Abstract
In this article, based on the previous works, a new fractional-order financial model is put up. The chaotic behavior of the fractional-order financial model is suppressed by designing an appropriate controller. By choosing the delay as the bifurcation parameter, we establish the sufficient condition to guarantee the stability and the existence of Hopf bifurcation of fractional-order financial model. Also, the influence of the delay and the fractional order on the stability and the existence of Hopf bifurcation of fractional-order financial model is revealed. An example is given to confirm the effectiveness of the analysis results. The main findings of this article play an important role in maintaining economic stability.
1. 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 [5–9].
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 denotes the saving amount, denotes the cost per investment, denotes the elasticity of demand of commercial markets, represents interest rate, represents investment demand, and represents price index. In 2008, Chen [13] designed a suitable time delayed 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 [5–7, 20–30].
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 [31–38]. 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 fractional-order financial systems. One can see [11, 12, 39–47].
In view of the above analysis and based on system (1), we modify system (1) as the following fractional-order financial model:where stands for the fractional order. The study reveals that chaotic phenomenon will appear if 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 fractional-order 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 fractional-order financial model. In Section 4, an example is given to show the effectiveness of the main findings. A conclusion is presented in Section 5.
2. Basic Results
In this section, we give some basic results on fractional calculus.
Definition 1 (see [48]). Define Caputo fractional-order derivative as follows:where . and is a positive integer such that If , then
Definition 2 (see [49]). The point is called an equilibrium point if the equationshold.
Lemma 3 (see [50]). Let be the root of the characteristic equation of the autonomous system , where Then system is asymptotically stable if and only if . The system is stable if and only if and those critical eigenvalues that satisfy have geometric multiplicity one.
Lemma 4 (see [51]). The system , where . Then the characteristic equation of the system is If all the roots of the characteristic equation of the system have negative real roots, then the zero solution of the system is asymptotically stable.
3. Chaos Control by Controller
If , then system (2) has a unique equilibrium point:If , then model (2) has three equilibrium points: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
Add the following feedback controller to the first equation of system (2):where and are the proportional control parameter and the derivative control parameter, respectively, and is time delay; then system (2) becomesThat is,Let ; then linear system of system (10) takes the formThe corresponding characteristic equation of (11) is given byThenwhere Assume that is a root of (13); then one haswhereBy (15), one hasLetThenIt follows from (17) thatNotice thatwhereBy (20), one haswhereLetand
Lemma 5. (a) If , then (13) has no root with zero real parts; (b) if and there exists a constant such that , then (13) has at least two pairs of purely imaginary roots.
Proof. (a) It follows from (25) thatNotice that ; then ,. By , one knows that (26) has no positive real root. In addition, is not the root of (13). We complete the proof of (a).
(b) Notice that and ; then and such that , and then (25) has at least two positive real roots. Thus (13) has at least two pairs of purely imaginary roots. We complete the proof of (b).
Without loss of generality, one can assume that (23) has ten positive real roots . By (13), we obtainwhere Then is a pair of purely imaginary roots of (10) when LetNow we give the following assumption:
, where
Lemma 6. If is the root of (13) near which satisfies , then
Proof. By (13), one hasNotice that and then one haswhereThenThusIt follows from thatTh proof of Lemma 6 is completed.
LetNext we give an assumption as follows:
.
Lemma 7. If and holds true, then system (9) is locally asymptotically stable.
Proof. If , then (13) takes the formBy , all roots of (39) satisfy By Lemma 3, we know that the conclusion of Lemma 4.3 holds. The proof of Lemma 3.4 is completed.
According to the analysis above, we have the following conclusion.
Theorem 8. In addition to condition (b) of Lemma 6. If and are fulfilled, then the equilibrium point of system (9) is locally asymptotically stable when and a Hopf bifurcation will appear around the equilibrium point when
Remark 9. In [4–7, 13–20], 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.
4. An Example
We give the following controlled financial system:Clearly, system (40) has an equilibrium point . Let . Then the critical frequency and the bifurcation point . We can check that the assumptions in Theorem 8 hold true. Figures 11–20 indicate that when , the equilibrium point 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 , 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 of (40) with Table 1.
5. 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 fractional-order 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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors were supported by the National Natural Science Foundation of China (no. 61673008), Project of High-Level Innovative Talents of Guizhou Province ([]5651), Major Research Project of the Innovation Group of the Education Department of Guizhou Province ([]039), Project of Key Laboratory of Guizhou Province with Financial and Physical Features ([]004), Foundation of Science and Technology of Guizhou Province ([]1025 and []1020), and Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Changsha University of Science & Technology) (2018MMAEZD21).