zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Global dynamics of Hopfield neural networks involving variable delays. (English) Zbl 0990.34036
The following system of delayed functional-differential equations used as a model describing the dynamics of Hopfield neural networks is considered $$\dot u(t)=-Bu(t)+Ag(u(t-\tau(t)))+J,\quad t\ge 0,\qquad u(t)=\phi(t),\quad -\tau\le t\le 0,$$ with $u(t)=\text{col}\{u_i(t)\} \in \bbfR^n$, $B=\text{diag}\{ b_i \}$, $A=(a_{ij})_{n\times n}$, $\tau(t)=(\tau_{ij}(t))$, $g(u)=\text{col}(g_i(u_i))$ with $g(0)=0$ continuous, $J=\text{col}\{J_i\}$, $\varphi=\text{col}\{\varphi_i\}$. Conditions for the uniform boundedness of the solutions are given. Existence and uniqueness of an equilibrium point under general conditions are established. Further, sufficient criteria for the global asymptotic stability are derived using a technique based on properties of nonnegative matrices and matrix inequalities. In particular, sufficient criteria for the global asymptotic stability independent of the delay are obtained.

34C11Qualitative theory of solutions of ODE: growth, boundedness
92B20General theory of neural networks (mathematical biology)
Full Text: DOI
[1] Baldi, P.; Atiya, A. F.: How delays affect neural dynamics and learning. IEEE trans. Neural networks 5, 612-621 (1994)
[2] Marcus, C. M.; Westervelt, R. M.: Stability of analog neural networks with delay. Phys. rev. A 39, 347-359 (1989)
[3] Den Driessche, P. Van; Zou, X. F.: Global attractivity in delayed Hopfield neural networks models. SIAM J. Appl. math. 58, 1878-1890 (1998) · Zbl 0917.34036
[4] Hou, C. H.; Qian, J. E.: Stability analysis for neural dynamics with time-varying delays. IEEE transactions on neural networks 9, 221-223 (1998)
[5] Lasalle, J. P.: The stability of dynamical system. (1976) · Zbl 0364.93002
[6] Horn, R. A.; Johnson, C. R.: Matrix analysis. (1991) · Zbl 0729.15001
[7] Xu, D. Y.: Integro-differential equations and delay integral inequalities. TĂ´hoki math. J. 44, 365-378 (1992) · Zbl 0760.34059
[8] Ma, Z. X.; Guo, Q. Y.; Xu, D. Y.: Stability and domain of attraction for a class of integral equations. J. of math. 19, 299-303 (1999) · Zbl 0952.45004
[9] Li, S. Y.; Xu, D. Y.; Zhao, H. Y.: Stability region of nonlinear integro-differential equations. Appl. math. Lett. 13, No. 2, 77-82 (2000) · Zbl 0974.45006
[10] Xu, D. Y.; Li, S. Y.; Pu, Z. L.; Guo, Q. Y.: Domain of attraction of nonlinear discrete systems with delays. Computers math. Applic. 38, No. 5/6, 155-162 (1999) · Zbl 0939.39013
[11] Berman, A.; Plemmons, R. J.: Nonnegative matrices in the mathematical sciences. 133-137 (1979) · Zbl 0484.15016