Understanding Dynamics and Bifurcation Control Mechanism for a Fractional-Order Delayed Duopoly Game Model in Insurance Market

Recently, the insurance industry in China has been greatly developed. The number of domestic insurance companies and foreign investment insurance companies has greatly increased. Competition between different insurance companies is becoming increasingly fierce. Grasping the internal competition law of different insurance companies is a very meaningful work. In this present work, we set up a novel fractional-order delayed duopoly game model in insurance market and discuss the dynamics including existence and uniqueness, non-negativeness, and boundedness of solution for the established fractional-order delayed duopoly game model in insurance market. By selecting the delay as a bifurcation parameter, we build a new delay-independent condition ensuring the stability and creation of Hopf bifurcation of the built fractional-order delayed duopoly game model. Making use of a suitable definite function, we explore the globally asymptotic stability of the involved fractional-order delayed duopoly game model. By virtue of hybrid controller which includes state feedback and parameter perturbation, we can effectively control the stability and the time of creation of Hopf bifurcation for the involved fractional-order delayed duopoly game model. The research indicates that time delay plays an all-important role in stabilizing the system and controlling the time of onset of Hopf bifurcation of the involved fractional-order delayed duopoly game model. To check the rationality of derived primary conclusions, Matlab simulation plots are explicitly presented. The established results in this manuscript are wholly novel and own immense theoretical guiding significance in managing and operating insurance companies.


Introduction
With the rapid development of the insurance market in China, different kinds of domestic insurance companies and foreign investment companies come forward in large numbers. In order to survive and better serve the people, the competition among various insurance companies is very fierce. The level of monopoly of the insurance market in China has gradually declined, but it still remains at the state of oligopoly. The monopoly competition game between oligarchs has become a very important research topic. In order to reveal the inherent law of competition among different oligarchs, it is necessary for us to set up mathematical models on competition and explore the quantitative relation of competition models among different oligarchs. Plenty of excellent and meaningful works on this topic have been published. For instance, Elabbsy et al. [1] set up a nonlinear where y 1 (t) stands for the price of the first insurance company, y 2 (t) stands for the price of the second insurance company, ρ 1 stands for the speed of price adjustment of the first insurance company, ρ 2 stands for the speed of price adjustment of the second insurance company, α represents the possible largest demand, β 1 denotes the effect of which the price of product 1 has on its quantity, β 2 denotes the effect of which the price of product 2 has on its quantity, δ 1 stands for substitution rate which the products of 1 shows to the products of 2, δ 2 stands for substitution rate which the products of 2 shows to the products of 1, µ ∈ (0, 1) denotes the weight of the current price at time t, 1 − µ denotes the weight of the current price at time t − θ, θ is a time delay, and all the parameters ρ i , β i , γ i , δ i , α, µ, θ, (i = 1, 2) are positive constants. For more details, one can refer to Refs. [18,19]. Taking advantage of stability criterion and bifurcation theory of delayed differential equation, Xu and Ma [18] set up a sufficient condition which guarantees the stability and the onset of Hopf bifurcation for model (1). It is noteworthy that all the involved literature above on game model in insurance market (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]) are basically concerned with the integer-order dynamical models. A large number of studies indicate that fractional-order differential equation has been deemed as a more valid tactics to describe the authentic natural phenomenon in the world than the conventional integer-order counterparts. At present, fractional dynamical systems have been applied in many areas such ascomplex networks, biological systems, artificial intelligence, various waves in physics, viscoelasticity, capacitor principle, biomedical treatment, electrical engineering, economics, and so on [20][21][22][23][24][25][26]. Its great application value comes from the powerful memory trait and hereditary superiority for various materials and evolutionary process [27,28]. In recent years, fractional dynamical systems have attracted great attention from many scientific circles and great achievements have been acquired. For example, Xu et al. [29] revealed the impact of delay on Hopf bifurcation of a class of fractional-order delayed bidirectional associate memory neural networks. Eshaghi et al. [30] explored the Hopf bifurcation, chaos control, and synchronization issues for a chaotic fractional-order dynamical model. Zhou et al. [31] probed into the Hopf bifurcation control problem of fractional-order prey-predator system involving delays via hybrid controller. For more concrete publications, one can see [32][33][34][35][36].
Considering that the fractional-order delayed duopoly game model can better reflect the memory trait and hereditary superiority in price of two insurance companies and is motivated by the investigation above and based on model (1), in the current work, we will set up the following fractional-order delayed duopoly game model in insurance market: where η ∈ (0, 1]. All other parameters and variables own the same economic meaning as those in model (1).
In this manuscript, we principally probe into the following four problems: (a) Investigate the existence and uniqueness, non-negativeness, and boundedness of the solution for system (2); (b) Set up the delay-independent condition guaranteeing the stability and the occurrence of Hopf bifurcation of model (2); (c) Build the sufficient condition to ensure the globally asymptotic stability of model (2); and (d) Control the time of onset of Hopf bifurcation of model (2) via hybrid controller.
The chief contributions of this manuscript are elaborated as follows: (1) Based on the earlier works, a novel fractional-order delayed duopoly game model in insurance market is proposed. (2) The sufficient condition ensuring the globally asymptotic stability of model (2) is set up via constructing an appropriate positive definite function. (3) Hopf bifurcation of model (2) is successfully dominated via hybrid control strategy. So far, very few scholars focus on the Hopf bifurcation control issue of fractional-order models by utilizing hybrid controller. (4) The influence of time delay on the stability behavior and the occurrence of Hopf bifurcation of model (2) and its controlled system is revealed. (5) The research approach can be applied to study the bifurcation control issue of lots of fractional dynamical models in numerous areas.
The novelty of this research lies in the design of hybrid controller for the fractionalorder delayed duopoly game model in insurance market. By designing a suitable hybrid controller, we can successfully control the stability region and Hopf bifurcation of model (2). The obtained results play a vital role in controlling the price of two insurance companies.
The structure of this research is arranged as follows. Some necessary basic knowledge about fractional-order differential equation is given in Section 2. Section 3 proves the existence and uniqueness, non-negativeness, and boundedness of the solution for model (2). A new delay-independent sufficient criterion which ensures the stability and the creation of Hopf bifurcation for model (2) is set up in Section 4. Section 5 explores the globally asymptotic stability of model (2) via a definite function. Hybrid control tactics are executed to control the stability and creation of Hopf bifurcation of model (2) in Section 6. Software simulation results are distinctly displayed to support the established key conclusions in Section 7. Section 8 draws a simple conclusion to complete this research.
Proof. Set up the following mapping: where For each Y,Ỹ ∈ Ψ, we get where Then h(Y) obeys Lipschitz condition with respect to Y (one can see [39]). Taking advantage of Banach fixed point theorem, one concludes that Theorem 1 is true.
(a) Every solution to system (2) starting with R 2 + is non-negative; (b) If the following inequality holds, then every solution to system (2) starting with R 2 + is uniformly bounded.
Proof. Let the initial value of system (2) be Y(t 0 ) = (y 1 (t 0 ), y 2 (t 0 )). Assume that ∃ a constant t * satisfying t ∈ (t 0 , t * ) such that According to system (2), we have By Lemma of [40], one gets y 1 (t + * ) = 0. By (8), we find that it is contradiction. Therefore, The proof of (a) ends. In the sequel, we shall prove uniformly boundedness of system (2). Set Then where κ > 0 is a constant and According to Lemma 2, we get then The proof of Theorem 2 finishes.
Here we omit the concrete proof of Lemma 1, one can consult [41].
Making use of Lemma 1, one gets the following result.

Global Asymptotic Stability Exploration
In this part, we will explore the global stability issue of the positive equilibrium point Y(y 1 * , y 2 * ) of model (2). Firstly, we give the following assumption:  (2) is globally asymptotically stable.
Proof. Setting up the following positive definite function: Then By (S 5 ), we can know that D η V (t) ≤ 0, which completes the proof.

Hybrid Control Technique for Bifurcation Control
In this part, we will make use of an appropriate hybrid controller which consists of state feedback and parameter perturbation to control the stability and Hopf bifurcation for model (2). By virtue of the research ideas in [42][43][44], we get the fractional-order controlled duopoly game model: where σ 1 , σ 2 represent feedback gain parameters. Models (40) and (2) own the same equilibrium points If (S 1 ) holds, then Y 4 (y 1 * , y 2 * ) is positive equilibrium point. The linear system of model (41) near Y 4 (y 1 * , y 2 * ) owns the expression: where where The characteristic equation of system (41) owns the following expression: which generates where When θ = 0, then Equation (45) becomes: is fulfilled, then the two roots λ 1 , λ 2 of Equation (47) obey |arg(λ 1 )| > ηπ 2 , |arg(λ 2 )| > ηπ 2 . It follows from Lemma 1 that the positive equilibrium point Y 4 (y 1 * , y 2 * ) of model (40) involving θ = 0 is locally asymptotically stable.
Making use of Lemma 1, one gets the following result.
Theorem 6. In 2012, Xu and Ma [18] explored the local stability and the creation of Hopf bifurcation of integer-order (1). In this current work, we mainly explore the various dynamics including the existence and uniqueness, non-negativeness, boundedness of the solution, local stability, onset of Hopf bifurcation, and Hopf bifurcation control problem for the established fractional-order delayed duopoly game model (2), which comes from the modified version of integer-order delayed duopoly game model (1). All investigation approaches and ideas practically differ from those in Xu and Ma [18]. The exploration idea of Xu and Ma [18] can not be applied to study the dynamical characteristics of model (2) in this work. From this viewpoint, we hold that our works replenish the research of [18] and expedite the development of bifurcation principle of fractional differential system.
To test the correctness of the key conclusions of Theorem 3, in the sequel, we will fix two delay values. Firstly, set θ = 1.52 which is less than θ 0 = 1.7, namely, θ falls into the range of value [0, θ 0 ). For this case, the Matlab simulation plots are provided in Figure 1. Apparently, Figure 1 demonstrates that the price of the first insurance company y 1 will approach to 0.5978 and the price of the second insurance company y 2 will approach to 0.5359 with the increase of time t. Secondly, set θ = 1.94 which is greater than θ 0 = 1.7, namely, θ exceeds the key value θ 0 ). For this case, the Matlab simulation plots are provided in Figure 2. Apparently, Figure 2 demonstrates that the price of the first insurance company y 1 will oscillate around the value 0.5978 and the price of the second insurance company y 2 will oscillate around the value 0.5359 with the increase of time t. That is to say, a Hopf bifurcation (a limit cycle) will take place near the equilibrium point Y(0.5978, 0.5359). In addition, in order to intuitively display the bifurcation value of delay, we give the bifurcation diagrams that show the bifurcation point θ 0 ≈ 1.7 (see Figures 3 and 4).

Example 2.
Consider the following fractional-order controlled delayed duopoly game model: where ρ 1 = 0.2, ρ 2 = 0.2, α = 5, β 1 = 4.5, β 2 = 5, µ = 0.2, δ 1 = 0.7, δ 2 = 0.6, γ 1 = 1 90 , γ 2 = 0.0001. Let σ 1 = 0.2, σ 2 = 0.4. By algebraic operation, one can derive the unique positive equilibrium point of system (63) takes the value Y(0.5978, 0.5359). It is checked that the hypotheses (S 1 )-(S 8 ) in Theorem 5 are all met. Utilizing Matlab software, one can determine that ξ 0 = 4.007, θ 0 * = 1.33. To test the correctness of the key conclusions of Theorem 5, in the sequel, we will fix two delay values. Firstly, set θ = 1.1 which is less than θ 0 * = 1.33, namely, θ falls into the range of value [0, θ 0 * ). For this case, the Matlab simulation plots are provided in Figure 5. Apparently, Figure 5 demonstrates that the price of the first insurance company y 1 will approach to 0.5978 and the price of the second insurance company y 2 will approach to 0.5359 with the increase of time t. Secondly, set θ = 1.45 which is greater than θ 0 * = 1.33, namely, θ exceeds the key value θ 0 ). For this case, the Matlab simulation plots are provided in Figure 6. Apparently, Figure 6 demonstrates that the price of the first insurance company y 1 will oscillate around the value 0.5978 and the price of the second insurance company y 2 will oscillate around the value 0.5359 with the increase of time t. Thai is to say, a Hopf bifurcation (a limit cycle) will take place near the equilibrium point Y(0.5978, 0.5359). In addition, in order to intuitively display the bifurcation value of delay, we give the bifurcation diagrams that show the bifurcation point θ 0 * ≈ 1.7 (see Figures 7 and 8). provided in Figure 6. Apparently, Figure 6 demonstrates that the price of the first insurance company y 1 will oscillate around the value 0.5978 and the price of the second insurance company y 2 will oscillate around the value 0.5359 with the increase of time t. Thai is to say, a Hopf bifurcation (a limit cycle) will take place near the equilibrium point Y (0.5978, 0.5359). In addition, in order to intuitively display the bifurcation value of delay, we give the bifurcation diagrams that show the bifurcation point θ 0 * ≈ 1.7(see .  . Numerical simulation results of model (7.1) concerning θ = 1.94 > θ 0 = near the positive equilibrium point Y (1.0221, 16.0947).      Numerical simulation results of model (7.2) concerning θ = 1.45 > θ 0 * = ear the positive equilibrium point Y (1.0221, 16.0947).

Remark 1.
By making use of a suitable hybrid controller, we can narrow the stability region and advance the onset of Hopf bifurcation of the fractional-order duopoly game model (2). From an economic point of view, we can advance the cyclic state of the price of the tow insurance companies via adjusting the delay and feedback gain parameters.

Conclusions
The price of insurance companies plays an important role in dominating the market and attracting the consumers. The price competition of insurance companies is a vital topic. In this current research, we propose a new fractional-order duopoly game model with delays in insurance market. The existence and uniqueness, non-negativeness, boundedness of solution, stability, Hopf bifurcation, globally asymptotic stability, and Hopf bifurcation control of the involved fractional-order delayed duopoly game model in insurance market have been systematically explored. A series of sufficient conditions which guarantee the existence and uniqueness, non-negativeness, boundedness of solution, stability of the positive equilibrium, onset of Hopf bifurcation, and globally asymptotic stability of the addressed fractional-order delayed duopoly game model in insurance market, are derived. By virtue of hybrid control technique, we successfully control the stability domain and the time of generation of Hopf bifurcation of the involved fractional-order delayed duopoly game model in insurance market. The obtained study results own great theory value and praxis function use for reference in administering and running insurance companies. In addition, the research approach is also used to probe into bifurcation dynamics and its control issue of a number of other fractional-order systems appearing in many areas. Here we must point out that although we can control the stability region and the time of onset of Hopf bifurcation of the fractional-order duopoly game model via hybrid controller, we may not be able to other fractional-order delayed models via the same hybrid controller. We must take some adequate measures to control the stability region and the time of onset of Hopf bifurcation according to different fractional-order delayed models. We will address this aspect in the near future.