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)
Chaos and asymptotical stability in discrete-time neural networks. (English) Zbl 0889.68122
Summary: By applying Marotto’s theorem this paper aims to prove that both transiently chaotic neural networks (TCNN) and discrete-time recurrent neural networks (DRNN) have chaotic structure. A significant property of TCNN and DRNN is that they have only one bounded fixed point, when absolute values of the self-feedback connection weights in TCNN and the difference time in DRNN are sufficiently large. We show that this unique fixed point can actually evolve into a snap-back repeller which generates a chaotic structure, if several conditions are satisfied. On the other hand, by using the Lyapunov functions, we also derive sufficient conditions on asymptotical stability for symmetrical versions of both TCNN and DRNN, under which TCNN and DRNN asymptotically converge to a fixed point. Furthermore, related bifurcations are also considered in this paper. Since both TCNN and DRNN are not special but simple and general, the obtained theoretical results hold for a wide class of discrete-time neural networks. To demonstrate the theoretical results of this paper better, several numerical simulations are provided as illustrating examples.

68T05Learning and adaptive systems
37D45Strange attractors, chaotic dynamics
Full Text: DOI
[1] Aihara, K.; Takabe, T.; Toyoda, M.: Chaotic neural networks. Phys. lett. A 144, No. 6/7, 333-340 (1990)
[2] Amari, S.: Characteristics of random nets of analog neuron-like elements. IEEE trans. On SMC 2, 643-657 (1972) · Zbl 0247.92006
[3] Chen, L.; Aihara, K.: Chaotic simulated annealing and its application to a maintenance scheduling problem n a power system. Int. symp. On nonlinear theory and its appl. 2, 695-700 (1993)
[4] Chen, L.; Aihara, K.: Transient chaotic neural networks and chaotic simulated annealing. Towards the harnessing of chaos, 347-352 (1994)
[5] Chen, L.; Aihara, K.: Chaotic simulated annealing for combinatorial optimization. Dynamic systems and chaos 1, 319-322 (1995)
[6] Chen, L.; Aihara, K.: Chaotic simulated annealing by a neural network model with transient chaos. Neural networks 8, No. 6, 915-930 (1995)
[7] Chen, L.: Chaos in transiently chaotic neural networks (one-dimension case). Technical report on power engineering, IEE Japan, PE-95-151 (1995)
[8] Golubitsky, M.; Stewart, I.; Schaeffer, D. G.: Singularities and groups in bifurcation theory. (1988) · Zbl 0691.58003
[9] Guckenheimer, J.; Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. (1983) · Zbl 0515.34001
[10] Hata, M.: Euler’s finite difference scheme and chaos in rn. Proc. Japan acad. A 58, 178-181 (1982) · Zbl 0544.58015
[11] Hirsch, M. W.; Smale, S.: Differential equations, dynamical systems, and linear algebra. (1974) · Zbl 0309.34001
[12] Hopfield, J.: Neural networks and physical systems with emergent collective computational abilities. Proc. nalt. Acad. sci. (USA) 79, 2554-2558 (1982)
[13] Hopfield, J.: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. natl. Acad. sci. (USA) 81, 3088-3092 (1984)
[14] Hopfield, J.; Tank, D.: Neural computation of decisions in optimization problems. Biological cybernetics 52, 141-152 (1985) · Zbl 0572.68041
[15] Kirkpatrick, S.; Jr, C. D. Gelatt; Vecchi, M. P.: Optimization by simulated annealing. Science 220, 671-680 (1983) · Zbl 1225.90162
[16] Komuro, M.: Definitions of chaos. Chaos seminar (1994)
[17] Kuznetsov, Y. A.: Elements of applied bifurcation theory. (1995) · Zbl 0829.58029
[18] Li, T. Y.; York, J. A.: Period three implies chaos. Amer. math. Monthly 82, 985-992 (1975) · Zbl 0351.92021
[19] Marcus, C. M.; Westervelt, R. M.: Dynamics of iterated-map neural networks. Phys. rev. A 40, 501-504 (1989)
[20] Marotto, F. R.: Snap-back repellers imply chaos in rn. J. math. Anal. appl. 63, 199-223 (1978) · Zbl 0381.58004
[21] Peterson, C.; Anderson, J. R.: A mean field theory algorithm for neural networks. Complex systems 1, 995-1019 (1989) · Zbl 0657.68082
[22] Peterson, C.; Soderberg, B.: Artificial neural networks. Modern heuristic techniques for combinatorial problems (1993)
[23] Pismen, L. M.; Rubinstein, B. Y.; Velarde, M. G.: On automated derivation of amplitude equations in nonlinear bifurcation problems. Int. J. Bifurt. and chaos (1996) · Zbl 1298.35180
[24] Rumelhart, D. E.; Group, J. L. Mcclelland The Pdp Research: Parallel distributed processing. 1 and 2 (1986)
[25] Sato, M.; Ishii, S.: Bifurcations in mean field theory annealing. ATR technical report, TR-H-167 (1995)
[26] Shiraiwa, K.; Kurata, M.: A generalization of a theorem of marotto. Proc. Japan acad. 55, 286-289 (1980) · Zbl 0451.58031
[27] Smale, S.: Diffeomorphisms with many periodic points. Differential and combinatorial topology, 63-80 (1968)
[28] Tong, H.: Non-linear time series -- A dynamical system approach. (1990) · Zbl 0716.62085
[29] Urabe, M.: Galerkin’s procedure for nonlinear periodic systems. Arch. rat. Mech. anal. 72, 121-152 (1965) · Zbl 0133.35502
[30] Ushio, T.; Hirai, K.: Chaos in non-linear sampled-data control systems. Int. J. Control 38, 1023-1033 (1983) · Zbl 0525.93046
[31] Ushio, T.; Hirai, K.: Chaotic behavior in pulse-width modulated feedback systems. Trans. society of instrument and control engineers 21, 539-545 (1985)
[32] Wells, D. M.: Solving degenerate optimization problems using networks of neural oscillators. Neural networks 5, 949-959 (1992)
[33] Yamaguti, M.; Matano, H.: Euler’s finite difference scheme and chaos. Proc. Japan acad. A 55, 78-80 (1979) · Zbl 0434.39003