Time Delay Stabilizes Chaos Dynamics in Economic System

Rashmi Bhardwaj; Anamika Ranjan

doi:https://doi.org/10.14445/22315373/IJMTT-V53P563

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 53 | Number 7 | Year 2018 | Article Id. IJMTT-V53P563 | DOI : https://doi.org/10.14445/22315373/IJMTT-V53P563

Time Delay Stabilizes Chaos Dynamics in Economic System

Rashmi Bhardwaj, Anamika Ranjan

Abstract

This paper discusses the role of time delay feedback which stabilizes the chaos of nonlinear financial model. The interest rate, investment demand and price index are modelled with the help of saving amount, cost per investment, demand elasticity of commercial markets and the strength of feedback. All these parameters are considered to be positive. The stability of financial model is studied by Routh-Hurwitz criterion. The distributed time delay feedback strength stabilized the unstable financial system. Bifurcation of parameter and Lypunove exponent is simulated for time delay feedback system. Using the numerical method, it is observed that the inappropriate combination of saving amount, cost per investment and elasticity of investment demand of commercial market in the financial system is the root cause of chaos. The unstable system is stabilized by introducing the distributed time delay feedback strength. The stability and chaotic behaviour of systems gives the condition and behaviour of economic implications. It is concluded that the system is chaotic and to ensure stability of state, it is controlled by time delay feedback controlled system.

Keywords

Nonlinear financial system, Distributed time delay, chaos, Routh-Hurwitz criterion, Lypunove exponent, Bifurcation and equilibrium point.

References

[1] Cai J. and Shen J., (2008). “Hopf Bifurcation Analysis and approximation of limit Cycle in Coupled Van Der Pol and Duffing Oscillators”, The Open Acoustics Journal,1,19-23
[2] Cai G,Hung J. and Jiangsu Y.,(2007)."A New Finance Chaotic Attractor. "International Journal of Nonlinear Science,3(3),213-220.
[3] Chen C.W (2008)."Nonlinear dynamics and chaos in a fractional order financial system " Chaos. Solition and Fractrals,36,1305-1314.
[4] Dumitru Li and Opris D, (2009) "Neimatk -Sacker bifurcation for the discrete-delay kaldor-kalecki model. "Chaos, Soliton and Fractal ,42,2405-2413.
[5] Gao Q. and Ma J. (2009)"Chaos and Hopf bifurcation of a finance system "Nonlinear Dynamics, 58,209-216.
[6] Kaddor A. and Alaoui H. T. (2008). “Hopf Bifurcation Analysis in a Delayed Kaldor-K alecki Model of Business Cycle”, Nonlinear Analysis, Modeling and Control,13(4),439-449.
[7] Kapitaniak T, (1998). "Chaos for Engineers Theory, Applications, and Control." Springer.
[8] Ma, J.-H. and Chen, Y.-S., (2001). “Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (I)”, Applied Mathematics and Mechanics, 22(11),1240–1251.
[9] Ranjan A. and Bhardwaj R., (2015). “Bifurcation analysis in economic system dynamics” Indian Journal of Industrial and Applied Mathematics, 6(1),49-56.
[10] Ranjan A. and Bhardwaj R, (2015). “Bifurcation Analysis of time delay in Economic System Dynamics”, Indian Journal of Industrial and Applied Mathematic 6(2) ,196-204.
[11] Wang L. and Wu. X. P (2009). “Bifurcation Analysis of Kaldor-Kalecki Model of Business Cycle with Time Delay”, Electronic Journal of Qualitative Theory of Differential Equations,27,1-20.
[12] Wang Y., Zhai, and Wang J. (2010). “Chaos and Hopf bifurcation of a finance system with Distributed Time Delay” International j. of Applied Mathematics and Mech. 6(20),1-13
[13] Wang Z., Huang X., and Shen H., (2012) “Control of an uncertain fractional order economic system via adaptive sliding mode,” Neurocomputing,83,83–88.
[14] Wolf A. ,Swift B. J., Swinney L. H. and Vastano A. J.,(1985) "Determining Lyapunov Exponents from a Time Series ,"Physica D,16,285-317.
[15] Xu C.and Zhang Q (2014). “Qualitative Analysis for A Lotka-Volterra Model with Time Delays”, Weas Transaction Mathematics,13,2224-2880.
[16] Yu H. Cai H.and Li Y. (2012)." Dynamic analysis and control of a new hyperchaotic finance system." Nonlinear Dynamics,67,2171-2182.
[17] Zen W.,Xia H.and Guodery S., (2011)."Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay " computer and mathematics with application ,62,1531-1539.
[18] Zhao X., Li Z. and Li S, (2011)"Synchronization of a chaotic finance system." Applied Mathematics and Computation.,217,6031-6039.

Citation :

Rashmi Bhardwaj, Anamika Ranjan, "Time Delay Stabilizes Chaos Dynamics in Economic System," International Journal of Mathematics Trends and Technology (IJMTT), vol. 53, no. 7, pp. 515-524, 2018. Crossref, https://doi.org/10.14445/22315373/IJMTT-V53P563