Double Delayed Feedback Control of a Nonlinear Finance System

1Fundamental Science Department, North China Institute of Aerospace Engineering, Langfang 065000, China 2Fundamental Education Department, Beijing Polytechnic College, Beijing 100042, China 3College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China 4State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China


Introduction
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][2][3][4][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][7][8][9][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 [11], PC method [12], fuzzy control [13], impulsive control [14][15][16][17][18], linear feedback control [19][20][21][22][23], delayed feedback method [24][25][26][27][28][29], multiple delayed feedback control [30][31][32][33][34][35], and so on.Nowadays, the delayed dynamic systems occupy a central position in many fields, such as biology, transport control, and chemistry [36][37][38][39].Since many economic processes have time-delay characteristics [40][41][42][43][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: Ẋ () =  () + ( () − )  () , Ẏ () = 1 −  () −  2 () , Ż () = − () −  () ,  where  > 0 is the saving amount,  > 0 is the cost per investment, and  > 0 is the elasticity of demand of commercial markets.For system (1), the feedback control with one delay had been studied by many authors [45][46][47].In 2008, Chen [48] firstly prevented the feedback control with three delays in system (1) where   represent the feedback strengths and   represent delays,  = 1,2,3.In [47], by the numerical simulations, Chen firstly gave the dynamic behavior of system (2) with only one   ̸ = 0 and then gave the dynamic behavior when all   ̸ = 0 ( = 1, 2, 3) and  1 =  2 =  3 = .After that, Woo-Sik Son et al. [49] 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  3 = 0 and  1 ,  2 ̸ = 0, the other cases are similar to analyze, then system (2) becomes and the delayed feedback control graph is shown in Figure 1, where x(t) is the state vector of the system.
We choose the initial conditions for system (3) as where () = ( 1 (),  2 (),  3 ())  ∈ ,  ∈ [−, 0],  = max{ 1 ,  2 } and  denotes the Banach space ([−, 0], R 3 ).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.

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][52][53][54].Let the right sides of system (3) be zero; it can obtain the equilibrium as follows.Remark .If the cost per investment is smaller than some value (/(1 + )), then E * ± is feasible.
In this paper, it always assumes that  −  −  > 0 holds and only considers the stability of E * + and the other one can be analyzed in the same way.
Remark .When  2 = 0 and ( 2 ) hold, Theorem 8 tells us that it can still adjust the cost per investment , such that the system tends to E * + or vibrates around E * + under some conditions.Under this situation, the state of system goes from chaos to order; that is, the financial crisis may be eliminated.
For every fixed   , there exists a sequence { 2 , ±i 0 are a pair of roots of (7).
Hence, by Hopf bifurcation theorem [57], it has the next result.
(i) If ( ) has no positive roots, then for any  2 ≥ 0, all roots of ( ) have negatively real parts and E * + is locally asymptotically stable.
Remark .When  2 > 0 and ( 1 ) or ( 2 ) hold, Theorem 10 tells us that, through adjusting the parameters (, , ,  2 ), the system will tend to E * + or vibrates around E * + .Under this situation, the state of system goes from order to order or from chaos to order.

Property of Hopf Bifurcation
In the above section, the sufficient conditions that system (3) undergoes a Hopf bifurcation at E * + when  2 =  0 2 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  2 =  0 2 by using the method developed in [62].
Finally, it will investigate the effect of  for system (1) for chaos to generate.Firstly, we fix  = 0.9 and  > 2.852; it can obtain the Hopf bifurcation curve in (, ) plane (see Figure 13).When  > 2.852 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  = 0.9,  = 1.2, choosing  = 0.01, 0.1, 0.2, respectively.When  = 0.01, 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.

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 (see [63]).The normalized forward sensitivity index of a variable, u, that depends differentiably on a parameter, , is defined as To perform local sensitivity analysis, we set  = 0.9,  = 0.2,  = 1.2.
Tables 1-3 show the effect of parameters , ,  on equilibrium E * + ( * ,  * ,  * ).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    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 Table 2: Normalized sensitivity indexes and order of importance of  * to the three parameters evaluated at the values  = 0.9,  = 0.2, and  = 1.2.

Parameter
Local sensitivity index Order of importance a +0.5192 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 Table 3: Normalized sensitivity indexes and order of importance of  * to the three parameters evaluated at the values  = 0.9,  = 0.2, and  = 1.2.

Parameter
Local sensitivity index Order of importance a -0.1378 3 b -0.2653 2 c -0.8724 1 demand of commercial markets is the most important factor to the price index.

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  1 varies in the case of  2 = 0.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  1 in a stability interval, regarding  2 as parameter; it can show that there exists the first critical value of  2 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 [48] 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.

Figure 1 :
Figure 1: The delayed feedback control graph.

Table 1 :
Normalized sensitivity indexes and order of importance of  * to the three parameters evaluated at the values  = 0.9,  = 0.2, and  = 1.2.