, J.: Restrictions and unfolding of double Hopf bifurcation in functional differential equations. J. differential equations 189, No. 1, 234-266 (2003) · Zbl 1032.34068
[2] Campbell, S. A.; Yuan, Y.; Bungay, S.: Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling. Nonlinearity 18, 2827-2846 (2005) · Zbl 1094.34049
[3] S.A. Campbell, I. Ncube, J. Wu, Multistability and stable asynchronous periodic oscillation in a multiple-delayed neural system, preprint · Zbl 1100.34054
[4] Diekmann, O.; Van Gils, S. A.; Verduyn, S. M.; Walther, H. -O.: Delay equations: functional, complex, and nonlinear analysis. Appl. math. Sci. 110 (1995) · Zbl 0826.34002
[5] Faria, T.; Huang, W. Z.; Wu, J.: Smoothness of center manifolds for maps and formal adjoints for semilinear fdes in general Banach spaces. SIAM J. Math. anal. 34, No. 1, 173-203 (2002) · Zbl 1085.34064
[6] Golubitsky, M.; Stewart, I.; Schaeffer, D. G.: Singularities and groups in bifurcation theory, vol. II. Appl. math. Sci. 69 (1988) · Zbl 0691.58003
[7] Guo, S.: Spatio-temporal patterns of nonlinear oscillations in an excitatory ring network with delay. Nonlinearity 18, No. 5, 2391-2407 (2005) · Zbl 1093.34036
[8] Guo, S.; Huang, L.: Hopf bifurcating periodic orbits in a ring of neurons with delays. Phys. D 183, 19-44 (2003) · Zbl 1041.68079
[9] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H.: Theory and applications of Hopf bifurcation. London math. Soc. lecture note ser. 41 (1981) · Zbl 0474.34002
[10] Huang, L.; Wu, J.: Nonlinear waves in networks of neurons with delayed feedback: pattern formation and continuation. SIAM J. Math. anal. 34, No. 4, 836-860 (2003) · Zbl 1038.34076
[11] Peng, M.; Yuan, Y.: Synchronization and desynchronization in a delayed discrete neural network. Internat. J. Bifur. chaos appl. Sci. engrg. 17, No. 3 (2007) · Zbl 1146.39015
[12] M. Peng, Y. Yuan, Complex dynamics in discrete delayed models with D4 symmetry, Chaos Solitons Fractals, in press · Zbl 1157.37323
[13] M. Peng, Y. Yuan, Stability, symmetry-breaking bifurcation and chaos in discrete delayed models, in preparation · Zbl 1147.39301
[14] Wu, J.: Symmetric functional-differential equations and neural networks with memory. Trans. amer. Math. soc. 350, 4799-4838 (1998) · Zbl 0905.34034
[15] Wu, J.; Faria, T.; Huang, Y. S.: Synchronization and stable phase-locking in a network of neurons with memory. Math. comput. Modelling 30, 117-138 (1999) · Zbl 1043.92500
[16] Y. Yuan, J. Wei, Multiple bifurcation analysis in a neural network model with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., in press · Zbl 1185.37136