# 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)
Multistability of neural networks with discontinuous activation function. (English) Zbl 1221.34131
Summary: The multistability is studied for two-dimensional neural networks with multilevel activation functions. It is shown that the system has $n^{2}$ isolated equilibrium points which are locally exponentially stable, where the activation function has $n$ segments. Furthermore, evoked by periodic external input, $n^{2}$ periodic orbits which are locally exponentially attractive, can be found. These results are extended to $k$-neuron networks, which really enlarge the capacity of the associative memories. Examples and simulation results are used to illustrate the theory.

##### MSC:
 34D05 Asymptotic stability of ODE 92B20 General theory of neural networks (mathematical biology)
Full Text:
##### References:
 [1] Cao, J.; Wang, J.: Global asymptotic and robust stability of recurrent neural networks with time delays, IEEE trans circuits syst part I 52, No. 2, 417-426 (2005) [2] Cao, J.; Wang, J.: Absolute exponential stability of recurrent neural networks with time delays and Lipschitz-continuous activation functions, Neural network 17, No. 3, 379-390 (2004) · Zbl 1074.68049 · doi:10.1016/j.neunet.2003.08.007 [3] Cao, J.; Liang, J.: Boundedness and stability for Cohen -- Grossberg neural networks with time-varying delays, J math anal appl 296, No. 2, 665-685 (2004) · Zbl 1044.92001 · doi:10.1016/j.jmaa.2004.04.039 [4] Cao, J.; Huang, D. S.; Qu, Y.: Global robust stability of delayed recurrent neural networks, Chaos solitons fractals 23, No. 1, 221-229 (2005) · Zbl 1075.68070 · doi:10.1016/j.chaos.2004.04.002 [5] Cao, J.; Chen, T.: Globally exponentially robust stability and periodicity of delayed neural networks, Chaos solitons fractals 22, No. 4, 957-963 (2004) · Zbl 1061.94552 · doi:10.1016/j.chaos.2004.03.019 [6] Yu, W.; Cao, J.: Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with time delays, Nonlinear anal 351, No. 1 - 2, 64-78 (2006) · Zbl 1234.34047 [7] Guo, S.; Huang, L.: Linear stability and Hopf bifurcation in a two-neuron network with three delays, Int J bifur chaos 14, No. 8, 2790-2810 (2004) · Zbl 1062.34078 · doi:10.1142/S0218127404011016 [8] Song, Y.; Han, M.; Wei, J.: Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays, Physica D 200, 185-204 (2005) · Zbl 1062.34079 · doi:10.1016/j.physd.2004.10.010 [9] Wei, J.; Ruan, S.: Stability and bifurcation in a neural network model with two delays, Physica D 130, 255-272 (1999) · Zbl 1066.34511 · doi:10.1016/S0167-2789(99)00009-3 [10] Forti, M.; Manetti, S.; Marini, M.: Necessary and sufficient condition for absolute stability of neural networks, IEEE trans circuits syst part I 41, 491-494 (1994) · Zbl 0925.92014 · doi:10.1109/81.298364 [11] Forti, M.; Tesi, A.: New conditions for global stability of neural networks with application to linear and quadratic programming problems, IEEE trans circuits syst part I 42, 354-366 (1995) · Zbl 0849.68105 · doi:10.1109/81.401145 [12] Tank, D. W.; Hopfield, J. J.: Simple ”neural” optimization networks: an AD converter signal decision circuit and a linear programming network, IEEE trans circuits syst 33, 533-541 (1986) [13] Hopfield, J. J.: Neurons with graded response have collective computational properties like those of two-state neurons, Proc natl acad sci 81, 3088-3092 (1984) [14] Forti, M.; Nistri, P.: Global convergence of neural networks with discontinuous neuron activation, IEEE trans circuits syst part I 50, 1421-1435 (2003) [15] Forti, M.; Nistri, P.; Papini, D.: Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain, IEEE trans neural network 16, 1449-1463 (2005) [16] Lu, W.; Chen, T.: Dynamical behaviors of Cohen -- Grossberg neural networks with discontinuous activation functions, Neural network 18, 231-242 (2005) · Zbl 1078.68127 · doi:10.1016/j.neunet.2004.09.004 [17] Juang, J.; Lin, S.: Cellular neural networks: mosaic pattern and spatial chaos, SIAM J appl math 60, 891-915 (2000) · Zbl 0947.34038 · doi:10.1137/S0036139997323607 [18] Shih, C.: Pattern formation and spatial chaos for cellular neural networks with asymmetric templates, Int J bifur chaos appl sci eng 8, 1907-1936 (1998) · Zbl 1002.92512 · doi:10.1142/S0218127498001601 [19] Shih, C.: Influence of boundary conditions on pattern formation and spatial chaos in lattice systems, SIAM J appl math 61, 335-368 (2000) · Zbl 0985.37091 · doi:10.1137/S0036139998340650 [20] Zeng, Z.; Wang, J.; Liao, X.: Stability analysis of delayed cellular neural networks described using cloning templates, IEEE trans circuits syst part I 51, 2313-2324 (2004) [21] Zeng, Z.; Wang, J.: Multiperiodicity and exponential attractivity evoked by periodic external inputs in delayed cellular neural networks, Neural comput 18, 848-870 (2006) · Zbl 1107.68086 · doi:10.1162/neco.2006.18.4.848 [22] Cheng, C.; Lin, K.; Shih, C.: Multistability in recurrent neural networks, SIAM J appl math 66, 1301-1320 (2006) · Zbl 1106.34048 · doi:10.1137/050632440 [23] Forti, M.: A note on neural networks with multiple equilibrium points, IEEE trans circuits syst part I 43, 487-491 (1996) [24] Yi, Z.; Tan, K.: Multistability of discrete-time recurrent neural networks with unsaturating piecewise linear activation functions, IEEE trans neural network 15, 329-336 (2004)