×

zbMATH — the first resource for mathematics

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:
34K20 Stability theory of functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34A08 Fractional ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
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
[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), Art. No. 108651 · Zbl 1195.34013
[7] Barabási, A.-L.; Albert, R., Emergence of scaling in random networks, Science, 286, 509-512, (1999) · Zbl 1226.05223
[8] Benchohra, M.; Slimani, B., Existence and uniqueness of solutions to impulsive fractional differential equations, Electronic journal of differential equations, 1-11, (2009) · Zbl 1178.34004
[9] Boroomand, A.; Menhaj, M., Fractional-order Hopfield neural networks, (), 883-890
[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
[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
[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
[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
[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
[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), Now Publishers Inc. · Zbl 1143.15305
[18] Guo, S., Spatio-temporal patterns of nonlinear oscillations in an excitatory ring network with delay, Nonlinearity, 18, 2391-2407, (2005) · Zbl 1093.34036
[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
[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
[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
[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
[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
[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) · Zbl 1369.92007
[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
[31] Kilbas, A.; Srivastava, H.; Trujillo, J., Theory and applications of fractional differential equations, (2006), Elsevier · Zbl 1092.45003
[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), Springer-Verlag · Zbl 0914.58025
[34] Lakshmikantham, V.; Leela, S.; Devi, J.V., Theory of fractional dynamic systems, (2009), Cambridge Scientific Publishers · Zbl 1188.37002
[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)
[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) · Zbl 1080.26505
[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
[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, E75-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), Academic Press · Zbl 0918.34010
[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
[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
[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).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.