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Bifurcation Analysis and Chaos Control in a Modified Finance System with Delayed Feedback

Department of Mathematics and Computer Science, Ningxia Normal University, Xueyuan Rd., Guyuan 756000, P. R. China

Corresponding author.

Department of Information Engineering, Zhengzhou Institute of Finance and Economics, Tianhe Rd. 36, Zhengzhou 475000, P. R. China

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Mei Liu

Department of Mathematics and Computer Science, Ningxia Normal University, Xueyuan Rd., Guyuan 756000, P. R. China

Department of Information Engineering, Zhengzhou Institute of Finance and Economics, Tianhe Rd. 36, Zhengzhou 475000, P. R. China

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https://doi.org/10.1142/S0218127416501054Cited by:14 (Source: Crossref)

Abstract

We investigate the effect of delayed feedback on the finance system, which describes the time variation of the interest rate, for establishing the fiscal policy. By local stability analysis, we theoretically prove the existences of Hopf bifurcation and Hopf-zero bifurcation. By using the normal form method and center manifold theory, we determine the stability and direction of a bifurcating periodic solution. Finally, we give some numerical solutions, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable equilibrium or periodic orbit.

Keywords:

References

Berezansky, L., Braverman, E. & Idels, L. [2013] “ Mackey–Glass model of hematopoiesis with non-monotone feedback: Stability, oscillation and control,” Appl. Math. Comput. 21, 6268–6283. Google Scholar
Bianca, C. & Rondoni, L. [2009] “ The non-equilibrium Ehrenfest gas: A chaotic model with flat obstacles,” Chaos 19, 013121. Crossref, Web of Science, Google Scholar
Bianca, C., Ferrara, M. & Guerrini, L. [2013] “ The time delays’ effects on the qualitative behavior of an economic growth model,” Abstr. Appl. Anal. 2013, 901014. Crossref, Google Scholar
Cesare, L. D. & Sportelli, M. [2005] “ A dynamic ISCLM model with delayed taxation revenues,” Chaos Solit. Fract. 25, 233–244. Crossref, Web of Science, Google Scholar
Chen, X. L., Liu, H. H. & Xu, C. L. [2014] “ The new result on delayed finance system,” Nonlin. Dyn. 78, 1989–1998. Crossref, Web of Science, Google Scholar
Chian, A. C., Rempel, E. L. & Rogers, C. [2006] “ Complex economic dynamics: Chaotic saddle, crisis and intermittency,” Chaos Solit. Fract. 29, 1194–1218. Crossref, Web of Science, Google Scholar
Ding, J., Yang, W. & Yao, H. [2009] “ A new modified hyperchaotic finance system and its control,” Int. J. Nonlin. Sci. 8, 59–66. Google Scholar
Du, J., Huang, T. & Sheng, Z. [2010] “ A new method to control chaos in an economic system,” Appl. Math. Comput. 217, 2370–2380. Crossref, Web of Science, Google Scholar
Gao, Q. & Ma, J. [2009] “ Chaos and Hopf bifurcation of a finance system,” Nonlin. Dyn. 58, 209–216. Crossref, Web of Science, Google Scholar
Ghosh, D., Roy Chowdhury, A. & Saha, P. [2008] “ Multiple delay Rossler system: Bifurcation and chaos control,” Chaos Solit. Fract. 35, 472–485. Crossref, Web of Science, Google Scholar
Guo, Y., Jiang, W. & Niu, B. [2013] “ Bifurcation analysis in the control of chaos by extended delay feedback,” J. Franklin Instit. 350, 155–170. Crossref, Web of Science, Google Scholar
Hale, J. [1977] Theory of Functional Differential Equations (Springer-Verlag, NY). Crossref, Google Scholar
Hanene, M. & Olfa, B. [2012] “ Chaos synchronization for master slave piecewise linear systems: Application to Chua’s circuit,” Commun. Nonlin. Sci. Numer. Simulat. 17, 1292–1302. Crossref, Web of Science, Google Scholar
Hassard, B. D., Kazarinoff, N. D. & Wan, Y. H. [1981] Theory and Applications of Hopf Bifurcation (Cambridge University Press, Cambridge). Google Scholar
He, X., Li, K., Wei, J. & Zheng, M. [2009] “ Market stability switches in a continuous time financial market with heterogeneous beliefs,” Econ. Model. 26, 1432–1442. Crossref, Web of Science, Google Scholar
Lancaster, P. & Tismenetsky, M. [1985] The Theory of Matrices with Applications (Academic Press, London). Google Scholar
Lorenz, H. W. & Nusse, H. E. [2002] “ Chaotic attractors, chaotic saddles, and fractal basin boundaries: Goodwin’s nonlinear accelerator model reconsidered,” Chaos Solit. Fract. 13, 957–965. Crossref, Web of Science, Google Scholar
Ma, J. & Chen, Y. [2001a] “ Study for the bifurcation topological structure and the global complicated character of a kind of non-linear finance system (I),” Appl. Math. Mech. 22, 1240–1251. Crossref, Web of Science, Google Scholar
Ma, J. & Chen, Y. [2001b] “ Study for the bifurcation topological structure and the global complicated character of a kind of non-linear finance system (II),” Appl. Math. Mech. 22, 1375–1382. Crossref, Web of Science, Google Scholar
Ma, J. H., Ren, B. & Chen, Y. S. [2004] “ Impulsive control of chaotic attractors in nonlinear chaotic systems,” Appl. Math. Mech. 25, 889–894. Google Scholar
Ma, J., Cui, Y. & Liu, L. [2008] “ Hopf bifurcation and chaos of financial system on condition of specific combination of parameters,” J. Syst. Sci. Complex. 21, 250–259. Crossref, Web of Science, Google Scholar
Ma, C. & Wang, X. Y. [2012] “ Hopf bifurcation and topological horseshoe of a novel finance chaotic system,” Commun. Nonlin. Sci. Numer. Simulat. 17, 721–730. Crossref, Web of Science, Google Scholar
Ott, E., Grebogi, C. & Yorke, J. A. [1990] “ Controlling chaos,” Phys. Rev. Lett. 64, 1196–1205. Crossref, Web of Science, Google Scholar
Pyragas, K. [1992] “ Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170, 421–429. Crossref, Web of Science, Google Scholar
Radelet, S., Sachs, J. & Cooper, R. [1998] “The east Asian financial crisis: Diagnosis, remedies, prospects,” Brookings Papers Econ. Activity 1, 1–74. Google Scholar
Ruan, S. & Wei, J. [2003] “ On the zeros of transcendental functions with applications to stability of delay differential equations with two delays,” Dyn. Contin. Discr. Impuls. Syst. Ser. A: Math. Anal. 10, 863–879. Web of Science, Google Scholar
Son, W. S. & Park, Y. J. [2011] “ Delayed feedback on the dynamical model of a financial system,” Chaos Solit. Fract. 44, 208–217. Crossref, Web of Science, Google Scholar
Song, Y. & Wei, J. [2004] “ Bifurcation analysis for Chen’s system with delayed feedback and its application to control of chaos,” Chaos Solit. Fract. 22, 75–91. Crossref, Web of Science, Google Scholar
Yu, H. J., Cai, G. L. & Li, Y. X. [2012] “ Dynamic analysis and control of a new hyperchaotic finance system,” Nonlin. Dyn. 67, 2171–2182. Crossref, Web of Science, Google Scholar
Zhang, H., Li, C. & Zhang, J. [2004] “ Controlling chaotic Chua’s circuit based on piecewise quadratic Lyapunov functions method,” Chaos Solit. Fract. 22, 1053–1061. Crossref, Web of Science, Google Scholar
Zhao, X., Li, Z. & Li, S. [2011] “ Synchronization of a chaotic finance system,” Appl. Math. Comput. 217, 6031–6039. Crossref, Web of Science, Google Scholar