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Symmetric functional-differential equations and neural networks with memory. (English) Zbl 0905.34034
Summary: The author establishes an analytic local Hopf bifurcation theorem and a topological global Hopf bifurcation theorem to detect the existence and to describe spatial-temporal pattern, asymptotic form and global continuation of bifurcations of periodic wave solutions to functional-differential equations with symmetry. The author applies these general results to obtain the coexistence of multiple large-amplitude wave solutions for the delayed Hopfield-Cohen-Grossberg model of neural networks with a symmetric circulant connection matrix.

MSC:
34C23Bifurcation (ODE)
34K13Periodic solutions of functional differential equations
34K18Bifurcation theory of functional differential equations
34C25Periodic solutions of ODE
34K20Stability theory of functional-differential equations
92B20General theory of neural networks (mathematical biology)