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Discrete Dynamics in Nature and Society
Volume 2019, Article ID 7254121, 17 pages
https://doi.org/10.1155/2019/7254121
Research Article

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

Correspondence should be addressed to Tongqian Zhang; nc.ude.tsuds@naiqgnotgnahz

Received 16 September 2018; Revised 16 November 2018; Accepted 26 November 2018; Published 5 February 2019

Academic Editor: Douglas R. Anderson

Copyright © 2019 Zhichao Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. 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 [15], 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 [610]. 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 [1418], linear feedback control [1923], delayed feedback method [2429], multiple delayed feedback control [3035], and so on. Nowadays, the delayed dynamic systems occupy a central position in many fields, such as biology, transport control, and chemistry [3639]. Since many economic processes have time-delay characteristics [4044], 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 [4547]. In 2008, Chen [48] firstly prevented the feedback control with three delays in system (1) as follows:where represent the feedback strengths and represent delays, . In [47], 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. [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 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.

Figure 1: The delayed feedback control graph.

We consider system (3) with the parameters , and [50]. When , system (3) has a chaotic attractor (see Figure 2).

Figure 2: Chaos phenomenon exists for system (3) when .

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 [5154]. Let the right sides of system (3) be zero; it can obtain the equilibrium as follows.

Lemma 1. (i) If the condition holds, then system (3) has a unique boundary equilibrium .
(ii) If the condition holds, then system (3) has three equilibria: and

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.

Let , where , , and , then system (3) becomeswhose characteristic equation iswhereNow we analyze the distribution of roots of (7) by using the method in [55, 56].

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 be the root of (10), then satisfiesSquaring and adding in the both sides of (11), it haswhere

Let , then (12) becomesHence Let and , from Ruan and Wei [55], 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.

Lemma 3. (i) Equation (14) has at least one positive root if .
(ii) Equation (14) has no positive root if and .
(iii) If and , then (14) has one positive roots iff and , where

It assumes that (14) has three positive roots, denoted by . Then (12) has three positive roots . Substituting into (11) yields and furthermore, it haswhere is determined by the sign of . Define

Let be the root of (10) satisfying

Lemma 4. It assumes that . Then

Proof. Substituting into (10) and taking the derivative of both sides with respect to , it has Hence, using and (10), it has where . Since and , then it has

By Lemmas 3 and 4 and applying the Hopf bifurcation theorem for FDE [5761], it has the following theorem.

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.

It has known that guarantees that all roots of (9) have negatively real parts. Now we assume that is violated. For convenience, denote then (9) becomes

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 [57], 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 [62].

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

Let be the solution of system (28) at . Define , thenwhere Rewrite (39) aswhere

Substituting (3) and (39) into , it has 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 35.

Figure 3: is unstable and chaos still exists with and .
Figure 4: becomes stable and chaos disappears with and .
Figure 5: is unstable, and a stable periodic solution appears with and .

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).

Figure 6: is asymptotically stable with and .
Figure 7: is unstable, and a periodic solution appears with and .

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 [48].

Figure 8: is unstable and chaos exists with and .
Figure 9: becomes stable and chaos disappears with and .
Figure 10: is unstable, and a periodic solution appears with and .

Using the same methods, it can also obtain the similar results in the following systems by choosing suitable : and

In [48], 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.

Figure 11: The equilibrium is unstable and chaos exists for system (52) with .
Figure 12: The equilibrium becomes stable and chaos disappears for system (52) with , and .

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.

Figure 13: The bifurcation curve in (c,b) plane for system (1) with .

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.

Figure 14: For (a), . For (b), . For (c), .
Figure 15: For (a), . For (b), . For (c), .

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 [63]). The normalized forward sensitivity index of a variable, , that depends differentiably on a parameter, , is defined as

To perform local sensitivity analysis, we set

Tables 13 show the effect of parameters on equilibrium .

Table 1: Normalized sensitivity indexes and order of importance of to the three parameters evaluated at the values , , and
Table 2: Normalized sensitivity indexes and order of importance of to the three parameters evaluated at the values , , and
Table 3: Normalized sensitivity indexes and order of importance of to the three parameters evaluated at the values , , and

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 [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.

Data Availability

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.

Acknowledgments

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.

References

  1. Z. Zhang, H. Shao, Z. Wang, and H. Shen, “Reduced-order observer design for the synchronization of the generalized Lorenz chaotic systems,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7614–7621, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. Y. Li, X. Huang, Y. Song, and J. Lin, “A new fourth-order memristive chaotic system and its generation,” International Journal of Bifurcation and Chaos, vol. 25, no. 11, Article ID 1550151, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  3. S. A. David, J. A. Machado, D. D. Quintino, and J. M. Balthazar, “Partial chaos suppression in a fractional order macroeconomic model,” Mathematics and Computers in Simulation, vol. 122, pp. 55–68, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. M. Guo, Y. Xue, Z. Gao, Y. Zhang, G. Dou, and Y. Li, “Dynamic analysis of a physical sbt memristor-based chaotic circuit,” International Journal of Bifurcation and Chaos, vol. 27, no. 13, Article ID 1730047, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  5. F. X. An and F. Q. Chen, “Multipulse orbits and chaotic dynamics of an aero-elastic fgp plate under parametric and primary excitations,” International Journal of Bifurcation and Chaos, vol. 27, no. 04, Article ID 1750050, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  6. J. Ma and Y. Chen, “Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (i),” Applied Mathematics and Mechanics, vol. 22, no. 11, pp. 1240–1251, 2001. View at Google Scholar
  7. J. H. Ma and Y. S. Chen, “Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (ii),” Applied Mathematics and Mechanics, vol. 22, pp. 1375–1382, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  8. Z. Wang, X. Huang, and G. D. Shi, “Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1531–1539, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  9. Baogui Xin and Yuting Li, “0-1 Test for Chaos in a Fractional Order Financial System with Investment Incentive,” Abstract and Applied Analysis, vol. 2013, Article ID 876298, 10 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  10. M. Yang, B. Cai, and G. Cai, “Projective synchronization of a modified three-dimensional chaotic finance system,” International Journal of Nonlinear Science, vol. 10, no. 1, pp. 32–38, 2010. View at Google Scholar · View at MathSciNet
  11. E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Physical Review Letters, vol. 64, no. 11, pp. 1196–1199, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. K. Tanaka, T. Ikeda, and H. O. Wang, “A unified approach to controlling chaos via an lmi-based fuzzy control system design,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 45, no. 10, pp. 1021–1040, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  14. T. Zhang, X. Meng, Y. Song, and T. Zhang, “A stage-structured predator-prey SI model with disease in the prey and impulsive effects,” Mathematical Modelling and Analysis, vol. 18, no. 4, pp. 505–528, 2013. View at Publisher · View at Google Scholar · View at Scopus
  15. F. Bian, W. Zhao, Y. Song, and R. Yue, “Dynamical analysis of a class of prey-predator model with beddington-deangelis functional response, stochastic perturbation, and impulsive toxicant input,” Complexity, vol. 2017, Article ID 3742197, 18 pages, 2017. View at Google Scholar
  16. T. Zhang, W. Ma, X. Meng, and T. Zhang, “Periodic solution of a prey-predator model with nonlinear state feedback control,” Applied Mathematics and Computation, vol. 266, pp. 95–107, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. H. Liu and H. Cheng, “Dynamic analysis of a prey–predator model with state-dependent control strategy and square root response function,” Advances in Difference Equations, vol. 2018, no. 63, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  18. J. Wang, H. Cheng, H. Liu, and Y. Wang, “Periodic solution and control optimization of a prey-predator model with two types of harvesting,” Advances in Difference Equations, vol. 2018, no. 1, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  19. M. Rafikov and J. M. Balthazar, “On control and synchronization in chaotic and hyperchaotic systems via linear feedback control,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 7, pp. 1246–1255, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. N. J. Peruzzi, F. R. Chavarette, J. M. Balthazar, A. M. Tusset, A. L. Perticarrari, and R. M. Brasil, “The dynamic behavior of a parametrically excited time-periodic mems taking into account parametric errors,” Journal of Vibration and Control, vol. 22, no. 20, pp. 4101–4110, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  21. Y. Li, H. Cheng, J. Wang, and Y. Wang, “Dynamic analysis of unilateral diffusion Gompertz model with impulsive control strategy,” Advances in Difference Equations, vol. 2018, no. 32, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  22. J. Wang, H. Cheng, X. Meng, and B. G. Pradeep, “Geometrical analysis and control optimization of a predator-prey model with multi state-dependent impulse,” Advances in Difference Equations, vol. 2017, no. 252, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  23. J. Wang, H. Cheng, Y. Li, and X. Zhang, “The geometrical analysis of a predator-prey model with multi-state dependent impulses,” Journal of Applied Analysis and Computation, vol. 8, no. 2, pp. 427–442, 2018. View at Google Scholar · View at MathSciNet
  24. X. Meng, F. Li, and S. Gao, “Global analysis and numerical simulations of a novel stochastic eco-epidemiological model with time delay,” Applied Mathematics and Computation, vol. 339, pp. 701–726, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  25. V. Pyragas and K. Pyragas, “Delayed feedback control of the lorenz system: An analytical treatment at a subcritical hopf bifurcation,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 73, Article ID 036215, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  26. H. Chang, Z. Wang, Y. Li, and G. Chen, “Dynamic analysis of a bistable bi-local active memristor and its associated oscillator system,” International Journal of Bifurcation and Chaos, vol. 28, no. 08, Article ID 1850105, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  27. X. Guan, G. Feng, and C. Chen, “A stabilization method of chaotic systems based on full delayed feedback controller design,” Physics Letters A, vol. 348, no. 3, pp. 210–221, 2006. View at Publisher · View at Google Scholar
  28. J. Zhang, X. Wu, L. Xing, C. Zhang, H. Iu, and T. Fernando, “Bifurcation analysis of five-level cascaded H-bridge inverter using proportional-resonant plus time-delayed feedback,” International Journal of Bifurcation and Chaos, vol. 26, no. 11, Article ID 1630031, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  29. Y. Song, A. Miao, T. Zhang, X. Wang, and J. Liu, “Extinction and persistence of a stochastic SIRS epidemic model with saturated incidence rate and transfer from infectious to susceptible,” Advances in Difference Equations, vol. 2018, no. 293, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  30. A. Ahlborn and U. Parlitz, “Stabilizing unstable steady states using multiple delay feedback control,” Physical Review Letters, vol. 93, Article ID 264101, 2004. View at Google Scholar · View at MathSciNet
  31. A. Ahlborn and U. Parlitz, “Controlling dynamical systems using multiple delay feedback control,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 72, Article ID 016206, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  32. Z. Yan, G. Zhang, J. Wang, and W. Zhang, “State and output feedback finite-time guaranteed cost control of linear itô stochastic systems,” Journal of Systems Science & Complexity, vol. 28, no. 4, pp. 813–829, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  33. J. Wang, K. Liang, X. Huang, Z. Wang, and H. Shen, “Dissipative fault-tolerant control for nonlinear singular perturbed systems with Markov jumping parameters based on slow state feedback,” Applied Mathematics and Computation, vol. 328, no. 1, pp. 247–262, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  34. J. Chen, T. Zhang, Z. Zhang, C. Lin, and B. Chen, “Stability and output feedback control for singular Markovian jump delayed systems,” Mathematical Control and Related Fields, vol. 8, no. 2, pp. 475–490, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  35. T. Zhang, X. Liu, X. Meng, and T. Zhang, “Spatio-temporal dynamics near the steady state of a planktonic system,” Computers & Mathematics with Applications, vol. 75, no. 12, pp. 4490–4504, 2018. View at Publisher · View at Google Scholar
  36. W. Wang and T. Zhang, “Caspase-1-mediated pyroptosis of the predominance for driving CD4 + T cells death: A nonlocal spatial mathematical model,” Bulletin of Mathematical Biology, vol. 80, no. 3, pp. 540–582, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  37. T. Zhang, W. Ma, and X. Meng, “Global dynamics of a delayed chemostat model with harvest by impulsive flocculant input,” Advances in Difference Equations, vol. 2017, no. 115, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  38. T. Zhang, X. Meng, and T. Zhang, “Global analysis for a delayed SIV model with direct and environmental transmissions,” Journal of Applied Analysis and Computation, vol. 6, no. 2, pp. 479–491, 2016. View at Google Scholar · View at MathSciNet
  39. T. Zhang, X. Meng, and T. Zhang, “Global dynamics of a virus dynamical model with cell-to-cell transmission and cure rate,” Computational and Mathematical Methods in Medicine, vol. 2015, Article ID 758362, 8 pages, 2015. View at Google Scholar
  40. L. D. Cesare and M. Sportelli, “A dynamic IS-LM model with delayed taxation revenues,” Chaos, Solitons & Fractals, vol. 25, no. 1, pp. 233–244, 2005. View at Google Scholar
  41. L. Fanti and P. Manfredi, “Chaotic business cycles and fiscal policy: an IS-LM model with distributed tax collection lags,” Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 736–744, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  42. M. Neamtu, D. Opris, and C. Chilarescu, “Hopf bifurcation in a dynamic IS-LM model with time delay,” Chaos, Solitons & Fractals, vol. 34, no. 2, pp. 519–530, 2007. View at Google Scholar
  43. I. SenGupta, “Generalized BN–S stochastic volatility model for option pricing,” International Journal of Theoretical and Applied Financee, vol. 19, no. 02, Article ID 1650014, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  44. A. Issaka and I. SenGupta, “Analysis of variance based instruments for Ornstein-Uhlenbeck type models: swap and price index,” Annals of Finance, vol. 13, no. 4, pp. 401–434, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  45. Y. Ding, W. Jiang, and H. Wang, “Hopf-pitchfork bifurcation and periodic phenomena in nonlinear financial system with delay,” Chaos, Solitons & Fractals, vol. 45, no. 8, pp. 1048–1057, 2012. View at Publisher · View at Google Scholar · View at Scopus
  46. R. Zhang, “Bifurcation analysis for a kind of nonlinear finance system with delayed feedback and its application to control of chaos,” Journal of Applied Mathematics, vol. 2012, Article ID 316390, 18 pages, 2012. View at Google Scholar
  47. Q. Gao and J. Ma, “Chaos and Hopf bifurcation of a finance system,” Nonlinear Dynamics, vol. 58, no. 209, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  48. W.-C. Chen, “Dynamics and control of a financial system with time-delayed feedbacks,” Chaos, Solitons & Fractals, vol. 37, no. 4, pp. 1198–1207, 2008. View at Publisher · View at Google Scholar · View at Scopus
  49. W.-S. Son and Y.-J. Park, “Delayed feedback on the dynamical model of a financial system,” Chaos, Solitons & Fractals, vol. 44, no. 4, pp. 208–217, 2011. View at Publisher · View at Google Scholar · View at Scopus
  50. M. Yang and G. Cai, “Chaos control of a non-linear finance system,” Journal of Uncertain Systems, vol. 5, no. 4, pp. 263–270, 2011. View at Google Scholar · View at Scopus
  51. X. Fan, Y. Song, and W. Zhao, “Modeling Cell-to-Cell Spread of HIV-1 with Nonlocal Infections,” Complexity, vol. 2018, Article ID 2139290, 10 pages, 2018. View at Publisher · View at Google Scholar
  52. A. Q. Khan and A. Sharif, “Some 3 × 6 systems of exponential difference equations,” Discrete Dynamics in Nature and Society, vol. 2018, Article ID 8362837, 35 pages, 2018. View at Google Scholar
  53. T. Zhang, T. Zhang, and X. Meng, “Stability analysis of a chemostat model with maintenance energy,” Applied Mathematics Letters, vol. 68, pp. 1–7, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  54. Z. Chang, X. Meng, and T. Zhang, “A new way of investigating the asymptotic behaviour of a stochastic SIS system with multiplicative noise,” Applied Mathematics Letters, vol. 87, pp. 80–86, 2019. View at Publisher · View at Google Scholar · View at MathSciNet
  55. S. Ruan and J. Wei, “On the zeros of a third degree exponential polynomial with applications to a delayed model for the control of testosterone secretion,” IMA Journal of Maths with Applications in Medicine & Biology, vol. 18, pp. 41–52, 2001. View at Google Scholar
  56. S. Ruan and J. Wei, “On the zeros of transcendental functions with applications to stability of delay differential equations with two delays,” Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, vol. 10, no. 6, pp. 863–874, 2003. View at Google Scholar
  57. J. H. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 1977. View at Publisher · View at Google Scholar
  58. L. Li, Z. Wang, Y. Li, H. Shen, and J. Lu, “Hopf bifurcation analysis of a complex-valued neural network model with discrete and distributed delays,” Applied Mathematics and Computation, vol. 330, pp. 152–169, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  59. Z. Jiang and L. Wang, “Global Hopf bifurcation for a predator-prey system with three delays,” International Journal of Bifurcation and Chaos, vol. 27, no. 07, Article ID 1750108, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  60. Z. Wang, X. Wang, Y. Li, and X. Huang, “Stability and hopf bifurcation of fractional-order complex-valued single neuron model with time delay,” International Journal of Bifurcation and Chaos, vol. 27, no. 13, Article ID 1750209, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  61. A. Miao, T. Zhang, J. Zhang, and C. Wang, “Dynamics of a stochastic SIR model with both horizontal and vertical transmission,” Journal of Applied Analysis and Computation, vol. 8, no. 4, pp. 1108–1121, 2018. View at Google Scholar · View at MathSciNet
  62. B. D. Hassard, N. D. Kazarinoff, and Y.-H. Wan, Theory and Application of Hopf Bifurcation, Cambridge University Press, Cambridge, UK, 1981. View at MathSciNet
  63. N. Chitnis, J. M. Hyman, and J. M. Cushing, “Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model,” Bulletin of Mathematical Biology, vol. 70, no. 5, pp. 1272–1296, 2008. View at Publisher · View at Google Scholar · View at MathSciNet