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A model in a coupled system of simple neural oscillators with delays. (English) Zbl 1176.34087
A coupled system of simple neural oscillators with delays is studied. The authors prove that there is fully symmetric solution which loses stability as a parameter varies, and this loss of stability is due to the crossing of imaginary eigenvalues through the imaginary axis. So, Hopf bifurcation to periodic solutions appears. Moreover, spatio-temporal patterns appear as mirror-reflecting waves, standing waves, etc. which can be seen from the computer simulations.

34K18 Bifurcation theory of functional-differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
34K20 Stability theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
Full Text: DOI
[1] Atay, F.M., Oscillator death in coupled functional differential equations near Hopf bifurcation, Journal of differential equations, 221, 190-209, (2006) · Zbl 1099.34066
[2] Dias, Ana Paula S.; Lamb, Jeroen S.W., Local bifurcation in symmetric coupled cell networks: linear theory, Physica D, 223, 1, 93-108, (2006) · Zbl 1112.34025
[3] Golubitsky, M.; Stewart, I.N.; Schaeffer, D.G., ()
[4] Guo, S.; Huang, L., Hopf bifurcating periodic orbits in a ring of neurons with delays, Physica D, 183, 19-44, (2003) · Zbl 1041.68079
[5] Krawcewicz, W.; Wu, J., Theory and applications of Hopf bifurcations in symmetric functional differential equations, Nonlinear analysis, 35, 37, 845-870, (1999) · Zbl 0917.58027
[6] Peng, M., Bifurcation and stability analysis of nonlinear waves in symmetric delay differential systems, J. differential equations, 232, 2, 521-543, (2007) · Zbl 1118.34067
[7] Drubi, Fátima; Ibáñez, Santiago; Ángel Rodríguez, J., Coupling leads to chaos, J. differential equations, 239, 371-385, (2007) · Zbl 1133.34027
[8] Tachikawa, M., Specific locking in populations dynamics: symmetry analysis for coupled heteroclinic cycles, Journal of computational and applied mathematics, 201, 374-380, (2007) · Zbl 1121.34044
[9] Wu, J., Symmetric functional differential equations and neural networks with memory, Transactions of the American mathematical society, 350, 12, 4799-4838, (1998) · Zbl 0905.34034
[10] Wang, L.; Zou, Xi., Hopf bifurcation in bidirectional associative memory neural networks with delays: analysis and computation, Journal of computational and applied mathematics, 167, 73-90, (2004) · Zbl 1054.65076
[11] Yuri, A.K., Elements of applied bifurcation theory, (1995), Springer-Verlag New-York · Zbl 0829.58029
[12] Zhang, C.; Zheng, B., Stability and bifurcation of a two-dimension discrete neural network model with multi-delays, Chaos, solitons and fractals, 31, 1232-1242, (2007) · Zbl 1141.39013
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