CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 1,Number 2, Spring 1993

GLOBAL HOPF BIFURCATION THEORY FOR CONDENSING FIELDS AND NEUTRAL EQUATIONS WITH APPLICATIONS TO LOSSLESS TRANSMISSION PROBLEMS

W.KRAWCEWICZ,J. W U AND H. XIA

1. Introduction. In [19, 25, 401, a purely topological approach, based on the S1-equivariant degree theory [18]and the Ize's comple- mentary function method [35], has been employed to develop an equiv- ariant bifurcation theory for the following parametrized equivariant co- incidence problem

where E and F are real Banach isometric representations of the group S1 { z E C : IzI = I), L : Dom ( L ) C E -, F is a closed equivariant F'redholm operator of index zero, and F : R2 XE + F is equivariant and completely continuous. Application of this bifurcation theory provides extensions to retarded functional differential equations and parabolic partial differential equations with or without delays of the well-known Alexander-Yorke's global bifurcation theorem which exploits very sharp information about global continua of periodic solutions.

One of the purposes of this paper is to generalize this equivariant bifurcation theory to (1.1) where F is equivariant and condensing with respect to a certain measure of noncompactness. The primary motivation for this generalization is to establish an analog of the Alexander-Yorke's global bifurcation theory and to develop a powerful tool in order to obtain the existence and multiplicity of periodic solutions for the following one-parameter family of neutral functional differential equations

when the parameter (Y is far away from those values at which the linearization of (1 .2 ) has a pair of purely imaginary characteristic

Received by the editors on November 22, 1991. The research by the first two authors was partially supported by NSERC-Canada.

Copyright 01993 Rocky Mountain Mathematics Consortium

168 W. KRAWCEWICZ, J. WU AND H. XIA

values, where x E Rn, a 2 0, xt(6) = x(t +9) for 9 E (-oo,a], cu E R, F and b : C((-oo, a]; Rn) x R Rn are continuously differentiable, and b satisfies the following Lipschitz condition (1.3)

Ib(cp,cu)--b($,ff)I Ik SUP I - , V,$E ((-00,aI;R") a€(-co,a]

for a constant k E (0,l). A standard abstract formulation of (1.2) leads to the following fully nonlinear coincidence problem

where L : Dom (L) E E -+ F is an equivariant Fredholm operator of index zero between E and F which are isometric Banach representations of S1, B : Ex R2 --, E is equivariant and condensing, N : Ex R2 --r F is equivariant and completely continuous, and ?r is the natural projection of E x R2 onto E.

In the nonequivariant case, (1.4) can be reduced to a fixed point problem for a condensing map, and, consequently, a topological de- gree (called composite coincidence degree) can be developed, based on Mawhin's coincidence degree and Nussbaum's fixed point index. For details, we refer to [20, 23, 53, 541. In this paper, we will prove that the idea of [20] can be generalized to show that (1.4) in the equivariant case can also be reduced to a fixed point problem for a certain equiv- ariant condensing map, or equivalently, to a problem of a zero for a certain equivariant condensing field.

One major technical tool is an equivariant version of the Nussbaum's bijection theorem which allows us to extend the S1-degree of Dylawer- ski et al. [18] from compact fields to condensing fields (see Section 2) and to obtain an equivariant version of composite coincidence degree for (1.4) (see Section 3). In Section 4, we will apply the equivariant com- posite coincidence degree in conjunction with the Ize's complementary function method to study the bifurcation problem for (1.4). We will obtain an equivariant analog of the local bifurcation theorem of Kras- nosel'skii type [38] and the global bifurcation theorem of Rabinowitz type [61]. This equivariant bifurcation theory for (1.4) will then, in Section 5, be applied to investigate the Hopf bifurcation problem for neutral equations (1.2). To each isolated center, we will assign a cross- ing number which is defined by the Brouwer degree of certain analytic

GLOBAL HOPF BIFURCATION THEORY 181

K = KO $ Io, where KO E C R ~ ( L O ) and lo is the identity mapping R -t R.

Consider the following nonlinear problem:

where no : V x R2 + V is the natural projection, Bo : V x R2 + V, and No : V x R2 -+ W are two equivariant mappings of class C1 such that (Bo, No) and its derivative (DBo, DNo) are both Lo-condensing G-pairs.

To describe our bifurcation problem, we assume that there exists a 2- dimensional submanifold M C vG x R2, where vG {v E V : yv = v for all y E S1), satisfying the following conditions:

( A ) For every x E M , no(x) - BO (x) E Dom (LO) and LO [no(x) - Bo(x)] = No (x), i . e., x is a solution to the equation (BP).

(B) If (vo,Ao) E M, vo E vG, A. E R2, then there exist an open neighborhood Ux, of A. in R2, an open neighborhood Uvo of vo in vG, and a c'-map 77 : UA, -+ vG such that

Since all points (v, A) E M are solutions of (BP), we call those points trivial solutions. All other solutions of (BP) will be called nontrivial. A point (vo, Xo) E M is called a bifircation point if in any neighborhood of (vo, Ao) there exists a nontrivial solution for (BP).

The problem (BP) is equivalent to the equation

where OK, (Bo, No) : V x R2 -t V is given by QKo (Bo, No) = Bo + R K ~ [No + KO(TO - Bo)]. We define a mapping f : V x R2 + V by f (v, A) = v - 6 3 ~ ~ (Bo, No) (v, A). Clearly, f is an equivariant condensing field of class C1. Therefore, by assumption,

D, f (v, A) = Id - OK,,(D,BO(V, A), DVNo(v, A)) : V + V,

where D, denotes the derivative with respect to v E V, is a condensing linear field, and hence is a Fredholm operator of index zero on the