Hopf bifurcation analysis of two neurons with three delays. (English) Zbl 1149.34046
The authors study linear stability and give conditions and direction for Hopf bifurcations in the following system of coupled delay differential equations (two neurons with three delays)
\[
\begin{aligned} \dot{x}(t) & = - x(t) +a_{11}f(x(t-\tau)) +a_{12}f(y(t-\tau_1)) ,\\ \dot{y}(t) & = - y(t) +a_{21}f(x(t-\tau_2)) +a_{22}f(y(t-\tau)). \end{aligned}
\]
Here \(x\) and \(y\) are scalar variables corresponding to two neurons, \(\tau_j\) denote the transmission delays, \(a_{ij}\) are synaptic weights. Additionally, \(f(0)=0\) so that the zero solution is the equilibrium, which produces Hopf bifurcations studied.
Reviewer: Sergiy Yanchuk (Berlin)
MSC:
| 34K18 | Bifurcation theory of functional-differential equations |
| 34K20 | Stability theory of functional-differential equations |
| 34K13 | Periodic solutions to functional-differential equations |
Software:
dde23References:
| [1] | Baptistini, M.; Táboas, P., On the existence and global bifurcation of periodic solutions to planar differential delay equations, J. Differential Equations, 127, 391-425 (1996) · Zbl 0849.34053 |
| [2] | 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 |
| [3] | 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 |
| [4] | 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 |
| [5] | Chen, Y.; Wu, J., The asymptotic shapes of periodic solutions of a singular delay differential systems, J. Differential Equations, 169, 614-632 (2001) · Zbl 0976.34060 |
| [6] | Faria, T., On a planar system modelling a neuron network with memory, J. Differential Equations, 168, 129-149 (2000) · Zbl 0961.92002 |
| [7] | Fotios, G.; Andreas, Z., Bifurcations in a planar system of differential delay equations modeling neural activity, Physica D, 159, 215-232 (2001) · Zbl 0984.92505 |
| [8] | Godoy, S. M.S.; Dos Reis, J. G., Stability and existence of periodic solutions of a functional differential equation, J. Math. Anal. Appl., 198, 381-398 (1996) · Zbl 0851.34074 |
| [9] | Gopalsamy, K.; Leung, I., Delay induced periodicity in a neural netlet of excitation and inhibition, Physica D, 89, 395-426 (1996) · Zbl 0883.68108 |
| [10] | Guo, S.; Huang, L., Linear stability and Hopf bifurcation in a two-neuron network with three delays, Internat. J. Bifurcation Chaos, 8, 2799-2810 (2004) · Zbl 1062.34078 |
| [11] | Hale, J.; Lunel, S. V., Introduction to Functional Differential Equations (1993), Springer: Springer New York · Zbl 0787.34002 |
| [12] | B. Hassard, N.Kazarinoff, Y.H. Wan, Theory of applications of Hopf bifurcation, London Math, Society Lecture Notes, Series, vol. 41, Cambridge University Press, Cambridge, 1981.; B. Hassard, N.Kazarinoff, Y.H. Wan, Theory of applications of Hopf bifurcation, London Math, Society Lecture Notes, Series, vol. 41, Cambridge University Press, Cambridge, 1981. · Zbl 0474.34002 |
| [13] | Li, C.; Chen, G.; Liao, X.; Yu, J., Hopf bifurcation in an Internet congestion control model, Chaos Solitons Fractals, 19, 853-862 (2004) · Zbl 1058.34090 |
| [14] | Li, C.; Chen, G.; Liao, X.; Yu, J., Hopf bifurcation and chaos in a single inertial neuron model with time delay, European Phys. J. B, 41, 337-343 (2004) |
| [15] | Li, S.; Liao, X.; Li, C.; Wong, K., Hopf bifurcation of a two-neuron network with different discrete-time delays, Internat. J. Bifurcation Chaos, 15, 5, 1589-1601 (2005) · Zbl 1092.34563 |
| [16] | Liao, X., Hopf and resonant codimension two bifurcation in van der Pol equation with two time delays, Chaos Solitons Fractals, 23, 857-871 (2005) · Zbl 1076.34087 |
| [17] | Mahaffy, J.; Joiner, K.; Zak, P., A geometric analysis of stability regions for a linear differential equation with two delays, Internat. J. Bifurcations Chaos, 5, 779-796 (1995) · Zbl 0887.34070 |
| [18] | Olien, L.; Bélair, J., Bifurcations, stability, and monotonicity properties of a delayed neural network model, Physica D, 102, 349-363 (1997) · Zbl 0887.34069 |
| [19] | Qin, Y.; Wang, L.; Liu, Y.; Zheng, Z., Stability of the Dynamics Systems (1989), Science Press: Science Press Beijing |
| [20] | Ruan, S.; Wei, J., Periodic solutions of planar systems with two delays, Proc. R. Soc. Edinburgh, 129A, 1017-1032 (1999) · Zbl 0946.34062 |
| [21] | Shampine, L. F.; Thompson, S., Solving DDEs in Matlab, Appl. Numer. Math., 37, 441-458 (2001) · Zbl 0983.65079 |
| [22] | Shayer, L.; Campbell, S., 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 |
| [23] | Táboas, P., Periodic solution of a planar delay equation, Proc. R. Soc. Edinburgh, 116A, 85-101 (1990) · Zbl 0719.34125 |
| [24] | Wei, J.; Ruan, S., Stability and bifurcation in a neural network model with two delays, Physica D, 130, 255-272 (1999) · Zbl 1066.34511 |
| [25] | Wu, J., Symmetric functional differential and neural networks with memory, Trans. Amer. Math. Soc., 350, 4799-4838 (1998) · Zbl 0905.34034 |
| [26] | Wu, J., Introduction to Neural Dynamics and Signal Transmission Delay (2001), Walter de Cruyter: Walter de Cruyter Berlin · Zbl 0977.34069 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
