TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 350, Number 12, December 1998, Pages 4799–4838 S 0002-9947(98)02083-2 SYMMETRIC FUNCTIONAL DIFFERENTIAL EQUATIONS AND NEURAL NETWORKS WITH MEMORY JIANHONG WU Abstract. We establish an analytic local Hopf bifurcation theorem and a topological global Hopf bifurcation theorem to detect the existence and to describe the spatial-temporal pattern, the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differen- tial equations in the presence of symmetry. We apply 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. 1. Introduction The purpose of this paper is to study the spatial-temporal patterns of solutions for systems of functional differential equations in the presence of symmetry. Of major concern is the existence, the asymptotic form, the isotropy group and the global continuation of periodic wave solutions. The well-known Hopfield-Cohen- Grossberg model of neural networks with delay provides the motivation and the illustration of our main general results. We will start with the symmetric local Hopf bifurcation problem of the following parametrized system of functional differential equations ˙ x(t)= L(α)x t + f (α,x t ), (1.1) where f (α, 0) = 0 and ∂ ∂φ f (α, 0) = 0 for α ∈ R and φ ∈ C := C([−τ, 0]; R n ), τ ≥ 0, is a given constant, x t is the usual notation for an element of C defined by x t (s)= x(t + s) with s ∈ [−τ, 0], L : R × C → R n is continuous and linear in the second argument. Moreover, there exists a compact Lie group Γ acting on R n such that f (α,γφ)= γf (α,φ) and L(α)γφ = γL(α)φ for (α,γ,φ) ∈ R × Γ × C, where γϕ ∈ C is given by (γφ)(s)= γφ(s) for s ∈ [−τ, 0]. We assume that there exists a critical value α 0 such that at α = α 0 , (i). The infinitesimal generator A(α) of the C 0 -semigroup generated by the linear system ˙ x(t)= L(α)x t has a pair of purely imaginary eigenvalues ±iβ 0 ; (ii) the generalized eigenspace U iβ0 (A(α 0 )) associated with iβ 0 consists of eigenvectors of A(α 0 ) only and the restricted action of Γ on U iβ0 (A(α 0 )) is isomorphic to V ⊕ V for some absolutely irreducible representation V of Γ. Received by the editors September 13, 1995. 1991 Mathematics Subject Classification. Primary 34K15, 34K20, 34C25. Key words and phrases. Periodic solution, delay differential equation, wave, symmetry, neural network, equivariant degree, global bifurcation. Research partially supported by the Natural Sciences and Engineering Research Council of Canada. c 1998 American Mathematical Society 4799 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use