## Multistability and stable asynchronous periodic oscillations in a multiple-delayed neural system.(English)Zbl 1100.34054

Here, neural networks with multiple delays are studied. Local stability and bifurcations of the trivial solution of such systems are analyzed. It is proved that the existence of standard Hopf bifurcations gives rise to synchronized periodic solutions, as well as $$D_3$$ equivariant Hopf bifurcations give rise to phase-locked, mirror-reflecting and standing waves.

### MSC:

 34K13 Periodic solutions to functional-differential equations 92B20 Neural networks for/in biological studies, artificial life and related topics 34K20 Stability theory of functional-differential equations 34K18 Bifurcation theory of functional-differential equations
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### References:

 [1] Barbălat, I., Systèmes d’équations differentielles d’oscillations non linéaires, Rev. math. pures appl., 4, 267-270, (1959) · Zbl 0090.06601 [2] Bélair, J., Stability in delayed neural networks, (), 6-9 · Zbl 0788.34073 [3] Bélair, J., Stability in a model of a delayed neural network, J. dyn. syst. differential equations, 5, 607-623, (1993) · Zbl 0796.34063 [4] Bélair, J.; Campbell, S.A., Stability and bifurcations of equilibria in a multiple delayed differential equation, SIAM J. appl. math., 54, 1402-1423, (1994) · Zbl 0809.34077 [5] Bélair, J.; Campbell, S.A.; van den Driessche, P., Frustration, stability and delay-induced oscillations in a neural network model, SIAM J. appl. math., 46, 245-255, (1996) · Zbl 0840.92003 [6] Bélair, J.; Dufour, S., Stability in a three-dimensional system of delay-differential equations, Canad. appl. math. quart., 4, 136-156, (1996) · Zbl 0880.34075 [7] S. Bungay, S.A. Campbell, Y. Yuan, Bifurcation interactions in a ring of identical cells with delayed coupling, Preprint, 2006 [8] Campbell, S.A., Stability and bifurcation of a simple neural network with multiple time delays, Fields inst. commun., 21, 65-79, (1999) · Zbl 0926.92003 [9] Campbell, S.A.; Bélair, J.; Ohira, T.; Milton, J., Limit cycles, tori, and complex dynamics in a second-order differential equations with delayed negative feedback, J. dynam. differential equations, 7, 1, 213-236, (1995) · Zbl 0816.34048 [10] Campbell, S.A.; Bélair, J.; Ohira, T.; Milton, J., Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback, Chaos, 5, 4, 640-645, (1995) · Zbl 1055.34511 [11] Campbell, S.A.; Ruan, S.; Wei, J., Qualitative analysis of a neural network mode with multiple time delays, Internat. J. bifur. chaos, 9, 8, 1585-1595, (1999) · Zbl 1192.37115 [12] Campbell, S.A.; Yuan, Y.; Bungay, S., Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling, Nonlinearity, 18, 2827-2846, (2005) · Zbl 1094.34049 [13] Chen, Y.; Wu, J., Existence and attraction of a phase-locked oscillation in a delayed network of two neurons, Differential integral equations, 14, 1181-1236, (2001) · Zbl 1023.34065 [14] Chen, Y.; Wu, J., Slowly oscillating periodic solutions for a delayed frustrated network of two neurons, J. math. anal. appl., 259, 188-208, (2001) · Zbl 0998.34058 [15] Chen, Y.; Wu, J., The asymptotic shapes of periodic solutions of a singular delay differential system, Celebration of Jack K. hale’s 70th birthday, part 4 (Atlanta, GA/Lisbon, 1998), J. differential equations, 169, 614-632, (2001), (special issue) · Zbl 0976.34060 [16] Chen, Y.; Wu, J., Minimal instability and unstable set of a phase-locked periodic orbit in a delayed neural network, Physica D, 134, 185-199, (1999) · Zbl 0942.34062 [17] Chen, Y.; Wu, J.; Krisztin, T., Connecting orbits from synchronous periodic solutions in phase-locked periodic solutions in a delay differential system, J. differential equations, 163, 130-173, (2000) · Zbl 0955.34058 [18] Faria, T.; Magalhães, L.T., Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcations, J. differential equations, 122, 181-200, (1995) · Zbl 0836.34068 [19] Golubitsky, M.; Stewart, I.; Schaeffer, D.G., Singularities and groups in bifurcation theory, (1988), Springer-Verlag New York · Zbl 0691.58003 [20] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Publishers The Netherlands · Zbl 0752.34039 [21] Gopalsamy, K.; Leung, I., Delay induced periodicity in a neural netlet of excitation and inhibition, Physica D, 89, 395-426, (1996) · Zbl 0883.68108 [22] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1983), Springer-Verlag New York · Zbl 0515.34001 [23] Haddock, J.; Terjeki, J., Liapunov – razumikhin functions and an invariance principle for functional differential equations, J. differential equations, 48, 95-122, (1983) · Zbl 0531.34058 [24] Halanay, A., () [25] Hale, J., Theory of functional differential equations, (1977), Springer-Verlag New York [26] Hale, J.; Huang, W., Global geometry of the stable regions for two delay differential equations, J. math. anal. appl., 178, 344-362, (1993) · Zbl 0787.34062 [27] Hale, J.; Lunel, S.M.V., Introduction to functional differential equations, (1993), Springer-Verlag New York [28] Huang, L.; Wu, J., Nonlinear waves in networks of neurons with delayed feedback: pattern formation and continuation, SIAM J. math. anal., 34, 4, 836-860, (2003) · Zbl 1038.34076 [29] Kolmanovskii, V.B.; Nosov, V.R., Stability of functional differential equations, () · Zbl 0593.34070 [30] Krawcewicz, W.; Wu, J., Theory and applications of Hopf bifurcations in symmetric functional-differential equations, Nonlinear anal., 35, 845-870, (1999) · Zbl 0917.58027 [31] Mahaffy, J.M.; Zak, P.J.; Joiner, K.M., A geometric analysis of stability regions for a linear differential equation with two delays, Internat. J. bifur. chaos, 5, 779-796, (1995) · Zbl 0887.34070 [32] Marcus, C.M.; Waugh, F.R.; Westervelt, R.M., Nonlinear dynamics and stability of analog neural networks, Physica D, 51, 234-247, (1991) · Zbl 0800.92059 [33] Marcus, C.M.; Westervelt, R.M., Basins of attraction for electronic neural networks, (), 524-533 [34] Marcus, C.M.; Westervelt, R.M., Stability of analog neural networks with delay, Phys. rev. A, 39, 347-359, (1989) [35] Milton, J., Dynamics of small neural populations, () · Zbl 0879.92005 [36] Ncube, I.; Campbell, S.A.; Wu, J., Change in criticality of synchronous Hopf bifurcation in a multiple-delayed neural system, Fields inst. commun., 36, 179-194, (2003) · Zbl 1162.92301 [37] Nussbaum, R., Differential delay equations with two time delays, Mem. amer. math. soc., 16, (1978) [38] Olien, L.; Bélair, J., Bifurcations, stability and monotonicity properties of a delayed neural network, Physica D, 102, 349-363, (1997) · Zbl 0887.34069 [39] Orosz, G.; Stépán, G., Hopf bifurcation calculations in delayed systems with translational symmetry, J. nonlinear sci., 14, 6, 505-528, (2004) · Zbl 1123.37049 [40] Orosz, G.; Wilson, R.E.; Krauskopf, B., Global bifurcation investigation of an optimal velocity traffic model with driver reaction time, Phys. rev. E, 70, 2, 026207, (2004) [41] Pakdaman, K.; Malta, C.P.; Grotta-Ragazzo, C.; Vibert, J.-F., Delay-induced transient oscillations in a two-neuron network, Resenhas, 45-54, (1997) · Zbl 1098.92504 [42] Pakdaman, K.; Malta, C.P.; Grotta-Ragazzo, C.; Vibert, J.-F., Effect of delay on the boundary of the basin of attraction in a self-excited single neuron, Neural comput., 9, 319-336, (1997) · Zbl 0869.68086 [43] Pakdaman, K.; Malta, C.P.; Grotta-Ragazzo, C.; Arino, O.; Vibert, J.-F., Transient oscillations in continuous-time excitatory ring neural networks with delay, Phys. rev. E, 55, 3234-3248, (1997) [44] Pakdaman, K.; Grotta-Ragazzo, C.; Malta, C.P., Transient regime duration in continuous-time neural networks with delay, Phys. rev. E, 58, 3623-3627, (1998) [45] Pakdaman, K.; Grotta-Ragazzo, C.; Malta, C.P.; Arino, O.; Vibert, J.-F., Effect of delay on the boundary of the basin of attraction in a system of two neurons, Neural netw., 11, 509-519, (1998) [46] Sattinger, D.H., () [47] Shayer, L.P.; Campbell, S.A., Stability, bifurcation and multistability in a system of two coupled neurons with multiple time delays, SIAM J. appl. math., 61, 2, 673-700, (2000) · Zbl 0992.92013 [48] Stépán, G., () [49] Stépán, G.; Haller, G., Quasiperiodic oscillations in robot dynamics, Nonlinear dynam., 8, 513-528, (1995) [50] Wei, J.; Ruan, S., Stability and bifurcation in a neural network model with two delays, Physica D, 130, 255-272, (1999) · Zbl 1066.34511 [51] Wu, J., Symmetric functional differential equations and neural networks with memory, Trans. amer. math. soc., 350, 4799-4838, (1998) · Zbl 0905.34034 [52] Wu, J., Introduction to neural dynamics and signal transmission delay, (2001), De-Gruyter Berlin · Zbl 0977.34069 [53] Wu, J.; Faria, T.; Huang, Y.S., Synchronization and stable phase-locking in a network of neurons with memory, Math. comput. modelling, 30, 117-138, (1999) · Zbl 1043.92500
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