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)
Nonlinear dynamics and chaos in fractional-order neural networks. (English) Zbl 1254.34103
Summary: Several topics related to the dynamics of fractional-order neural networks of Hopfield type are investigated, such as stability and multi-stability (coexistence of several different stable states), bifurcations and chaos. The stability domain of a steady state is completely characterized with respect to some characteristic parameters of the system, in the case of a neural network with ring or hub structure. These simplified connectivity structures play an important role in characterizing the network’s dynamical behavior, allowing us to gain insight into the mechanisms underlying the behavior of recurrent networks. Based on the stability analysis, we are able to identify the critical values of the fractional order for which Hopf bifurcations may occur. Simulation results are presented to illustrate the theoretical findings and to show potential routes towards the onset of chaotic behavior when the fractional order of the system increases.
MSC:
34K20Stability theory of functional-differential equations
34K18Bifurcation theory of functional differential equations
34A08Fractional differential equations
References:
[1]Ahmad, B.; Sivasundaram, S.: Some existence results for fractional integro-differential equations with nonlinear conditions, Communications in applied analysis 12, 107-112 (2008) · Zbl 1179.45009
[2]Ahmad, B.; Sivasundaram, S.: Existence of solutions for impulsive integral boundary value problems of fractional order, Nonlinear analysis: hybrid systems 4, 134-141 (2010) · Zbl 1187.34038 · doi:10.1016/j.nahs.2009.09.002
[3]Anastasio, T.: The fractional-order dynamics of brainstem vestibulo-oculomotor neurons, Biological cybernetics 72, 69-79 (1994)
[4]Arena, P.; Fortuna, L.; Porto, D.: Chaotic behavior in noninteger-order cellular neural networks, Physical review E 61, 776-781 (2000)
[5]Baldi, P.; Atiya, A. F.: How delays affect neural dynamics and learning, IEEE transactions on neural networks 5, 612-621 (1994)
[6]Baleanu, D.; Sadati, S.; Ranjbar, A.; Ghaderi, R.; Abdeljawad, T.: Mittag-Leffler stability theorem for fractional nonlinear systems with delay, Abstract and applied analysis (2010)
[7]Barabási, A. -L.; Albert, R.: Emergence of scaling in random networks, Science 286, 509-512 (1999) · Zbl 1226.05223 · doi:10.1126/science.286.5439.509
[8]Benchohra, M.; Slimani, B.: Existence and uniqueness of solutions to impulsive fractional differential equations, Electronic journal of differential equations, 1-11 (2009)
[9]Boroomand, A.; Menhaj, M.: Fractional-order Hopfield neural networks, Lncs 5506, 883-890 (2009)
[10]Bungay, S. D.; Campbell, S. A.: Patterns of oscillation in a ring of identical cells with delayed coupling, International journal of bifurcation and chaos 17, 3109-3125 (2007) · Zbl 1185.37180 · doi:10.1142/S0218127407018907
[11]Campbell, S. A.; Ruan, S.; Wei, J.: Qualitative analysis of a neural network model with multiple time delays, International journal of bifurcation and chaos in applied sciences and engineering 9, 1585-1595 (1999) · Zbl 1192.37115 · doi:10.1142/S0218127499001103
[12]Cottone, G.; Paola, M. D.; Santoro, R.: A novel exact representation of stationary colored Gaussian processes (fractional differential approach), Journal of physics A: mathematical and theoretical 43, 085002 (2010) · Zbl 1187.82043 · doi:10.1088/1751-8113/43/8/085002
[13]Diethelm, K.; Ford, N.; Freed, A.: A predictor–corrector approach for the numerical solution of fractional differential equations, Nonlinear dynamics 29, 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341
[14]El-Saka, H.; Ahmed, E.; Shehata, M.; El-Sayed, A.: On stability, persistence, and Hopf bifurcation in fractional order dynamical systems, Nonlinear dynamics 56, 121-126 (2009) · Zbl 1175.37084 · doi:10.1007/s11071-008-9383-x
[15]Elwakil, A.: Fractional-order circuits and systems: an emerging interdisciplinary research area, IEEE circuits and systems magazine 10, 40-50 (2010)
[16]Engheia, N.: On the role of fractional calculus in electromagnetic theory, IEEE antennas and propagation magazine 39, 35-46 (1997)
[17]Gray, R. M.: Toeplitz and circulant matrices: a review, (2005) · Zbl 1143.15305 · doi:10.1561/0100000006
[18]Guo, S.: Spatio-temporal patterns of nonlinear oscillations in an excitatory ring network with delay, Nonlinearity 18, 2391-2407 (2005) · Zbl 1093.34036 · doi:10.1088/0951-7715/18/5/027
[19]Guo, S.; Huang, L.: Hopf bifurcating periodic orbits in a ring of neurons with delays, Physica D: Nonlinear phenomena 183, 19-44 (2003) · Zbl 1041.68079 · doi:10.1016/S0167-2789(03)00159-3
[20]Guo, S.; Huang, L.: Non-linear waves in a ring of neurons, IMA journal of applied mathematics (Institute of mathematics and its applications) 71, 496-518 (2006) · Zbl 1117.37042 · doi:10.1093/imamat/hxl002
[21]Guo, S.; Huang, L.: Stability of nonlinear waves in a ring of neurons with delays, Journal of differential equations 236, 343-374 (2007) · Zbl 1132.34048 · doi:10.1016/j.jde.2007.01.027
[22]Guo, S. J.; Huang, L. H.: Pattern formation and continuation in a trineuron ring with delays, Acta Mathematica sinica, English series 23, 799-818 (2007) · Zbl 1132.34051 · doi:10.1007/s10114-005-0842-8
[23]Henry, B.; Wearne, S.: Existence of Turing instabilities in a two-species fractional reaction–diffusion system, SIAM journal on applied mathematics 62, 870-887 (2002) · Zbl 1103.35047 · doi:10.1137/S0036139900375227
[24]Heymans, N.; Bauwens, J. -C.: Fractal rheological models and fractional differential equations for viscoelastic behavior, Rheologica acta 33, 210-219 (1994)
[25]Hirsch, M.: Convergent activation dynamics in continuous-time networks, Neural networks 2, 331-349 (1989)
[26]Hopfield, J.: Neural networks and physical systems with emergent collective computational abilities, Proceedings of the national Academy of sciences 79, 2554-2558 (1982)
[27]Ichise, M.; Nagayanagi, Y.; Kojima, T.: An analog simulation of non-integer order transfer functions for analysis of electrode processes, Journal of electroanalytical chemistry 33, 253-265 (1971)
[28]Kaslik, E. (2009). Dynamics of a discrete-time bidirectional ring of neurons with delay. In Proceedings of the international joint conference on neural networks. Atlanta, GA, USA (pp. 1539–1546).
[29]Kaslik, E.; Balint, S.: Complex and chaotic dynamics in a discrete-time-delayed Hopfield neural network with ring architecture, Neural networks 22, 1411-1418 (2009)
[30]Kaslik, E.; Sivasundaram, S.: Dynamics of fractional-order neural networks, , 611-618 (2011)
[31]Kilbas, A.; Srivastava, H.; Trujillo, J.: Theory and applications of fractional differential equations, (2006)
[32]Kitajima, H.; Kurths, J.: Bifurcation in neuronal networks with hub structure, Physica A: statistical mechanics and its applications 388, 4499-4508 (2009)
[33]Kuznetsov, Y. A.: Elements of applied bifurcation theory, (1998)
[34]Lakshmikantham, V.; Leela, S.; Devi, J. V.: Theory of fractional dynamic systems, (2009)
[35]Lu, X.; Guo, S.: Complete classification and stability of equilibria in a delayed ring network, Electronic journal of differential equations 2008, 1-12 (2008) · Zbl 1170.34351 · doi:emis:journals/EJDE/Volumes/2008/85/abstr.html
[36]Lundstrom, B.; Higgs, M.; Spain, W.; Fairhall, A.: Fractional differentiation by neocortical pyramidal neurons, Nature neuroscience 11, 1335-1342 (2008)
[37]Mainardi, F.: Fractional relaxation–oscillation and fractional phenomena, Chaos, solitons fractals 7, 1461-1477 (1996)
[38]Matignon, D. (1996). Stability results for fractional differential equations with applications to control processing. In Computational engineering in systems applications (pp. 963–968).
[39]Matsuzaki, T.; Nakagawa, M.: A chaos neuron model with fractional differential equation, Journal of the physical society of Japan 72, 2678-2684 (2003)
[40]Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics reports 339, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[41]Milo, R.; Shen-Orr, S.; Itzkovitz, S.; Kashtan, N.; Chklovskii, D.; Alon, U.: Network motifs: simple building blocks of complex networks, Science, 298 (2002)
[42]Nakagawa, M.; Sorimachi, K.: Basic characteristics of a fractance device, IEICE transactions on fundamentals of electronics, communications and computer sciences 75-A, 1814-1819 (1992)
[43]NanoDotTek (2007). What is fractance and why is it useful? Technical report. NDT24-11-2007.
[44]Petras, I. (2006). A note on the fractional-order cellular neural networks. In IEEE international conference on neural networks (pp. 1021–1024).
[45]Podlubny, I.: Fractional differential equations, (1999)
[46]Sugimoto, N.: Burgers equation with a fractional derivative: hereditary effects on nonlinear acoustic waves, Journal of fluid mechanics 225, 631-653 (1991) · Zbl 0721.76011 · doi:10.1017/S0022112091002203
[47]Wei, J.; Jiang, W.: Stability and bifurcation analysis in a neural network model with delays, Dynamics of continuous, discrete and impulsive systems series A: mathematical analysis 13, 177-192 (2006) · Zbl 1099.34069
[48]Zhou, S., Hu, P., & Li, H. (2009). Chaotic synchronization of a fractional neuron network system with time-varying delays. In 2009 international conference on communications, circuits and systems. ICCCAS 2009 (pp. 863–867).
[49]Zhou, S.; Li, H.; Zhu, Z.: Chaos control and synchronization in a fractional neuron network system, Chaos, solitons and fractals 36, 973-984 (2008) · Zbl 1139.93320 · doi:10.1016/j.chaos.2006.07.033
[50]Zhu, H., Zhou, S., & Zhang, W. (2008). Chaos and synchronization of time-delayed fractional neuron network system. In Proceedings of the 9th international conference for young computer scientists. ICYCS 2008 (pp. 2937–2941).