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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 368652, 18 pages

http://dx.doi.org/10.1155/2014/368652
Research Article

Bifurcation Analysis and Spatiotemporal Patterns of Nonlinear Oscillations in a Ring Lattice of Identical Neurons with Delayed Coupling

1Department of Mathematics, Shanghai Maritime University, Shanghai 201306, China

2Department of Mathematics, Tongji University, Shanghai 200092, China

Received 2 May 2013; Revised 15 March 2014; Accepted 17 March 2014; Published 29 April 2014

Academic Editor: Patricia J. Y. Wong

Copyright © 2014 Jiao Jiang and Yongli Song. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the dynamics of a delayed neural network model consisting of identical neurons. We first analyze stability of the zero solution and then study the effect of time delay on the dynamics of the system. We also investigate the steady state bifurcations and their stability. The direction and stability of the Hopf bifurcation and the pitchfork bifurcation are analyzed by using the derived normal forms on center manifolds. Then, the spatiotemporal patterns of bifurcating periodic solutions are investigated by using the symmetric bifurcation theory, Lie group theory and -equivariant degree theory. Finally, two neural network models with four or seven neurons are used to verify our theoretical results.

1. Introduction

Growing biological evidence suggests that propagation delay in axons and dendrites may play a key role in the processing of sensory information by the brain. For instance, spatiotemporal patterns of synchronous oscillations in and frequency bands emerge in the hippocampus when a rat goes through previously visited places. These oscillations and their interaction form a coding scheme that is used to readout from long-time memory (see, e.g., [1]). Although there are many experimental results pointing to the phase-locked firing of pyramidal cells (see, e.g., [2]), the mechanisms underlying this phenomenon are fairly unknown. The hippocampus has a laminar stratificated structure, where each pyramidal cell receives and integrates a large amount of spikes arriving at different time instants to different parts of the cell. Then their active integration by the cell, to produce an output spike, depends on the relative time delays of the incoming spikes. Thus the cell output is conditioned by delays occurring in incoming electric wave, which justifies the need of mathematical modeling of these complex phenomena [3].

On the other hand, artificial neural networks including delays have been known to be useful for mimicking various neuroprocesses like in the image processing (see, e.g., [4]). Their further expansion and exploitation are limited by the added complexity of the mathematical analysis brought by the delay (ordinary or partial differential equations become functional differential equations thus implying an infinite number of degrees of freedom). There are, however, some advantages. For instance, a harmonic oscillator augmented with time delay shows stable “robust” oscillations [5]. The “fragile” character of linear oscillations and linear waves when unfolded in space precludes their utility in reliably carrying information. An interesting point, however, is that a harmonic oscillator augmented with white noise permits transferring the latter into a colored noise thus creating a memory-like stochastic process and so a kind of delay process. In [6] this idea has been used to model quasiharmonic oscillations observed in inferior olive. The spiking propagation process in the brain comes indeed from robust oscillations as known by many authors. This fact permits modeling the dynamics using harmonic oscillators with appropriate nonlinearity which is like in the case of a van der Pol-Bonhöcffer oscillator and the Fitzhugh-Nagummo excitability dynamics underlying the Hodgkin-Huxley equation. An alternative to van der Pol's idea was earlier suggested by Lord Ragleigh who also proposed augmenting the harmonic oscillator with suitable active fiction thus allowing maintained oscillations. In both cases we have systems where there is an appropriate input-output energy balance and thus driven-dissipative systems.

Focusing on robots or neuron inspired artifacts, the use of ring lattice models has shown potential in different applications, for example, from electron transport to the dynamical decentralized gait control in robotics [7]. In the latter case movement of each robot leg is associated (coupled) with the behavior of the corresponding unit in the ring. Then standing waves excited in the ring can be used to reproduce the leg movement without the need of a computer, unlike what happens in nature, which uses central pattern generators (CPG) designing artificial locomotion for this purpose [8]. It has been shown in [9] for a hexapod robot that a ring of six units is able to reproduce metachronal, caterpillar, and tripod gaits. However, ripple gait, widely employed by insects for medium speed movement, is difficult to mimic. We foresee that the use of delays may help in CPGs to control the robot walking behavior.

Since Marcus and Westervelt [10] incorporated a single time delay into the connection term of Hopfield's model and observed sustained oscillations resulting from this time delay, there has been a growing interest to the dynamical properties of neural networks when the time delay cannot be neglected. It is known that delay can significantly alter the dynamics of neural networks [1012], for example, leading to instability of a delay-free stable solution [13]. There have been lots of researches on the neural networks with delay (see, e.g., [1418] and references therein).

A ring with identical elements often has a kind of topological symmetry. Considering these symmetries helps in the analysis of wave solutions in a ring lattice. Golubitsky et al. [19] have shown that systems with symmetry can exhibit different oscillatory patterns, which are predictable based on the theory of equivariant bifurcations. Later, in a series of papers, Wu and coworkers [2022] have extended the theory of equivariant Hopf bifurcations to functional differential equations. These theoretical advances have led to a vast literature on the mechanisms of spatiotemporal activity in neural networks with symmetry and delays (see, e.g., [2235]). The majority of these studies have focused on a ring structure with bidirectional couplings between the neighboring elements. From the view point of group theory, the symmetry in a ring with such bidirectional coupling if there is a reflection symmetry is given by a dihedral group (the symmetry of a regular -gon: is the number of units). Another type of symmetry in a ring is a cyclic group (the symmetry of a direct -gon) if one direction is preferred; that is, the coupling goes in one direction. The latter type of symmetry is more natural for the neural networks. In 1994, Baldi and Atiya [11] proposed a neural network consisting of neurons connected in a ring by undirectional couplings with delays (Figure 1(a)): where is the state of the th unit (related to the membrane potential), is the transfer function, which describes chemical interaction between pre- and postsynaptic neurons, and accounts for the coupling strength. For positive or negative we can speak of excitatory or inhibitory coupling, respectively. The nonnegative constants represent the transmission delays between neighboring neurons.

fig1
Figure 1: Lattice ring. (a) Network architecture with units unidirectionally coupled. Each unit receives a delayed output from the preceding unit. (b) Transfer (synaptic) function.

Baldi and Atiya [11] investigated the effects of delays on the ring dynamics, in particular on the oscillatory properties, showing numerically that when the ring has odd number of units and the coupling is inhibitory ( ), the origin (the zero solution) is the only fixed point, and if it is unstable, the dynamics of the network always converges to a stable limit cycle. They also argued that, for even and , there are no stable limit cycles. But a strict mathematical proof is still lacking. Model (1) has been studied by several researchers. In [36, 37], the authors studied a ring with two and three units and obtained sufficient conditions for the asymptotic stability of the zero equilibrium and for the existence of Hopf bifurcations. Later Wei and Li investigated the global existence of multiple periodic solutions in a ring of three units [38]. The stability analysis and local Hopf bifurcation of the zero solution in a ring of four units have been given in [39]. In the above-mentioned studies the authors considered nonidentical elements, where the theory of equivariant bifurcations is not applicable. Besides, the properties of nonzero equilibria have not been addressed. To fill this gap Guo [40] studied a ring (1) consisting of four identical neurons coupled with the same delay. He provided conditions on the linear stability of zero solution, spatiotemporal patterns of nonlinear oscillations, albeit restricting consideration to the case of excitatory coupling ( ). The direction and stability of bifurcating periodic solutions and steady state bifurcations were not considered. Thus there is no result on the case of the undirectional ring consisting of arbitrary number of units. In particular, little is known about the patterns of nonlinear oscillations and their stability even in a ring of more than four identical neurons. The steady state bifurcation patterns are also lacking. This paper is a generalization of the previous investigation for the case of the undirectional ring consisting of two or three neurons with discrete delay in [37]. We obtain not only the stability and delay-induced Hopf bifurcation but also the spatiotemporal patterns of periodic oscillations. We would also like to mention that the rings of neural networks with unidirectional coupling and distributed delays have recently been considered in [4143]. The paper also can be considered the complement of the results in [4143] for the case of distributed delays.

In this paper, we consider model (1) consisting of (arbitrary) identical neurons; that is, , coupled with the same delay . We assume that the synaptic coupling is described by a sufficiently smooth sigmoidal function (Figure 1(b)). A widely used example in the literature is . However, here we only assume , for the stability analysis, and we require , , for the bifurcation analysis. We investigate the global and local stability, Hopf bifurcations, pitchfork bifurcations, and spatiotemporal patterns of bifurcating periodic solutions. We also derive normal forms on center manifolds and determine direction and stability of the Hopf bifurcations.

2. Background

For convenience, we recall results for the model (1) for the case of a ring composed of identical neurons System (2) admits the zero as an equilibrium solution , which we will refer to as the rest state.

Every initial state uniquely defines a solution of system (2) for all . Due to the uniqueness of the Cauchy initial value problem of system (2), every initial state satisfying (called synchronous phase point) gives a synchronous solution , that is, , and the solution of system (2) can be characterized by the scalar delay differential equation System (3) has been studied by many researchers. For example, Mallet-Paret and Nussbaum [44] have obtained some results on global continuation and asymptotical behavior of periodic solutions, and Krisztin et al. [45] and Krisztin and Walther [46] give a complete description of the global attractor of (3) as a three-dimensional smooth solid spindle when is in a certain range. A few results from Mallet-Paret and Nussbaum [44] are needed for our subsequent work.

Lemma 1. Assume that is a smooth sigmoid-like function, is a steady state of (3), and is defined by (13) with .(i)If , then is asymptotically stable for any .(ii)If , then is asymptotically stable for and unstable for .(iii)If , then is unstable for any .(iv)If , then (3) undergoes a Hopf bifurcation at when .(v)If , then (3) has at least periodic solution for with or with .

3. Stability and Hopf Bifurcations of the Rest State

In this section, we assume(H1) , , .Let us first study the global stability of the rest solution .

Lemma 2. If , then the solution of system (2) is globally asymptotically stable for any .

The proof is given in Appendix A.

Lemma 2 shows that the dynamics of the network (2) is simple when . Any perturbation of the network state decays in time. Therefore, in the remainder of this paper, we will investigate the dynamics of (2) with .

The linearization of system (2) around the rest state is given by The characteristic matrix associated with system (4) is where is identity matrix and Let . Then, we get It follows that the characteristic equation is where

An equilibrium solution of a delay differential equation is locally asymptotically stable if all roots of the corresponding characteristic equation have negative real parts and unstable if at least one root has positive real part (see, e.g., [47]). Thus, in order to investigate the stability of the rest state, it is necessary to study the distribution of roots of the characteristic equation (8) and when they cross the imaginary axis.

It is straightforward to see that if , then is not a root of for any and . Therefore, in what follows we assume that two complex roots cross the imaginary axis at some positive critical values of time delay; that is, . Then from (8) and (9) it follows that satisfies which gives Then for , with . Let us first consider . Set where and . It is easy to verify that has a purely imaginary root for and for . Since , one can conclude that the characteristic equation (8) with has a pair of simple purely imaginary roots at . Proceeding for the case exactly as done above we can get similar results. In fact, we can set with and such that . Then the characteristic equation (8) has a pair of simple purely imaginary roots at .

Differentiating with respect to we obtain that Thus, the implicit function theorem implies that there exist and a smooth curve such that and . Differentiating with respect to after some algebraic calculus we get

Now we can state the following results about the distribution of roots of (8).

Lemma 3. Assume that is defined by (13) with and such that .(I)If is even, then when the characteristic equation (8) has at least one root with positive real part for all .(II)If is odd, then(i)when or the characteristic equation (8) has at least one root with positive real part for all ;(ii)when all roots of the characteristic equation (8) have negative real parts for and all roots of the characteristic equation (8) except have negative real parts at , but the characteristic equation (8) has at least two roots with positive real parts for .(III)If , then when , the characteristic equation (8) has a pair of simple purely imaginary roots .

The proof is given in Appendix B.

From Lemma 3, the transversality condition (15), and the standard Hopf bifurcation theorem of delay differential equations (see, e.g., [47]), we can state the following theorem.

Theorem 4. Assume that and is defined by (13).(I)If is even, then the rest state of system (2) is unstable for all .(II)If is odd, then(i)when or the rest state of system (2) is unstable for all ;(ii)when the rest state of system (2) is asymptotically stable for and unstable for .(III)The system (2) undergoes a Hopf bifurcation at . That is, there exists a unique branch of periodic solutions with period , and as .

Theorem 5. (I) Assume that . Then for all the characteristic equation (8) has one simple root , and other roots have negative real parts.

(II) Assuming that and is even, then, for all , the characteristic equation (8) has one simple root , and other roots have negative real parts.

The proof is given in Appendix C.

4. Stability and Steady State Bifurcations

In this section, we study the properties of the equilibrium of the system (2) and assume(H2) , , .

From system (2), we immediately have the following lemma.

Lemma 6. If is an equilibrium of system (2) and there exists one component , then it must be the zero equilibrium .

Theorem 7. Assuming that , every equilibrium of system (2) must be synchronous and system (2) has exactly three synchronous equilibria: the zero equilibrium , the negative equilibrium , and the positive equilibrium , where are nonzero solutions of the equation . Moreover, these two nonzero equilibria and are both asymptotically stable for all .

The proof is given in Appendix D.

Theorem 8. Assuming that is odd and , system (2) has just one zero equilibrium .

The proof is given in Appendix E.

Theorem 9. If is even and , then system (2) has exactly three equilibria: , , , where are the nonzero solutions of . Moreover, these two nonzero equilibria are both asymptotically stable for all .

The proof is given in Appendix F. From Theorems 7, 8, and 9, we have the following.

Theorem 10. System (2) undergoes a pitchfork bifurcation at when is even and at when is odd.

Remark 11. From Theorems 7, 8, and 9, we also know that the size of the network does not affect the position of the steady states (see Figures 2 and 3 obtained from numerical simulations) and their stability.

368652.fig.002
Figure 2: Steady state pitchfork bifurcation of system (2) for odd.
368652.fig.003
Figure 3: Steady state pitchfork bifurcation of system (2) for even.

5. Center Manifold Reduction and Normal Forms

In this section, we will apply the method of Faria and Magalhães [48, 49] to obtain normal forms on center manifold. After that, we will study the properties of Hopf and steady state bifurcations. Without loss of generality, we assume(H3) , , , .Firstly, we introduce a new time scale to normalize the delay and give universalities to our analysis. Then system (2) can be written as in the phase space . For , we have Under the assumption (H3), we can expand in the Taylor series where and h.o.t. stands for the higher order terms.

5.1. Properties of Hopf Bifurcations:

From Lemma 3 and Theorem 4, we know that for the characteristic equation (8) has a pair of simple purely imaginary roots and system (2) undergoes a Hopf bifurcation at the critical delay values . In this subsection, we will determine the direction and stability of Hopf bifurcations by calculating explicitly the normal form of system (2) on the associated two-dimensional (2D) center manifold. For a Hopf bifurcation, the normal form on the center manifold is given in polar coordinates by Here, is the bifurcation parameter, , and the coefficients and are real numbers. The qualitative behavior of the asymptotic solutions of (16) (or equivalently of (2)) is the same as the behavior of solutions of (19), which, in turn, only depends on the signs of the two coefficients and . The sign of the product determines the direction of the bifurcation (supercritical if and subcritical if ), and the sign of determines the stability of the nontrivial periodic orbits (stable if and unstable if ) (see, e.g., [50]).

The derivation of the normal form coefficients and are given in Appendix G. According to Appendix G, we have Consequently, we have the following theorem.

Theorem 12. If (resp., ), then the Hopf bifurcations occurring on the center manifold of system (2) at are supercritical (resp., subcritical), with nontrivial periodic stable (resp., unstable) orbits on the center manifold.

Remark 13. From Lemma 3, we know that the characteristic equation (8) has at least one root with positive real part if one of the following conditions is satisfied:(i) is even and ;(ii) is odd and either or ;(iii) is odd, , and .Therefore, if one of the above three conditions is satisfied, then the periodic solutions bifurcating from the rest state at the critical value of must be unstable in the whole phase space although they are stable on the center manifold.

However, when is odd and , all roots of the characteristic equation (8) except have negative real parts at . Thus in this case the stability of periodic solutions bifurcating at on the center manifold is equivalent to that of periodic solutions in the whole phase space.

5.2. The Steady State, Pitchfork Bifurcation:

We first consider the case when is even and . From Theorem 5, we know that the characteristic equation (8) has a simple root for all . To study the bifurcation in detail we introduce a new bifurcation parameter . Then the normal form becomes (see Appendix H) This normal form, together with the results in [51], permits the following theorem to hold.

Theorem 14. Suppose that is even and . Then the rest state of system (2) is stable if and unstable if for any .

Next if , letting , one can obtain and then get the same normal form (21). So, we can obtain the following theorem.

Theorem 15. The rest state of system (2) is stable if and unstable if for any and .

Now we can explicitly determine the stability of the zero solution of the system (2) on the plane. According to the results stated in Theorems 4, 5, 14, and 15, the stable regions of the rest state of system (2) with can be illustrated graphically in Figures 4 and 5.

368652.fig.004
Figure 4: Stable region (shaded area) of the rest state of system (2) for even.
368652.fig.005
Figure 5: Stable region (shaded area) of the rest state of system (2) for odd.

6. Spatiotemporal Patterns of Bifurcating Periodic Solutions

Earlier we have described the Hopf bifurcation at the critical value leading to a family of periodic solutions. In this section, we will investigate the spatiotemporal patterns of these bifurcating periodic solutions and refer to [22] for explanations of notations involved.

Let us first introduce notations from the theory of compact groups.(i) is the cyclic group of order , which corresponds to rotations of . Denoting the generator of this group by , its action on is given by (ii)Let and be a compact group. The system is said to be -equivariant if for all .It is clear that system (2) is equivariant. Let . Denote by the Banach space of all continuous periodic mappings from into , equipped with the supremum norm. Then, for the circle group , acts on by Denote by the subspace of of all periodic solutions of (4) with . Then where and are dimensional vector functions with the th components defined by , respectively. For each subgroup , it is clear that the fixed point set is a subspace.

Lemma 16. Consider

The proof is given in Appendix I.

It is known from [19] that the subgroups of , up to conjugacy, describe the symmetry of periodic solutions of system (2) which exhibit certain spatiotemporal patterns given by Form Lemma 16, it follows that which means that This, together with Lemma 16, allows us to apply the symmetric Hopf bifurcation theorem for delay differential equations due to Wu [22] to obtain the following results.

Theorem 17. Suppose that and is defined by (13) with and such that . Then near there exists a branch of small-amplitude periodic solutions of system (2) with period near , satisfying

7. Examples and Numerical Simulations

In this section, we consider two examples with four neurons and seven neurons, respectively, to justify our theoretical results.

7.1. Four-Neural Network Model ( Even)

Consider a four-neuron network model From (13), we get the Hopf bifurcation curves with and such that . It follows that these Hopf bifurcation curves are symmetric about -axis (see Figure 6). From Sections 2 and 3, we can obtain the following results.

368652.fig.006
Figure 6: The curves of Hopf and pitchfork bifurcations of system (31).

Corollary 18. (i) If , then the rest state of system (31) is stable for all .

(ii) If , then the rest state of system (31) is unstable but the other two nonzero steady states are both stable for all .

(iii) System (31) undergoes a Hopf bifurcation at (as shown in Figure 6) and bifurcating periodic solutions are all unstable in the phase space (although stable on the center manifold).

For , the system (31) has three equilibria , and . Figure 7 shows the evolution of system (31) starting with the initial value and .

fig7
Figure 7: Evolution of system (31) with . The rest state of system (2) is unstable and the other two nonzero equilibria are stable. (a) shows the time evolution for system (33) and (b) is the phase portrait for the first three components , and of system (2).

Taking , system (31) has three equilibria , and . The zero equilibrium is unstable and the other two nonzero equilibria are both stable (see Figure 8).

fig8
Figure 8: Evolution of (31) with . The rest state of system (2) is unstable and the other two nonzero equilibria are stable. (a) shows the time evolution for system (33) and (b) is the phase portrait for the first three components , and of system (2).
7.2. Seven-Neuron Network Model ( , Odd)

Consider a seven-neuron network model Let with and such that . Then from Sections 2 and 3, we have the following results.

Corollary 19. (i) If , then the rest state of system (33) is stable for all .

(ii) If or , then the rest state of system (33) is unstable for all .

(iii) If , then the rest state of system (33) is asymptotically stable for and unstable for .

(iv) System (33) undergoes a Hopf bifurcation at (as shown in Figure 9) and the bifurcating periodic solutions satisfy

Moreover, the bifurcating periodic solutions at the first bifurcating point are stable and others unstable in the phase space.

368652.fig.009
Figure 9: Stable region and curves of Hopf and pitchfork bifurcations of system (33) for the rest state .

To perform the numerical calculation, we consider the parameter value . It follows from (34) that

Taking , Figure 10 shows that the rest state of system (33) is asymptotically stable.

fig10
Figure 10: (a) shows the time evolution for system (33) and (b) is the phase portrait for the first three components , and of system (33) at .

For , it follows from Corollary 19 that the rest state of system (33) becomes unstable and there exists a small-amplitude phase-locked oscillation with period satisfying . This result is illustrated in Figure 11.

fig11
Figure 11: (a) shows the time evolution for system (33) and (b) is the phase portrait for the first three components , and of system (33) at .

Taking far away from the first critical value , the numerical simulation shows the existence of phase-locked oscillations with period satisfying , (see Figure 12). The amplitude of this periodic oscillation is larger than that of periodic oscillation shown in Figure 11. This large-amplitude phase-locked oscillation is very similar to the square waves reported in [52, 53] for singularly perturbed delay equations. This phase-locked oscillation with period satisfying also shows that this stable bifurcating periodic solution comes from the first critical value not from other critical values.

fig12
Figure 12: (a) shows the time evolution for system (33) and (b) is the phase portrait for the first three components , and of system (33) at far away from the first critical value .

Taking , system (33) has three synchronous equilibria , and . The zero equilibrium is unstable and the other two nonzero equilibria are both stable (see Figure 13).

fig13
Figure 13: Evolution of system (33) with starting from different initial values. (a) shows the time evolution for system (33) and (b) is the phase portrait for the first three components , and . The rest state of system (2) is unstable and two nonzero equilibria and are stable.

8. Conclusions

In this paper, we have studied a lattice ring of identical neurons coupled with the same delay. We have analyzed the global stability and delay dependent local stability regions and conditions and explored the different types of bifurcation after the stability is lost. We also have shown that the stability of the zero equilibrium state depends not only on the synaptic strength and time delay but also on the parity of the network size. However, if the nonzero equilibrium exists, it is always stable.

We have also studied the stability of the zero equilibrium on the pitchfork values and have given easy-to-check conditions on the stability and direction of Hopf bifurcations. For example, if , then when is odd the periodic orbits bifurcating from the first critical value are stable in the phase space, but when is even all bifurcating periodic orbits are unstable in the phase space.

On the other hand, we have analyzed the spatiotemporal patterns of nonlinear oscillations by using the symmetric bifurcation theory of delay differential equations coupled with representation theory of cyclic groups. Finally, using MATLAB software, we have done some numerical simulations showing the existence of the stable equilibrium and phase-locked periodic waves arising from Hopf bifurcations.

We have generalized recently results for lattice rings with a few units to an arbitrary number. The results obtained are expected to be of interest to neurodynamics and to scientists interested in robots. Indeed we have shown that there are significant features different for even at qualitative level and odd which do not disappear when goes to large. A finding of practical interest is that for what enough time intervals these differences are not so relevant.

Appendices

A. Proof of Lemma 2

Using the assumption (H1), we can write as where and find such that for all and . Thus, system (2) becomes Using the Lyapunov functional we obtain For and , , which completes the proof.

B. Proof of Lemma 3

Conclusion (III) follows from the fact that the characteristic equation (8) has a pair of simple purely imaginary roots at and the transversality condition (15) holds.

Denote by a zero of with . From (9), we have Clearly, for . If is even, then for . These, together with (15) and the Rouché-Froebenius theorem, imply that conclusion (I) is true.

If is odd with , then . If is odd with , then Just then from (B.1), we have It follows that the root of has negative real parts for any . In addition, from (13), we have which means From (13), we also have, for being odd with , but Consequently, is the first positive critical value for the occurrence of purely imaginary roots . Thus, by (15) and the Rouché-Froebenius theorem, conclusion (II) is proved.

C. Proof of Theorem 5

It can be verified that is a simple zero of only for when . When and is even, is a simple zero of with .

On the other hand, suppose that is a root of . Then from (9) For , the latter implies which can be only satisfied for . This completes the proof.

D. Proof of Theorem 7

For , every equilibrium of system (2) must satisfy Using the monotonicity of and the assumption , we obtain that if , say , then . Repeating the above procedure, we have which is a contradiction, implying . Thus, is an equilibrium of system (2) if and only if , satisfy the equation . According to the assumption (H2) when it has just one zero root, and when it has exactly three roots and . Consequently, when system (2) has exactly three equilibria: In addition, note that the characteristic equation of system (2) at is where Thus, from Section 3 and the fact that , we have these two equilibria and both asymptotically stable for all . This completes the proof.

E. Proof of Theorem 8

Lemma 2 says that system (2) has just the zero equilibrium for . In fact, from the proof of Theorem 7 we also obtain that system (2) has just the zero equilibrium when . We now consider the case . From system (2), is an equilibrium of system (2) if and only if satisfies This means that , satisfy Let Then we have and Note that for all , is odd, and . Thus, we have for all and hence is the unique root of . This completes the proof.

F. Proof of Theorem 9

Clearly, is an equilibrium. Let be a nonzero equilibrium of system (2). At first, suppose that , which, together with (D.1), implies , then , and so on. Consequently, we obtain that . Thus, , satisfy . Using the monotonicity of and , we get for any . This contradiction leads to . In fact, we can also obtain . By (E.4), we know that when is even and . Therefore, there exists a sufficiently small positive number such that for . From we have for and for . On the other hand, since is bounded, we have as and as . This means that there exist and such that , for and for , and . In addition, by for and for , we obtain, for , So, Similarly, we can obtain These imply that when is even and , has exactly three roots: . Thus, from Lemma 6 and it follows that system (2) has exactly three roots: the zero equilibrium and From (E.1) and (F.4) we can get , , and are roots of the following equation: So, Again using the assumption (H1), we have Therefore, and then Letting , , we can obtain that the characteristic equation of the system at is where Thus, from Section 3 and (F.9) we get that the equilibrium is asymptotically stable for all . Similarly, we can also obtain that the equilibrium is asymptotically stable for all . This completes the proof.

G. Derivation of Coefficients of the Normal Form (19)

The linearized equation at the zero equilibrium of (16) is where The linear operator can be expressed in the integral form where is a function of bounded variation.

Denoting and introducing the new parameter so that corresponds to the Hopf bifurcation, we rewrite (16) as where

Let and . It is clear from Lemma 3 that the characteristic equation of has a pair of simple imaginary roots and no other roots in the imaginary axis which are multiple of . Thus, the nonresonance conditions relative to are satisfied.

Let be a matrix whose columns form a basis of the center space of with , where the bar means complex conjugation and is a vector in such that Suppose is a matrix whose rows form a basis for the adjoint space with (the identity matrix) for the adjoint bilinear form on defined by with , where is the dimensional space of row vectors.

Then we have , for such that From (7), (G.6), and (G.7), we can choose where .

Following the procedure [48, 49] very closely, we can obtain the normal form associated with the Hopf singularity where Moreover, the normal form (G.10) can be written in real coordinates through the change of variables . Transformed to polar coordinates , this normal form becomes (19), where and .

H. Derivation of the Normal Form (21)

It is convenient to write (16) as the following delay system: with the phase space , where, for , with .

Let and consider the center space for and its dual space , as in the previous appendix. We can choose normalized dual bases of and of as follows: with , and .

As the procedure introduced in Section 5.1, decomposing in (H.1) according to the decomposition of as the form , with and , we can obtain the normal form of (H.1) on the center manifold of the origin where , h.o.t. stands for higher order terms, and are the second and third order terms in , respectively. Then it follows from [48, 49] that where