Stability and Bifurcations Analysis of Discrete Dynamical SystemsView this Special Issue
Double Delayed Feedback Control of a Nonlinear Finance System
In this paper, a class of chaotic finance system with double delayed feedback control is investigated. Firstly, the stability of equilibrium and the existence of periodic solutions are discussed when delays change and cross some threshold value. Then the properties of the branching periodic solutions are given by using center manifold theory. Further, we give an example and numerical simulation, which implies that chaotic behavior can be transformed into a stable equilibrium or a stable periodic solution. Also, we give the local sensitivity analysis of parameters on equilibrium.
Chaos applied to various disciplines of natural science and social science is a complex dynamic phenomenon. Chaotic systems with nonlinearity have been extensively investigated by many communities [1–5], and so on. In 1985, chaos was first discovered in the economic field, which had great impact imposed on the prominent economies. In economic field, chaos means that the economic operation has its intrinsic uncertainty.
Nonlinear methods are an important research method that has been widely used to explain complex economic phenomena [6–10]. In economic field, financial risks mean the possibilities of suffering losses caused by uncertainly endogenous factors in financial or investment activities, which displays irregularly fluctuations. The source of risks derives from the strange attractor, while the key of risk management is to control the chaotic attractor. In fact, one of the features of chaos in the economic system is financial crisis. In the passed few decades, a lot of ways to control or synchronize chaos have been proposed, such as OGY method , PC method , fuzzy control , impulsive control [14–18], linear feedback control [19–23], delayed feedback method [24–29], multiple delayed feedback control [30–35], and so on. Nowadays, the delayed dynamic systems occupy a central position in many fields, such as biology, transport control, and chemistry [36–39]. Since many economic processes have time-delay characteristics [40–44], they are unsuitable using ordinary differential equations (ODEs) to describe. Some authors concluded that the chaotic behavior of a microeconomic system could be stabilized to periodic orbits by using delayed feedback control, which seemed more applicable to experimental systems and avoided heavy data processing. In [6, 7], the authors proposed a financial system including four parts (production, money, stock, and labor force). Furthermore, they proposed a simplified model which is described using three variables: represents the interest rate, represents the investment demand, and represents the price index. The three-dimensional model is given as follows:where is the saving amount, is the cost per investment, and is the elasticity of demand of commercial markets. For system (1), the feedback control with one delay had been studied by many authors [45–47]. In 2008, Chen  firstly prevented the feedback control with three delays in system (1) as follows:where represent the feedback strengths and represent delays, . In , by the numerical simulations, Chen firstly gave the dynamic behavior of system (2) with only one and then gave the dynamic behavior when all and . After that, Woo-Sik Son et al.  gave the theory analysis for the above results. We find that system (2) with one delay has obtained complete results and the delayed feedback control method is valid. But system (2) with multiple different delays has not completely been investigated. Therefore, our objective is to study system (2) with two different delays. Next, we assume that and , the other cases are similar to analyze, then system (2) becomes and the delayed feedback control graph is shown in Figure 1, where is the state vector of the system.
We choose the initial conditions for system (3) as where and denotes the Banach space .
This paper is arranged as follows. In Section 2, the stability of equilibrium and existence of Hopf bifurcations are obtained by investigating the distribution of roots of characteristic equation. In Section 3, an algorithm is derived for deciding the properties of the branching periodic solutions by computing center manifold. In Section 4, some numerical simulations are given for verifying the theoretical analyses. In Section 5, the local sensitivity analyses of parameters on equilibrium are given. At last, we give a brief conclusion and discussion.
2. Stability of Equilibrium and Local Hopf Bifurcation
The existence and uniqueness of solutions and stability of equilibrium have always been an important issue for differential and difference systems [51–54]. Let the right sides of system (3) be zero; it can obtain the equilibrium as follows.
Remark 2. If the cost per investment is smaller than some value , then is feasible.
In this paper, it always assumes that holds and only considers the stability of and the other one can be analyzed in the same way.
When , (7) becomesIt has if .
Let Then and under . By using Routh-Hurwitz criterion, all roots of (9) have negatively real parts if holds.
Next, let and use as parameter, then (7) becomes
Let , then (12) becomesHence Let and , from Ruan and Wei , it knows that (14) has at least one positive root if ; (14) has no positive roots if and ; if , then (14) has positive roots iff Hence, under the condition , it has the following lemma.
Let be the root of (10) satisfying
Lemma 4. It assumes that . Then
Theorem 5. When , suppose that is satisfied.
(i) If and , then, for any , all roots of (10) has negatively real parts and is locally asymptotically stable.
(ii) If either or and hold, then has at least a positive root , for , all roots of (10) have negatively real parts, and is locally asymptotically stable.
(iii) If conditions (ii) and hold, then system (3) undergoes Hopf bifurcations at when ,
Remark 6. When and hold, Theorem 5 tells us that, through adjusting the cost per investment, the system will tend to or vibrates around . Under this situation, the state of system goes from order to order, that is, the macroeconomic operation is definite.
Let , then (23) becomeswhere and
Define From Cardano’s formula, it has the following theorem.
Theorem 7. If , then (24) has a real root and a pair of complex roots , that is to say, (23) has a real root , and a pair of complex roots .
(ii) If , then (24) has three real roots and (23) has also three real roots.
Furthermore, it assumes that
Theorem 8. When , suppose that the condition is satisfied.
(i) If and , then, for any , (10) has at least one root with positively real parts and is unstable.
(ii) If either or and hold, then has at least one positive root , for , (10) has at least one root with positively real parts, and is unstable. In addition, if , then is locally asymptotically stable when , where is the second bifurcating value.
(iii) If conditions (ii) and hold, then system (3) undergoes Hopf bifurcations at when , ,
Remark 9. When and hold, Theorem 8 tells us that it can still adjust the cost per investment , such that the system tends to or vibrates around under some conditions. Under this situation, the state of system goes from chaos to order; that is, the financial crisis may be eliminated.
From the above discussions, it knows that system (3) possibly makes stability switches as varying when . Define is stable interval of . Let and be the root of (7), then it haswhere We know that (26) has finite positive roots . For every fixed , there exists a sequence satisfying (26). Define , . When , are a pair of roots of (7). Hence, by Hopf bifurcation theorem , it has the next result.
Theorem 10. Suppose that either or is satisfied and .
(i) If (26) has no positive roots, then for any , all roots of (7) have negatively real parts and is locally asymptotically stable.
(ii) If (26) has positive roots, then all roots of (7) have negatively real parts when and is locally asymptotically stable. In addition, if holds, then system (3) undergoes Hopf bifurcation at when
Remark 11. When and or hold, Theorem 10 tells us that, through adjusting the parameters (), the system will tend to or vibrates around . Under this situation, the state of system goes from order to order or from chaos to order.
3. Property of Hopf Bifurcation
In the above section, the sufficient conditions that system (3) undergoes a Hopf bifurcation at when have already been obtained. In this section, it assumes that Theorem 10 (ii) is satisfied and establishes the explicit formula for determining the properties of Hopf bifurcation at by using the method developed in .
For convenience, it assumes that and lets and drops the bar for simplifying. Then is the Hopf bifurcation value. Since , the phase space , system (3) is transformed into the following FDE in :where and are given, respectively, by where where .
By Riesz representation theorem, there exists a bounded variation function for , such that For , define and
For , it has and system (28) becomes
For and , define and the inner product where It knows that and are adjoint operators, then are eigenvalues of and when
By computations, it can obtain that is the eigenvector of corresponding to the eigenvalue , and is the eigenvector of corresponding to the eigenvalue . Therefore, it has that where
By comparing the coefficients, it can obtain where
Substituting and into and , then can be expressed. Thus we may compute the following important quantities:
Theorem 12. If , Hopf bifurcation is supercritical (subcritical). If , periodic solution is stable (unstable). If , the period of periodic solution is increase (decrease).
4. Numerical Simulations
In this section, we give an example:With these parameters, it can obtain and is satisfied. When , by computation, it can obtain that (12) has two positive roots , . Substituting them into (16) gives, respectively, Furthermore, , . By Theorem 8, is asymptotically stable when and unstable for , which means that the stability switches occur. These results are illustrated in Figures 3–5.
Let ; we obtain . By Theorem 10, it knows that is asymptotically stable for and . Furthermore, it has and . Therefore, at , the periodic solution is orbitally asymptotically stable, and the Hopf bifurcation is forward (see Figures 6 and 7).
Next, we investigate the effect of two different delays. Firstly, we choose and find system (3) has chaos phenomenon (see Figure 8). When choosing and , we find that chaos phenomenon disappears and the solutions of system (48) approach stable equilibrium (see Figure 9). When choosing and , chaos phenomenon disappears and appears stable periodic solutions (see Figure 10). The above results show that double delayed feedback control is superior to delayed feedback control. Hence, we improve the results in .
Using the same methods, it can also obtain the similar results in the following systems by choosing suitable : and
In , Chen investigated the dynamics of system (2) with three delays by numerical simulations with and , and Chen found that the dynamics of this case have become more complex (the inverse period doubling, period doubling routes, and chaos). Next, we further consider system (2) with three delays by numerical simulations with , , and . System (2) becomesWe find that when , system (52) has chaos phenomenon (see Figure 11), while when , chaos phenomenon disappears and the solutions of system (52) trend to stable equilibrium (see Figure 12). Using the same methods, we can also give the theory analysis of the above result for system (2) and the similar conclusions can be obtained.
Finally, it will investigate the effect of for system (1) for chaos to generate. Firstly, we fix and ; it can obtain the Hopf bifurcation curve in plane (see Figure 13). When is chosen, it can obtain value where Hopf bifurcation will occur. The conditions here are just sufficient for the existence of Hopf bifurcation about parameter.
Next, we fix , , choosing , respectively. When , system has a periodic solution. Increasing , system will produce period doubling bifurcation and ultimately lead to chaos (see Figures 14 and 15). These show that the cost per investment makes system change from order to chaos, which means the importance of the cost per investment to control chaos.
5. Local Sensitivity Analysis
Local sensitivity analysis index allows us to measure the relative change of a state variable as parameter changing. Next, we use the following definition of normalized forward sensitivity index to perform local sensitivity analysis and compute normalized sensitivity indices.
Definition 13 (see ). The normalized forward sensitivity index of a variable, , that depends differentiably on a parameter, , is defined as
To perform local sensitivity analysis, we set
Table 1 shows that decreasing (respectively, increasing) the savings amount by 1% will increase (respectively, decrease) the interest rate by 0.1378%. Decreasing (respectively, increasing) the cost per investment by 1% will increase (respectively, decrease) the interest rate by 0.2653%. Increasing (respectively, decreasing) the elasticity of demand of commercial markets by 1% will increase (respectively, decrease) the interest rate by 0.1276%. The conclusion is that the cost per investment is the most important factor to the interest rate.
Table 2 shows that increasing (decreasing) the savings amount by 1% will increase (decrease) the investment demand by 0.5192%. Decreasing (respectively, increasing) the cost per investment by 1% will increase (decrease) the investment demand by 0.4808%. The conclusion is that the savings amount is the most important factor to the investment demand.
Table 3 shows that decreasing (increasing) the savings amount by 1% will increase (decrease) the price index by 0.1378%. Decreasing (increasing) the cost per investment by 1% will increase (decrease) the price index by 0.2653%. Decreasing (increasing) the elasticity of demand of commercial markets by 1% will increase (decrease) the price index by 0.8724%. The conclusion is that the elasticity of demand of commercial markets is the most important factor to the price index.
6. Conclusions and Discussions
Bifurcation in nonlinear finance system with one delay has been studied by many researchers. However, there are few papers to focus on nonlinear finance system with multiple delay feedback control. In this paper, we analyze a chaotic finance system using double delayed feedback control and find that the stability switches can occur when varies in the case of . The conclusion shows that if the saving amount, cost per investment, and the elasticity of demand are fixed, the feedback control used on the interest rate term can cause periodic fluctuations of the system when the feedback strength is fixed and chaotic phenomenon vanish. That is, it is effective in eliminating financial crisis using delayed feedback control in the interest rate term.
Then fix in a stability interval, regarding as parameter; it can show that there exists the first critical value of at which the equilibrium loses its stability and the Hopf bifurcation occurs. These conclusions show that if the feedback control used on the interest rate term under some delay is invalid to remove chaos, then it may add the feedback control to the investment demand term at the same time, which can make chaos disappear and the system produces regular vibrations. The results tell us that the double delayed feedback control can be considered better method than single delayed feedback control for the control of chaotic attractor.
Our results show that, for a class of chaotic finance system, the chaos oscillation can be controlled by delays. In addition, by choosing different delays and numerical simulations, we improve the results in  and show that the multiple delayed feedback control is more effective than one delayed feedback control.
In addition, we also obtain that system can produce chaos by period doubling bifurcation when increasing the cost per investment , which means the importance of the cost per investment to control chaos. At last, local sensitivity analyses of parameters on equilibrium are given. The conclusions are that the cost per investment is the most important factor to the interest rate; the savings amount is the most important factor to the investment demand; the elasticity of demand of commercial markets is the most important factor to the price index.
All the data in this study is hypothetical to verify the correctness of the theoretical results.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Z. Jiang was supported by National Natural Science Foundation of China (nos. 11801014 and 11875001), Natural Science Foundation of Hebei Province (no. A2018409004), and Doctoral Research Fund Project of North China Institute of Aerospace Engineering from China (no. BKY-2016-01); Y. Guo was supported by Scientific Research Project of Beijing Polytechnic College from China (no. bgzyky201744z); T. Zhang was supported by SDUST Research Funds (no. 2014TDJH102) and Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents.
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