#### Abstract

Core and satellite structure is one of the common structures in enterprise clusters. In core and satellite structure, there are one core enterprise and at least two satellite enterprises. There exist a competitive relationship between satellite enterprises and a cooperative relationship between satellite enterprise and core enterprise. However, the dynamic evolution of competition-cooperation enterprise clusters with core-satellite structure is not well understood. In this paper, a novel competition-cooperation enterprise cluster model with core-satellite structure is proposed. The boundedness of the positive equilibrium is investigated. It is found that there exists upper bound of both core enterprise output and satellite enterprise output and the upper bound of core enterprise not only depends on its own production capacity but also depends on the production capacity of two satellite enterprises. Then, by selecting the production period as bifurcating parameter, the conditions of local stability and Hopf bifurcation are obtained. Once the production period passes a critical value, the output of both core enterprise and satellite enterprise loses stability and displays periodic fluctuations. This may lead to the decline of efficiency of enterprise and resource mismatch. Furthermore, the fluctuation properties are studied. Finally, a numerical example is presented to show the effectiveness of theorem.

#### 1. Introduction

Recently, enterprise cluster, as an effective form of industrial space organization, has gradually become a common phenomenon in the process of modern industry and internationalization [1–4]. The competition-cooperation relationship widely exists in the real enterprise clusters and has a major impact on the evolution of enterprise clusters [5–9]. To study the influence of competition and cooperation on the evolution of enterprise clusters, researchers propose a competition-cooperation enterprise cluster model [10] based on the ecology model [11–15], which is described as follows:where are the enterprise output; is the intrinsic growth; denotes the carrying capacity of market under nature unlimited conditions; is the initial output of core enterprise; and is the production period. Let , and system (1) can be rewritten as follows:

For this model, the dynamic behaviors including stability, Hopf bifurcation, and chaos have been widely studied [16–21].

In practical enterprise clusters, organization structure has a major impact on the production efficiency of enterprise clusters. The efficiency of overall operation is one of the important factors of enterprise's success. Thus, it is necessary to consider the structure in the enterprise cluster model. Among the many organization structures, core and satellite structure is one of the common structures in enterprise clusters, which is described in Figure 1. In core and satellite structure, there are one core enterprise and at least two satellite enterprises. There exist competitive relationship among satellite enterprises and cooperative relationship between satellite enterprise and core enterprise. For example, in automobile enterprise cluster, the core enterprise produces the motor vehicle and the satellite enterprise produces automobile parts for core enterprises. To reduce the cost and ensure the stability of supply chain, the core enterprise has at least two satellite enterprises for the same automobile part. It is easy to see that there exists competition between the two satellite enterprises and there exists cooperation between satellite enterprise and core enterprise. However, few works investigate the dynamic evolution of enterprise cluster model with core-satellite structure.

Inspired by the discussion, in this paper, a competition-cooperation enterprise cluster model composed of a core enterprise and two satellite enterprises is proposed, which is shown in Figure 1. In this enterprise cluster, there are one core enterprise and two satellite enterprises. There is a competitive relationship between two satellite enterprises. And there is a cooperative relationship between satellite enterprise and core enterprise. The model is described as follows:where is the satellite enterprise output; is the core enterprise output; is the self-regulation of enterprise *i*; is the intrinsic growth; is the completion rate of satellite enterprise; is the completion rate between satellite enterprise and core enterprise; is the rate of conversion of commodity into the reproduction of enterprise; is the initial output of core enterprise; is the initial output of satellite enterprise; is the initial output of satellite enterprise ; and is the production period. The main contributions of this paper are as follows:(1)A competition-cooperation enterprise cluster model is composed of a core enterprise and two satellite enterprises. There is a competitive relationship between two satellite enterprises. And there is a cooperative relationship between satellite enterprise and core enterprise.(2)The boundedness of positive equilibrium is investigated. And there exists a upper limit of output of enterprise cluster model.(3)The production period plays a key role in dynamics of the proposed enterprise cluster. When it passes a critical value, the output of the enterprise cluster system loses its stability and displays a periodic fluctuation, which may cause a drop in productivity of the enterprise cluster system.

The remainder of this paper is organized as follows. In Section 2, the boundedness analysis of positive equilibrium is given. In Section 3, the conditions of Hopf bifurcation are discussed. In Section 4, the normal form of Hopf bifurcation is given. In Section 5, an example is given to verify the theoretical analysis. In Section 6, we give the economic meaning.

#### 2. Boundedness of Positive Equilibrium

In this section, we investigate the boundedness of positive equilibrium. It can be seen that system (3) has more than three equilibria if any one of the enterprise output is zero. As it has no economic sense if one of the enterprise output is zero, we only study the property of positive equilibrium where all enterprise outputs are positive. Let be the positive equilibrium of system (3), where

From the perspective of enterprise management, the output of enterprise cannot be negative. Thus, the initial condition of enterprise output must satisfy , , and for . In respect of the boundedness of , we have the following lemma.

Lemma 1. *(see [21]). Let .*(1)*If is satisfied, then for all and .*(2)*If is satisfied, then for all and .*

Theorem 1. (1)*Suppose and ; there exists such that , where .*(2)*Suppose and ; there exists such that where .*

*Proof. *First, we investigate the boundedness of and . By (3), one can obtain for all . Then, we haveAccording to Lemma 1, one hasSimilarly, one hasIf there exists such that is local max, where , using the same method, one can obtain that has upper limit at . Thus, one has . Similarly, we also can obtain .

In this same way, it follows that there exists such that for , where . We complete the proof.

*Remark 1. *From Theorem 1, one can see that there are upper bounds on the output of core enterprise and two satellite enterprises. Moreover, the upper bound of core enterprise not only depends on its own production capacity but also depends on the production capacity of two satellite enterprises.

#### 3. Bifurcation Analysis

In this section, the conditions of Hopf bifurcation for (3) with are presented. By (3), one can obtain the Jacobian matrix for (3) with :

Then, we havewhere

If is a root of (9), one can obtain

Separating the real and imaginary parts of (11), we have

By (12) and (13), one can obtainwhere .

Let *z* = ; then, (14) becomes

Letting , , we get the following.

Lemma 2. *(see [22–24]).*(i)*If , (15) has at least one positive root.*(ii)*If and , (15) has no positive root.*(iii)*If , , , and , (15) has positive roots.**Assume (14) has three positive roots , k = 1, 2, 3. By (12) and (13), we have*

Thus, denotewhere *k* = 1, 2, 3; *j* = 0, 2, …. Define .

Note that when , (9) becomes

By using the Routh–Hurwitz criterion [16], one can obtain the condition that all roots of (18) have negative real parts.

Lemma 3. *(see [25]). Consider the exponential polynomialwhere and ( are constants. As vary, the sum of the order of the zeros of on the open right half plane can change only if a zero appears on or crosses the imaginary axis.*

Lemma 4. *If , then when and the sign of is same as the sign of .*

*Proof. *Differentiating (9) with respect to yieldsFor convenience, we denote and by and ; then, we haveThen, we getwhere .

Then, if , we have , and the sign of is the same as the sign of . We complete the proof.

Thus, from Lemmas 2, 3, and 4, one has the following.

Theorem 2. *Suppose that (H1) holds, and we have*(1)*If and , (3) is asymptotically stable at .*(2)*If either or and , holds, (3) is asymptotically stable at for and undergoes a Hopf bifurcation at when .*

#### 4. Direction of the Hopf Bifurcation

In this section, we study the properties of Hopf bifurcation by using the center manifold [23, 26–28]. Letting , (3) can be transformed into a FDE aswithwherewhere

According to Riesz representation theorem, there exists a function of bounded variation for , such that

Letwhere is Dirac delta function.

By [21], we define

Then, system (24) can be rewritten aswhere .

The adjoint operator of is defined bywhere is the transpose of the matrix .

For and , we definewhere . We know that is an eigenvalue of , so is also an eigenvalue of . We can get

By [21], we have

Hence, one can obtain

Assume that the eigenvector of is

By [21], we have

Hence, we obtain

Let

Then, we can compute

Hence, we obtain

Assume that is a solution of (33) with ; we define

On , one haswhere

In fact, and are local coordinates for in and . As , we have

Rewrite (39) aswhere

Letwhere

Following the method in [18–21], one can obtain

Since , one has

Thus, one can obtain

So,

By (27) and (43), one can obtainwhere

As , one has

Then, we have

Next, we compute and . According to the expression of , we have .

Comparing the coefficients of (63), one has

Substituting (64) and (65) into (55), one can obtain

So,

Now, we compute and . From the definition of in (31), one can obtain

From (50), (68), and (69), we have

Following the method in [18–21], we have

Then, we can obtainwhich leads toand

Then, by (72) and (74), we can obtain the following properties of Hopf bifurcation:

Theorem 3. *From (75), one has*(1)*The directions of the Hopf bifurcation is determined by: if , then the Hopf bifurcation is forward (backward) and the bifurcating periodic solutions exist for .*(2)*The stability of the bifurcating periodic solutions is determined by: the bifurcating periodic solutions are stable (unstable) if .*(3)*The period of the bifurcating periodic solutions is determined by: the period increases (decreases) if .*

#### 5. Numerical Examples

In this section, a numerical example is presented to support our obtained results. Consider system (3) with the following parameters: , , , , , , , , and . The positive equilibrium of (3) is. By (17), we can obtain .

First, we choose, and the outputs of three enterprises are shown in Figure 2. It is easy to see that output of enterprise is asymptotically stable.

Finally, we choose , and the outputs of three enterprises are shown in Figure 3; it is easy to see that the output of the enterprise displays a periodic fluctuation.

#### 6. Economic Meaning

In this paper, a novel competition-cooperation enterprise cluster model with core-satellite structure is proposed. The boundedness of the positive equilibrium is investigated. It is found that there exists upper bound of both core enterprise output and satellite enterprise output and the upper bound of core enterprise not only depends on its own production capacity but also depends on the production capacity of two satellite enterprises. Moreover, it is found that the production period plays a key role on the evolution of enterprise cluster. From simulation, we can obtain that the critical value of the production period is . When the production period passes , the output of the enterprise cluster system loses its stability and displays a periodic fluctuation, which means too long production cycle will lead to capacity fluctuations. From the viewpoint of enterprise management, output fluctuation may affect the stable development of enterprises, worker employment, and production efficiency. Thus, it is important for enterprise cluster to control the production period in a suitable region.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This study was supported in part by the Chongqing Social Science Planning Project under grant no. 2019BS053, in part by the Science and the Technology Research Program of Chongqing Municipal Education Commission under grant no. KJQN201900816, in part by the Social Science Research Program of Chongqing Municipal Education Commission under grant no. 20SKJD072 in part by the Research Funds of Chongqing Technology and Business University under grant no. 1955028, and in part by the Chongqing Basic and Frontier Research Project under grant nos. cstc2019jcyj-msxmX0470 and cstc2020jcyj-msxmX0139.