Theory and applications of Hopf bifurcations in symmetric functional differential equations

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Introduction

In [15], we provided an analytic construction of an equivariant topological degree for compact fields which preserve certain symmetries. The computation formula for such a degree in some special cases was given in [26], with the help of which a bifurcation theory was developed for multi-parameter equivariant fixed point equations.

One of the purposes of this paper is to apply the results in 15, 26 to establish a general theory of Hopf bifurcations in symmetric functional differential equations. This general theory provides some important bifurcation invariants, called crossing numbers, to detect the existence of periodic solutions and to describe their orbits and global continuation. We will show that these crossing numbers can be computed from the linearization around equilibria and from the isotypical decomposition of representation spaces.

In comparison to the general theory of symmetric Hopf bifurcations for ordinary differential equations and some parabolic partial differential equations developed in 17, 32, 13, we establish some local symmetric Hopf bifurcation theorems for retarded functional differential equations without requiring genericity conditions on vector fields, dimension restrictions on some fixed point subspaces and maximality assumptions on a certain isotropy group. Unfortunately, due to the topological nature of our approach, we are unable to study the stability of the obtained branch of periodic solutions. Hopf bifurcation problems have been extensively studied via degree-theoretical approach and our approach is based on the equivariant degree theory developed in 15, 26 and is motivated by the work of 1, 2, 3, 4, 5, 7, 8, 9, 10, 14, 21, 25, 29, 31 in the Hopf bifurcation theory without symmetries. It should be mentioned that Ize, Massabó and Vignoli 22, 23 have developed a competing equivariant bifurcation theory.

Another purpose of this paper is to demonstrate the application of the general theory of Hopf bifurcations of symmetric functional differential equations. Of particular interest is the delay-induced oscillation and the global existence of large-amplitude symmetric periodic oscillations. As an illustrative example, we will consider a ring of identical cells coupled by diffusion along a polygon. Such a Turing ring provides a model for many biological and chemical systems. We incorporate a time delay in the coupling of adjacent cells because in many biological and chemical oscillators the time needed for the transport or processing of chemical components or signals may be of considerable length. To the best of our knowledge, the delay-induced oscillations of a Turing ring has not been investigated. We will illustrate that the time delay provides an important resource for the occurrence and global continuation of oscillations in a Turing ring. In particular, we will show that the delay may give rise to phase-locked oscillations even when the state of each cell is described by a single variable. This is in sharp contrast with the observation of 16, 18 that bifurcations of phase-locked oscillations cannot occur in a Turing ring in which the state of a cell is described by one variable if the delay is not presented.

The remaining part of this paper is organized as follows. In Section 2, we collect some results from [26]. In Section 3, we establish some results on the existence, the minimal period and the global continuation of a branch of periodic solutions for general symmetric functional differential equations. These results are then applied, in Section 4, to a ring of identical cells coupled by delayed diffusion along the sides of a polygon.

Section snippets

Preliminary results

We start with a brief review about equivariant degrees and we refer to [15] for details. Suppose that G is a fixed compact Lie group. For a subgroup HG, we use (H) to denote the conjugacy class of H in G which consists of all subgroups conjugate to H. We will use O(G) to stand for the set of all conjugacy classes of closed subgroups of G. Also, for each nonnegative integer n, defineOn(G)={(H)∈O(G);dimWH=n},where WH is the Weyl group NH/H and NH is the normalizer of H in G. We say a compact Lie

Hopf bifurcation theory for symmetric FDEs

Let τ≥0 be a given constant, N a positive integer and CN,τ the Banach space of continuous functions from [−τ,0] into RN equipped with the usual supremum norm‖ϕ‖=sup−τ≤θ≤0|ϕ(θ)|,ϕ∈CN,τ.In what follows, if x:[−r,A]→RN is a continuous function with A>0 and if t∈[0,A], then xtCN,τ is defined byxt(θ)=x(t+θ),θ∈[−τ,0].

Consider the following retarded functional differential equationẋ(t)=f(xt),where f:CN,τRN is a continuously differentiable function preserving a certain symmetry described by the

An example: discrete waves caused by delays in identical cells coupled in a ring

In this section, we illustrate our main result for discrete waves with a ring of identical oscillators with identical coupling between adjacent cells. Such a ring was modelled in the seminar paper by [34] on morphogenesis and provides models for many situations in biology, chemistry and electrical engineering. The Hopf bifurcation of this Turing ring has been extensively studied in the literature. We refer to 2, 3, 13, 18, 20, 30, 32, 36 and references therein for the current state of the

For Further Reading:

The following references are also of interest to the reader: [24], [33] and [35].

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    1

    This research was partly supported by NSERC-Canada and by Alexander von Humboldt Foundation.

    2

    This research was partly supported by NSERC-Canada and by Faculty of Arts (York University) Fellowship.

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