PD θ Control Strategy for a Fractional-Order Chaotic Financial Model

1Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang 550004, China 2School of Mathematics and Physics, University of South China, Hengyang 421001, China 3School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, China 4Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha 410114, China 5School of Mathematics and Statistics, Central South University, Changsha 410083, China


Introduction
Establishing financial models to investigate the complex dynamical behavior of economic society has attracted more attention of scholars in numerous areas.To grasp the law of operation accurately, various financial models have been established to reveal the inherent characteristics of economic development and numerous fruitful results are achieved.For instance, Zhang et al. [1] discussed the stability of a financial hyperchaotic model, Serletic [2] investigated the chaos in economic system, Lin et al. [3] made an detailed analysis on chaotic behavior of a financial complex model, and Gao and Ma [4] studied the chaotic phenomenon and bifurcation of a finance model.For more information on financial models, one can see [5][6][7][8][9].
In many cases, chaotic behavior often happens in financial models.Chaotic phenomenon will have a serious effect on man's everyday life.Thus the research on chaos control of financial models becomes a hot issue in financial community.The appearance of chaotic phenomenon in economic system implies that the macroeconomic operation has its inherent indefiniteness and complexity.Of course, government departments can take measures for control or interference, but the effect is very limited [10].Thus, it is worthwhile to deal with the control of chaos in financial systems by theoretical analysis.
In 2001, Ma and Chen [11,12] studied the following financial model: where  1 ≥ 0 denotes the saving amount,  2 ≥ 0 denotes the cost per investment,  3 ≥ 0 denotes the elasticity of demand of commercial markets,  1 represents interest rate,  2 represents investment demand, and  3 represents price index.In 2008, Chen [13] designed a suitable time delayed 2 Complexity feedback controller to control the chaotic phenomena of model (1); computer simulations are presented to illustrate the effectiveness of designed controller.In 2011, Son and Park [14] further dealt with the chaos control issue of model (1) by applying delayed feedback method.Detailed theoretical analysis and numerical simulations are carried out to check the correctness of the controller.In 2009, Gao and Ma [4] discussed the chaos control of model ( 1) by adding a time delay feedback term to the second equation of system (1).Also the sufficient condition to guarantee the stability and the existence of Hopf bifurcation of involved controlled financial model is established.Considering the effect of time delay on the financial phenomena, Zhang and Zhu [15], Chen et al. [16], Mircea et al. [17], and Zhang [18] established different delayed financial models and analyzed their Hopf bifurcation or chaos control issue.In 2014, Zhao et al. [19] investigated the the anticontrol of Hopf bifurcation and chaos control of model ( 1) by applying delayed washout filters.For details, one can see [5][6][7][20][21][22][23][24][25][26][27][28][29][30].
Here we would like to point out that all the above works are only concerned with the integer-order differential systems.Nowadays, numerous scholars have found that fractional calculus, which is a generalization of ordinary differentiation and integration, has potential applications in numerous fields such as economics, physics, heat transfer, and chemical engineer [31][32][33][34][35][36][37][38].Many researchers argued that it is more reasonable to describe the object natural phenomena by fractional-order differential equations than integer-order differential equations, since fractional-order differential equations can better describe the memory characteristics and historical dependence.Noticing that financial coefficients possess very long memory and the variation of financial coefficients has close connection with previous and current time, we think that it is important for us to establish fractional-order financial systems.In recent years, there are numerous articles that investigate the fractionalorder financial systems.One can see [11,12,[39][40][41][42][43][44][45][46][47].
In view of the above analysis and based on system (1), we modify system (1) as the following fractional-order financial model: where 0 <  < 1 stands for the fractional order.The study reveals that chaotic phenomenon will appear if  = 0.73 and The results can be shown in Figures 1-10.
Our key task is concerned with two topics: (i) designing a suitable   controller to suppress the chaotic behavior of system (2) and (2) seeking the influence of delay and fractional order on the stability and bifurcation phenomenon of controlled system.
The superiority of this article is stated as follows:  (I) A new fractional-order financial model is established.
(II) A   controller is designed to suppress the chaos of the fractional-order financial model.Also, a set of  new sufficient conditions to guarantee the stability and the existence of Hopf bifurcation of fractionalorder financial model are obtained.In addition, the   effect of the delay and fractional order on the dynamics of fractional-order financial model is displayed.
(III) To the best of our knowledge, it is the first time to control chaos of fractional-order financial model by applying   controller.
We organize this article as follows.In Section 2, some basic knowledge on fractional calculus is presented.In Section 3,   controller is designed to control chaos of fractionalorder financial model.In Section 4, an example is given to show the effectiveness of the main findings.A conclusion is presented in Section 5.

Basic Results
In this section, we give some basic results on fractional calculus.

Chaos Control by 𝑃𝐷 𝜗 Controller
If  2 +  1  2  3 −  3 > 0, then system (2) has a unique equilibrium point: If  2 +  1  2  3 −  3 < 0, then model ( 2) has three equilibrium points: 3 ) , ) . ( During the past several decades, many different control strategies have been applied to control the chaos and bifurcation behavior.But they only involve the integer-order differential systems.Applying   control strategy to control the chaotic behavior of fractional-order chaotic system is rare.To make up for the deficiency, we try to design a   feedback controller to suppress the chaotic phenomenon of model (2).In this paper, we only consider the equilibrium point . The other equilibrium points can be discussed in the same way.Here we omit it.Throughout this paper, we always assume that  2 +  1  2  3 −  3 < 0.
Proof.By ( 13), one has Notice that and then one has where Then Thus It follows from Th proof of Lemma 6 is completed. Let Next we give an assumption as follows: Lemma 7. If  = 0 and (A2) holds true, then system ( ) is locally asymptotically stable.
According to the analysis above, we have the following conclusion.
Remark .In [4][5][6][7][13][14][15][16][17][18][19][20], the authors studied the various dynamics of integer-order financial models.They did not involve the fractional-order forms.In [39], Bhalekar and Gejji considered the chaos of fractional-order financial model by predictor-corrector method.In this article, we control the chaos of fractional-order financial model by applying   control strategy.All the derived results and analysis ways of [4-7, 13-20, 39] can not be transferred to (2) to control the chaotic behavior.Based on these viewpoints, the fruits of this paper are entirely innovative and supplement the previous publications.
Clearly, system (40) has an equilibrium point (0.6831, 2.6667, −0.4554).Let   = 0.07,   = 0.32,  = 0.672.Then the critical frequency  0 = 0.8653 and the bifurcation point  0 = 0.5262.We can check that the assumptions in Theorem 8 hold true.indicate that when  ∈ [0, 0.5262), the equilibrium point (0.6831, 2.6667, −0.4554) of model ( 40) is locally asymptotically stable.From the financial point of view, it means that as time goes on, the interest rate will tend to the constant 0.6831, investment demand will tend to the constant 2.6667, and price index will tend to the constant -0.4554.Figures 21-30 indicate that  ∈ [0.5262, +∞), system (40) becomes unstable, and a Hopf bifurcation emerges.From the financial point of view, it means that as time goes on, the interest rate, investment demand, and price index will keep a periodic cycle.In addition, we show the relation of parameters  and  0 of (40) with Table 1.

Conclusions
In this paper, we propose a new fractional-order financial system.To control the chaotic behavior of the fractional-order financial system, we successfully design a   controller to achieve our goal.By adjusting the proportional and derivative parameters, we can change the stability and Hopf bifurcation character of the considered fractional-order financial system.By regarding the time delay as bifurcation parameter, we have established a new sufficient condition to ensure the stability  and the existence of Hopf bifurcation of the fractionalorder financial model.Also, the effect of the fractional order and delay on the stability and Hopf bifurcation is revealed.The research idea and the obtained theoretical  results of this article enrich and develop the bifurcation and control theory of fractional-order differential equations.The obtained results can provide useful guidance to people in financial community.We can properly adjust the parameter of the   controller to apply the suggested fractional-order financial models to deal with financially chaotic problems.In addition, we point out that although the   controller can effectively control the chaos of the fractional-order financial model, it involves multiple parameters: proportional control parameter   , the derivative control parameter   , time delay , and fractional order .In the future, we will seek some more simple controllers with less parameters to suppress the chaotic behavior.

Table 1 :
(40)relation of parameters  and  0 of model(40).Figure 11: The relation of  and  1 in model (