Research Article | Open Access
Equivariant Hopf Bifurcation in a Ring of Identical Cells with Delay
A kind of delay neural network with n elements is considered. By analyzing the distribution of the eigenvalues, a bifurcation set is given in an appropriate parameter space. Then by using the theory of equivariant Hopf bifurcations of ordinary differential equations due to Golubitsky et al. (1988) and delay differential equations due to Wu (1998), and combining the normal form theory of functional differential equations due to Faria and Magalhaes (1995), the equivariant Hopf bifurcation is completely analyzed.
The works of Golubitsky et al. [1, Chapter XVIII] have shown that rings of identical cells can lead to many interesting patterns of oscillation, which are predictable based on the theory of equivariant bifurcations. In a series of papers, Wu et al. [2–4] have extended the theory of equivariant Hopf bifurcation to delay differential equations. Recently, there has been interest in applying these results to neural networks, primarily to models related to the Hopfield-Cohen-Grossberg neural networks with time delays [5–9]. In a series of papers [10–23], the authors studied the Hopf bifurcation to a network with one or two delays. Particularly, Wu et al.  have investigated the synchronization and stable phase-locking in a delayed network with three identical neurons: where is a sufficiently smooth sigmoid amplification function, normalized so that and , and are regarded as parameters. They have shown that in the region of the normalized parameters, system (4.1) is absolutely synchronous in the sense that every solution is convergent to the set of all synchronized phase states independent of the size of the time delay. The -limit set of a given orbit can be either a synchronized equilibrium or a synchronized periodic solution, depending on the connection topology of the network, the strength of the self-connection and the neighborhood-interaction, and the size of the delay. On the other hand, they have shown that, in the region , there exists a continuous surface where Hopf bifurcation of either a stable synchronized periodic solution or two stable phase-locked periodic solutions and six unstable periodic waves take place (more precisely, three mirror-reflecting waves and three standing waves). A question of mathematical and biological interest is whether the dynamics of (4.1) are possible for the following neural network model with elements: The purpose of the paper is to provide a detailed analysis of this question in the case that is an odd number. More precisely, we shall extend the main results of Wu et al.  to the system (1.2) with being an odd number. We are going to regard the delay as a parameter to investigate the dynamics of (1.2). According to the properties of the function , we assume that the transfer function is adequately smooth, for example, , and satisfies the following normalization, monotonicity, concavity, and boundedness conditions:(C1)(C2)(C3)(C4)
At first, by analyzing the distribution of the eigenvalues, we give a bifurcation set in an appropriate parameter space to describe the stability of the equilibrium of the system (1.2) of how to change as the parameters change. Meanwhile, the equivariant Hopf bifurcations are found. Then, by employing the center manifold and normal form theory, the direction of the bifurcation and the stability of the bifurcating periodic solutions are determined.
The rest of this paper is organized as follows. In Section 2, the characteristic equation of the linearization of system (1.2) at the zero equilibrium is derived. In Section 3, by analyzing the distribution of the eigenvalues, a bifurcation set is given in the -plane. Then by using the theory of equivariant Hopf bifurcation to ordinary and delay differential equations due to Golubitsky et al.  and Wu , respectively, the local equivariant Hopf bifurcation of the trivial solution is completely analyzed. The appendix contain the detailed calculations of the normal forms on center manifolds of system (1.2) near Hopf bifurcation points.
2. The Characteristic Equations
From conditions , we know that the linearization of system (1.2) at an equilibrium is given by The characteristic matrix is where is identity matrix and
Let From it follows that Hence, and the characteristic equation is where We observe that (2.7) can be simplified, further elucidating the structure of characteristic equation.(I)When is an odd number, we obtain (II) When is an even number, we obtain
In the case when is even, that is, , may be further simplified into two subcases.(i)When is an odd number, we have (ii)When is an even number, we have (1)When is an odd number, (2)When is an even number and is not the multiple of , (3)When is the multiple of ,
3. Bifurcation Analysis with being an Odd Number
In this section, we are going to analyze the distribution of the root of the characteristic equations given in previous section, respectively.
In this case, the characteristic equation of (1.2) is Clearly, is a root of (3.1) if and only if there exists a such that satisfies that is, This leads to This shows that the necessary condition for (3.1) possessing purely imaginary roots is that there exists such that , that is,
From (3.1), conclusion then follows immediately.
Proposition 3.1. Equation (3.1) has at most pairs of purely imaginary roots when is an odd number.
Without loss of generality, one assumes the (3.1) has pairs of purely imaginary roots exactly. Define Then are purely imaginary roots of (3.1) with , respectively. one notes that when
We can obtain the following result with no difficulty.
Proposition 3.2. Let be the root of (3.1) near satisfying , . Then
Proof. From (3.1) we have
Differentiating both sides with respect to , we have
Notice that , it follows that This completes the proof.
When , (3.1) becomes Its roots are given by
For convenience, denote and let . One can find out that the region is a sector, is a parallelogram, and is the outside of the parallelogram in the sector . These are shown in Figure 1.
Proposition 3.3. If and , that is , then all roots of (3.13) have negative real parts.
Proposition 3.4. If and , that is, , then all roots of (3.1) have negative real parts for all .
From Proposition and Corollary of Ruan and Wei , conclusion then follows immediately.
Proposition 3.5. If there exist such that , that is, is located outside of the sector , then (3.1) has at least one root with positive part for all .
Proposition 3.6. Suppose that one of the following conditions is satisfied (i.e., ):
(1)(2)(3) Then there exist at most sequences of critical value defined by (3.7) such that(1)all roots of (3.1) have negative real parts for all , where (2)equation (3.1) has at least one root with positive part when ;(3)at , (3.1) has a pair of purely imaginary roots
Theorem 3.7. (i) If and , then the zero equilibrium of (1.2) is asymptotically stable for all .
(ii) If exist such that , then the zero equilibrium of (1.2) is unstable for all .(iii) Suppose that one of the following conditions holds:(1)(2)(3) Then there exist at most sequences of critical value defined by (3.7) such that the zero equilibrium of (1.2) is asymptotically stable for , where , unstable for all , and the system undergoes a Hopf bifurcation with .
Proposition 3.6 shows that the region , which is the outside of the parallelogram in the sector shown as in Figure 1, is a conditionally stable region. This means that the stability of the zero equilibrium of (1.2) is dependent on the delay. Namely, the distribution of the roots of (3.1) is dependent on the delay. Particularly, for , (3.1) with some critical value has simple pure imaginary roots or multiply pure imaginary roots. It is well know that the Hopf bifurcations are different between the simple and multiple pure imaginary roots. So it is necessary to observe the region . In fact, we have the following conclusions on occurrence of purely imaginary roots of (3.1) for .
(1) In the region Equation (3.1) with has one pair of purely imaginary roots exactly, denoted by , which are simple, where and
(3) In the region Equation (3.1) has three pairs of purely imaginary roots exactly, denoted by , , and , when , , and , respectively. Here are simple, and are double, , and are defined by (3.20) and (3.22), respectively, and Furthermore, Naturally, the region (when is multiple of ) or (when is not multiple of ) can be defined by
: Obviously, in the region (or ), we have that
(4) In the region Equation (3.1) has a pair of purely imaginary roots exactly, denoted by , which are double, where and
(5) In the region Equation (3.1) has two pairs of purely imaginary roots exactly, denoted by and , when and , respectively, and the two pairs of purely imaginary roots are all double. Here is defined by (3.30), and Moreover, Naturally, the region (when is multiple of ) or (when is not multiple of ) can be defined by
: Obviously, in the region (or ), we have
To carry out our work we need some background from the theory of functional differential equation. Let denote the Banach space of continuous mappings from into equipped with the supernorm. Let be a solution of (1.2) and define . If is continuous, then . With this structure, we may write the model as the following functional differential equation: where is defined via Similarly, the linearization of (3.37) at equilibrium may be written as where the linear operator is defined via It is well known that a linear functional differential equation such as (3.39) generates a strongly continuous semigroup of linear operators with infinitesimal generator given by where .
To explore the possible (spatial) symmetry of (1.2), we need to introduce three compact Lie groups. One is the cycle group ; another is , the cyclic group of order , which corresponds to rotations of , denoting the generator of this group by , then action on is given by ; the third is the dihedral group of order , which corresponds to the group of symmetries of an -gon. It can be shown that is generated by and , where is the flip of order 2 or reflection, and it acts on by .
Definition 3.8. Let and be a compact group. The system (3.37) is said to be -equivariant if for all .
Proof. We begin with (3.37); that is, we let be as in (3.38) and . We need only check the equivariant condition on the generators, , , of :
Thus (3.37) is equivariant.
From (3.39), we begin by noting that may be written componentwise as follows: The rest of the proof is similar to that for (3.37). This completes the proof.
In the case where , we can apply the standard Hopf bifurcation theorem of delay differential equations to obtain a Hopf bifurcation of synchronous periodic solutions. In the case where (), the aforementioned standard Hopf bifurcation theorem does not apply since are double eigenvalues. On the other hand, the considered system is equivariant with respect to the -action where the subgroup acts by permutation (sending to ) and the flip acts by interchanging (sending to ). This allows us to apply the symmetric Hopf bifurcation theorem for delay differential equations established in  (as an extension of the well-known Golubitsky-Stewart Theorem  for symmetric ordinary differential equations) to obtain branches of asynchronous periodic solutions. More precisely, we have the following theorem.
Theorem 3.10. Assume that (, where the regions and are defined by (3.15) and (3.16), resp.). Then
(1)in case , near there exists a supercritical bifurcation of stable synchronous periodic solutions of period near , bifurcated from the zero solution of system (1.2);(2)in case (), near there exist branches of asynchronous periodic solutions of period near , bifurcated simultaneously from the zero solution of system (1.2), and there are(a)two stable phase-locked oscillations: , for and ,(b) unstable mirror-reflecting waves: for and , where ,(c) unstable standing waves: for and , where .
Proof. (1) The existence is an immediate application of the standard Hopf bifurcation theorem for functional differential equations. Let , . According to the calculations in Part 2 of the appendix, the normal form of (1.2) on the center manifold can be written in polar coordinates as
Conclusion then follows immediately.
(2) Let , , . We obtain from the calculations in Part 1 of the appendix the normal form of (1.2) on the center manifolds that is given by where Introducing the periodic-scaling parameter and letting with and using a computation process similar to that of Wu et al. , we obtain the normal form Let be given so that is the right-hand side of (3.50). Then (3.50) can be written as Note that Also note that is -equivariant with respect to the following -action on : Using a computation process similar to that of Wu et al. , we obtain By the results of [1, page 376], we know that the bifurcation of phase-locked oscillation is supercritical (resp., subcritical) and depends on whether (resp., ), and these are orbitally asymptotically stable if and .
Note that Consequently, the bifurcation of phase-locked oscillations is supercritical and orbitally asymptotically stable.
Note also that
We infer from the results of [1, page 376] again that the bifurcations of mirror-reflecting waves and standing waves are supercritical and unstable. This completes the proof.
Using a proof process similar to that of Yuan and Campbell , by using Liapunov's second method we can obtain the followings.
Theorem 3.11. If , then the trivial solution of (1.2) is global asymptotically stable.
4. Computer Simulations
Clearly, the origin is a fixed point to (4.1). Choosing and we obtain that and , thus .
From Theorem 3.7, it follows that the zero equilibrium is asymptotically stable if , it is unstable if , and the system undergoes a Hopf bifurcation with . Simulate the solutions of system (4.1) for and . In Figure 2, it is shown that the zero equilibrium is asymptotically stable for . In Figure 3, for the data , it is shown that there exists a periodic orbit which is orbitally asymptotically stable.
A. The Calculation of Normal Forms on Center Manifolds with being an Odd Number
We first rescale the time by to normalize the delay so that (1.2) can be written as in the phase space , where for , we have with . We also assume that with The linearized equation at zero for system (A.1) is where The characteristic equation of (A.5) at is , that is, where is the characteristic equation of the linearization of (1.2) at .
Part 1. Case 1 (). In this case, at the characteristic equation of (A.5) has imaginary zeros which are double, where and