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Hopf bifurcation analysis in a tri-neuron network with time delay. (English) Zbl 1149.34044
The paper studies the following three coupled delay differential equations (tri-neuron network) \align \dot{x}_{1}(t) & = \mu x_{1}(t) +f_{11}(x_{1}(t-\tau)) +f_{12}(x_{2}(t-\tau)) +f_{13}(x_{3}(t-\tau)),\\ \dot{x}_{2}(t) & = \mu x_{2}(t) +f_{21}(x_{1}(t-\tau)) +f_{22}(x_{2}(t-\tau)) +f_{23}(x_{3}(t-\tau)),\\ \dot{x}_{3}(t) & = \mu x_{3}(t) +f_{31}(x_{1}(t-\tau)) +f_{32}(x_{2}(t-\tau)) +f_{33}(x_{3}(t-\tau)),\endalign where $\tau>0$. Considering delay $\tau$ as a parameter, the paper investigates existence and direction of Hopf bifurcations as well as stability of bifurcating periodic solutions. The group of conditions is given for the system to have multiple periodic solutions when the delay becomes sufficiently large.

##### MSC:
 34K18 Bifurcation theory of functional differential equations 34K20 Stability theory of functional-differential equations 34K60 Qualitative investigation and simulation of models 34K13 Periodic solutions of functional differential equations
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##### References:
 [1] Baldi, P.; Atiya, A.: How delays affect neural dynamics and learning, IEEE trans. Neural networks 5, 610-621 (1994) [2] Campbell, S. A.; Belair, J.; Ohira, T.; Milton, J.: Limit cycles, tori, and complex dynamics in a second-order differential equation with delayed negative feedback, J. dynamic differential equations 7, No. 1, 213-236 (1995) · Zbl 0816.34048 · doi:10.1007/BF02218819 [3] 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 · doi:10.1016/S0167-2789(99)00111-6 [4] Chen, Y.; Wu, J.: Slowly oscillating periodic solutions for a delayed frustrated network for two neurons, J. math. Anal. appl. 259, 188-208 (2001) · Zbl 0998.34058 · doi:10.1006/jmaa.2000.7410 [5] Chen, Y.; Wu, J.: Existence and attraction of a phase-locked oscillation in a delayed frustrated network for two neurons, Differential integral equations 14, 1181-1236 (2001) · Zbl 1023.34065 [6] Faria, T.: On a planar system modelling a neuron network with memory, J. differential equations 168, 129-149 (2000) · Zbl 0961.92002 · doi:10.1006/jdeq.2000.3881 [7] Guo, S.; Huang, L.: Hopf bifurcating periodic orbits in a ring of neurons with delays, Pyhsica D 183, 19-44 (2003) · Zbl 1041.68079 · doi:10.1016/S0167-2789(03)00159-3 [8] Hale, J. K.: Theory of functional differential equations, (1977) · Zbl 0352.34001 [9] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H.: Theory and applications of Hopf bifurcation, (1981) · Zbl 0474.34002 [10] Li, Y.; Muldowney, J. S.: On Bendixson’s criterion, J. differential equations 106, 27-39 (1994) · Zbl 0786.34033 · doi:10.1006/jdeq.1993.1097 [11] Ruan, S.; Wei, J.: On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics continuous, discrete impulsive systems ser. A: math. Anal. 10, 863-874 (2003) · Zbl 1068.34072 [12] Shayer, L.; Campbell, S.: Stability, bifurcation, and multistability in a system of two coupled neurons with multiple time delays, SIAM J. Appl. math. 61, No. 2, 673-700 (2000) · Zbl 0992.92013 · doi:10.1137/S0036139998344015 [13] Song, Y.; Han, M.; Wei, J.: Stability and Hopf bifurcation 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 [14] Wang, L.; Zou, X.: Hopf bifurcation in bidirectional associative memory neural networks with delays: analysis and computation, J. comput. Appl. math. 167, 73-90 (2004) · Zbl 1054.65076 · doi:10.1016/j.cam.2003.09.047 [15] Wei, J.; Li, M. Y.: Global existence of periodic solutions in a tri-neuron network model with delays, Physica D 198, 106-119 (2004) · Zbl 1062.34077 · doi:10.1016/j.physd.2004.08.023 [16] 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 [17] Wei, J.; Velarde, M.: Bifurcation analysis and existence of periodic solutions in a simple neural network with delays, Chaos 14, No. 3, 940-952 (2004) · Zbl 1080.34064 · doi:10.1063/1.1768111 [18] Wei, J.; Yuan, Y.: Synchronized Hopf bifurcation analysis in a neural network model with delays, J. math. Anal. appl. 312, 205-229 (2005) · Zbl 1085.34058 · doi:10.1016/j.jmaa.2005.03.049 [19] Wu, J.: Symmetric functional differential equations and neural networks with memory, Trans. amer. Math. soc. 350, 4799-4838 (1998) · Zbl 0905.34034 · doi:10.1090/S0002-9947-98-02083-2 [20] 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 · doi:10.1016/S0895-7177(99)00120-X [21] Yan, X.: Hopf bifurcation and stability for a delayed tri-neuron neural network model, J. comput. Appl. math. 196, No. 2, 579-595 (2006) · Zbl 1175.37086 · doi:10.1016/j.cam.2005.10.012