Abstract
A reaction diffusion system is used to study the interaction between species in a population dynamic system. It is not only used in a population dynamic system with the diffusion phenomenon but also used in physical chemistry, medicine, and animal and plant protection. It has been studied by more and more scholars in recent years. The FitzHugh–Nagumo model is one of the most famous reaction-diffusion models. This article takes a deeper look at a FitzHugh–Nagumo model in a network with time delay. Firstly, we studied the linear stability of the equilibrium, then the existence of Hopf bifurcation is given, and finally, the stability of the Hopf bifurcation is introduced.
1. Introduction
Reaction-diffusion equations are widely used in ecology and biology fields [1–3]. The Hopf bifurcation is very useful in practice; in [4–7], the turing instability is analyzed. Turing [8] first introduced the coupled reaction-diffusion equations; he pointed out that the stable process could evolve into an instability with diffusive effects. As a time- and space-breaking symmetry, Turing–Hopf bifurcation is a well-known mechanism for generating spatiotemporal patterns [9, 10].
The FitzHugh–Nagumo model is one of the most famous reaction-diffusion models; Hodgkin and Huxle first introduced the FitzHugh–Nagumo model. They used the model to study electrical impulses on nerve fibers [11–13]. In the Fitzhugh–Nagumo model, the variables of equations for the voltage and have many forms, and the following form is commonly used:
The function is positive feedback, and it is a third-order polynomial; however, the function is negative feedback. In [14–18], the pattern formation in the FitzHugh–Nagumo system is studied frequently. In 2001, Medvedev and Kopell [19] introduced the FitzHugh–Nagumo oscillators with coupled electrically, and the synchronization and transience are obtained; in 2004, Kostova et al. [20] investigated the revised FitzHugh–Nagumo equation. The FitzHugh–Nagumo model with time delays is considered by C. J. in 2013 [21]. Based on the abovementioned research, we discussed the FitzHugh–Nagumo model with delay as follows:
In the abovementioned system, is the membrane potential, is the recovery variable, and are the diffusive coefficients of and , respectively, is the excitatory threshold, is the diffusion coefficient, is the excitability and are parameters, they can change the rest-state and dynamics, denotes the Laplacian operator which describes the diffusion, and represents the time delay. For detail, one can see [22]. This model is based on a spatially continuous domain, that is, the entire landscape is spatially continuous; in this paper, we introduce a network that owns system (2).where is an adjacency matrix, whose entry equals 1 if there exists an edge from node to node and 0 otherwise. It is obvious that for . We assume that is strongly connected, i.e., for any pair of distinct nodes, there exists a path from one node to the other. It is known that the adjacency matrix is irreducible [23]. The definition of the Laplacian matrix is as follows:
The rest of this paper is designed as follows. In Section 2, we give the linear stability of the equilibrium and the existence of Hopf bifurcation. In Section 3, the stability of the Hopf bifurcation is introduced.
2. Stability of the Equilibrium and Hopf Bifurcation
By simple calculation, we know that equation (3) has three equilibriums ; and , where
The interior equilibrium exists if and only if
Now, we focus on the stability properties of the coexisting equilibrium , we define the small perturbations , and system (3) can be expressed as follows:where andwhere .
As in [24], we use to express the eigenvalues of the Laplacian matrix and to express each eigenvector relevant to a topological eigenvalue , so we can get . Let be the transpose and express any small disturbance in the equilibrium state; we can have the following basis decomposition:
Inserting (9) into (7), using the orthogonality of the eigenvectors, we get
So, the characteristic equation can be written as
Substituting (8) into (11), the characteristic equation of (3) can be written as
Lemma 1. Ifand (6) holds, then we can get the following:(1)When , all characteristic roots of (12) have negative real parts(2)When , the characteristic equation (12) has a pair of purely imaginary roots at , where
Proof. (1) When , the characteristic equation (12) can be reduced toConsidering (13), we can obtain that and . Therefore, the characteristic equation (12) has a root with a negative real part (3). Suppose are a pair of pure imaginary roots of (12). Substituting into (12), we can getand through separating real and imaginary parts of the root, we can getIt is noticed that , and we can getEquation (13) has a unique positive real root if and only if (13) holds.
Theorem 1. (1)When , system (3) is locally stable at the equilibrium if (6) and (13) hold(2)When , system (3) is unstable at the equilibrium if (6) and (13) hold(3)We can find the constant and a smooth curve when and for all
Proof. We refer to the Theorem 3.6 in the work of Ruan [25]. In consideration of that is the smallest value when the root of equation (12) has zero real part, when , the real parts of the characteristic roots are all negative. Through using G. J. Buther’s Lemma [26], we know that is locally stable when .
Now, we check the transversality conditionSuppose is the eigenvalue of (12), and by substituting into (12), we can obtainDifferentiating (20) with respect to , we haveEliminating the terms from the abovementioned equation, we getTo verify , we just need to verify thatWhen , (20) becomesBy some simple calculations, we can getSo, (23) becomesWe must verifySubstituting (25) into (27) yieldsSince , we need only to verify thatThe abovementioned inequality holds, so system (3) undergoes a Hopf bifurcation at .
By applying the implicit function to , we can find the constant and a smooth curve when and for all .
Now, by using proposition 6.5 in [28], we obtain that is locally unstable for .
3. The Stability of the Hopf Bifurcation
As a matter of convenience, let . Then, is the Hopf bifurcation value for system (3). Let . For convenience, we leave off the bars of and . System (3) can be rewritten as
We rewrite the abovementioned equation as follows through using the orthogonality of the eigenvectors.
We use to denote the solution of (31) and let , where denotes the transpose, . System (31) can become the functional differential equation as follows:where ; they have the following forms:
We can letwhere is the Dirac function, and we can getwhere
Then, two operators and on can be defined by
Then, (32) is transformed intowhere . We can define the bilinear form by taking the solution space of (38) as the complex space .for , where .
From the proof in Lemma 1, we can get that are eigenvalues of (12); thus, are the eigenvalues of and , respectively. We setand we know that and . Letfor .
So, we have
In order to ensure , we should determine the value of , and from (40), we get
Therefore, we can let
Then, we construct a coordinate for the center manifold at by using and ; the method is similar to that in [27]. Let be a solution of (38), and we define
We further getwhere and are the local coordinates of the center manifold in directions and , respectively.
We only take real solutions into account. For a solution of (26) with , from (47), we have
Thus, (49) can become
Lemma 2. If we setthen can be explicitly expressed by (55).
Proof. Noticing (47) and (48), we obtainIn view of (52), substituting (53) into (51),where , and are the -th components of , and , respectively, for . The results in the theorem are valid.
Next, we will get the values of and . From (38) and (47), we obtainwhere
From (48), we have
Substituting (48) into (56), we can get that
By comparing the coefficients of and from (58) and (59), we get
For , it follows from (47), (48), (57), and (58) that
By comparing the coefficients of and from (57) and (61), we get
Thus, we can get
Since , we obtainwhere is a constant vector. We also can havewhere is a constant vector.
From the definition of and (60), we haveandwhere . From (56), when , we get
Then, we obtain
Therefore, we obtain
Comparing both sides of (70), we have
Since is the eigenvalue of and is the corresponding eigenvector, we get
Then, substituting (65) and (67) into (72), we obtainthat is,where
Thus,
Therefore, we can get the values of and from (64) and (65).
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Program for Innovative Research Team in Huainan Normal University (No. XJTD202008).