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)
Linear stability and Hopf bifurcation in a delayed two-coupled oscillator with excitatory-to-inhibitory connection. (English) Zbl 1205.34107
This paper deals with the dynamical behaviour of a delayed two-coupled oscillator with excitatory-to-inhibitory connection. Some parameter regions are given for linear stability, absolute synchronization and Hopf bifurcations by using the theory of functional differential equations. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. The authors also investigate the spatio-temporal patterns of bifurcating periodic oscillations by using symmetric bifurcation theory of delay differential equations with representation theory of Lie groups. The paper ends with some numerical simulations to illustrate the theoretical results.
MSC:
34K60Qualitative investigation and simulation of models
34K20Stability theory of functional-differential equations
34K18Bifurcation theory of functional differential equations
34K17Transformation and reduction of functional-differential equations and systems; normal forms
34K19Invariant manifolds (functional-differential equations)
34C15Nonlinear oscillations, coupled oscillators (ODE)
References:
[1]Aronson, D.; Ermentrout, G. B.; Kopell, N.: Amplitude response of coupled oscillators, Physica D 41, 403-449 (1990) · Zbl 0703.34047 · doi:10.1016/0167-2789(90)90007-C
[2]Ermentrout, G. B.; Kopell, N.: Multiple pulse interactions and averaging in systems of coupled neural oscillators, J. math. Biol. 29, 195-217 (1991) · Zbl 0718.92004 · doi:10.1007/BF00160535
[3]Kawato, M.; Sokabe, M.; Suzuki, R.: Synergism and antagonism of neurons caused by an electrical synapse, Biol. cybern. 34, 81-89 (1979) · Zbl 0407.92010 · doi:10.1007/BF00365472
[4]Hindmarsh, J. L.; Rose, R. M.: A model of the nerve impulse using two first-order differential equations, Nature 296, 162-164 (1982)
[5]Hansel, D.; Mato, G.; Meunier, C.: Phase dynamics for weakly coupled Hodgkin–Huxley neurons, Europhys. lett. 23, 367-372 (1993)
[6]Wilson, H. R.; Cowan, J. D.: Excitatory and inhibitory interactions in localized populations of model neurons, Biophys. J. 12, 1-24 (1972)
[7]Kryukov, V. I.; Borisyuk, G. N.; Borisyuk, R. M.; Kirillov, A. B.; Kovalenko, E. L.: Metastable and unstable states in the brain, Stochastic cellular systems: ergodicity, memory, morphogenesis, 225-357 (1990)
[8]Macgregor, R. J.: Neural and brain modelling, (1987) · Zbl 0643.92007
[9]Marcus, C. M.; Waugh, F. R.; Westerbelt, R. M.: Nonlinear dynamics and stability of analog neural net works, Physica D 51, 234-247 (1991) · Zbl 0800.92059 · doi:10.1016/0167-2789(91)90236-3
[10]Babcock, K. L.; Westerbelt, R. M.: Complex dynamics in simple neural circuits, Neural networks for computing, 288-293 (1986)
[11]Babcock, K. L.; Westerbelt, R. M.: Dynamics of simple electronic neural networks with added inertia, Physica D 23, 464-469 (1987)
[12]Gopalsamy, K.; Leung, I.: Delay induced periodicity in a neural netlet of excitation and inhibition, Physica D 89, 395-426 (1996) · Zbl 0883.68108 · doi:10.1016/0167-2789(95)00203-0
[13]Golubitsky, M.; Stewart, I. N.: Hopf bifurcation in the presence of symmetry, Arch. ration. Mec. anal. 87, 107-165 (1985) · Zbl 0588.34030 · doi:10.1007/BF00280698
[14]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
[15]Guo, S.; Lamb, J. S. W.: 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
[16]Hassard, B.; Kazarinoff, N.; Wan, Y. H.: Theory of application of Hopf bifurcation, London math, society lecture notes, series 41 (1981) · Zbl 0474.34002
[17]Hale, J. K.; Lunel, S. M. Verduyn: Introduction to functional differential equations, (1993)