×

zbMATH — the first resource for mathematics

Sustaining stable dynamics of a fractional-order chaotic financial system by parameter switching. (English) Zbl 1345.37095
Summary: In this paper, a simple parameter switching (PS) methodology is proposed for sustaining the stable dynamics of a fractional-order chaotic financial system. This is achieved by switching a controllable parameter of the system, within a chosen set of values and for relatively short periods of time. The effectiveness of the method is confirmed from a computer-aided approach, and its applications to chaos control and anti-control are demonstrated. In order to obtain a numerical solution of the fractional-order financial system, a variant of the Grünwald-Letnikov scheme is used. Extensive simulation results show that the resulting chaotic attractor well represents a numerical approximation of the underlying chaotic attractor, which is obtained by applying the average of the switched values. Moreover, it is illustrated that this approach is also applicable to the integer-order financial system.

MSC:
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34A08 Fractional ordinary differential equations and fractional differential inclusions
37N40 Dynamical systems in optimization and economics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Smale, S., The mathematics of time: essays on dynamical systems, economic processes, and related topics, (1980), Springer-Verlag Berlin · Zbl 0451.58001
[2] Chian, A. C., Nonlinear dynamics and chaos in macroeconomics, International Journal of Theoretical and Applied Finance, 3, 3, 601, (2000)
[3] Feichtinger, G., Economic evolution and demographic change, (1992), Springer Berlin
[4] Puu, T., Chaos in duopoly pricing, Chaos, Solitons & Fractals, 1, 6, 573-581, (1991) · Zbl 0754.90015
[5] Fanti, L.; Manfredi, P., Chaotic business cycles and fiscal policy: an IS-LM model with distributed tax collection lags, Chaos, Solitons & Fractals, 32, 2, 736-744, (2007) · Zbl 1133.91482
[6] Serletis, A., Is there chaos in economic time series ?, Canadian Journal of Economics/Revue Canadienne d’Economique, 29, 210-212, (1996)
[7] Lipton-Lifschitz, A., Predictability and unpredictability in financial markets, Physica D: Nonlinear Phenomena, 133, 1-4, 321-347, (1999) · Zbl 1194.91210
[8] Goodwin, R. M., Chaotic economic dynamics, (1990), Oxford University Press Oxford
[9] LeBaron, B., Chaos and nonlinear forecastability in economics and finance, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences, 348, 1688, 397-404, (1994) · Zbl 0863.90032
[10] Kopel, M., Improving the performance of an economic system: controlling chaos, Journal of Evolutionary Economics, 7, 3, 269-289, (1997)
[11] Hołyst, J. A.; Hagel, T.; Haag, G.; Weidlich, W., How to control a chaotic economy ?, Journal of Evolutionary Economics, 6, 1, 31-42, (1996)
[12] Hołyst, J. A.; Urbanowicz, K., Chaos control in economical model by time-delayed feedback method, Physica A: Statistical Mechanics and its Applications, 287, 3, 587-598, (2000)
[13] Chen, L.; Chen, G., Controlling chaos in an economic model, Physica A: Statistical Mechanics and its Applications, 374, 1, 349-358, (2007)
[14] Chen, L.; Chai, Y.; Wu, R., Control and synchronization of fractional-order financial system based on linear control, Discrete Dynamics in Nature and Society, 2011, (2011), 958393
[15] Ma, J.; Chen, Y., Study for the bifurcation topological structure and the global complicated character of a kind of nonlinear finance system (II), Applied Mathematics and Mechanics, 22, 12, 1375-1382, (2001) · Zbl 1143.91341
[16] Wang, Z.; Huang, X.; Shi, G., Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay, Computers & Mathematics with Applications, 62, 3, 1531-1539, (2011) · Zbl 1228.35253
[17] Ma, J.; Chen, Y., 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, (2001) · Zbl 1001.91501
[18] Chen, W. C., Nonlinear dynamics and chaos in a fractional-order financial system, Chaos, Solitons & Fractals, 36, 5, 1305-1314, (2008)
[19] Abd-Elouahab, M. S.; Hamri, N.-E.; Wang, J., Chaos control of a fractional-order financial system, Mathematical Problems in Engineering, 2010, 270646, (2010) · Zbl 1195.91185
[20] Diethelm, K., The analysis of fractional differential equations, (2004), Springer-Verlag Berlin
[21] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego, CA · Zbl 0918.34010
[22] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and applications of fractional differential equations, (2006), North-Holland Mathematics Studies Amsterdam · Zbl 1092.45003
[23] Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J., (Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, (2012), World Scientific) · Zbl 1248.26011
[24] Bhalekar, S.; Daftardar-Gejji, V.; Baleanu, D.; Magin, R., Transient chaos in fractional Bloch equations, Computers and Mathematics with Applications, 64, 10, 3367-3376, (2012) · Zbl 1268.34009
[25] Momani, S.; Rqayiq, A. A.; Baleanu, D., A nonstandard finite difference scheme for two-sided space-fractional partial differential equations, International Journal of Bifurcation and Chaos, 22, 4, 1-5, (2012) · Zbl 1258.35201
[26] Li, C.; Zeng, F., Finite difference methods for fractional differential equations, International Journal of Bifurcation and Chaos, 22, 4, (2012), 1230014, 28 · Zbl 1258.34018
[27] Scherer, R.; Kalla, S. L.; Tang, Y.; Huang, J., The Grünwald-Letnikov method for fractional differential equations, Computers & Mathematics with Applications, 62, 3, 902-917, (2011) · Zbl 1228.65121
[28] Mao, Y.; Tang, W. K.S.; Danca, M. F., An averaging model for chaotic system with periodic time-varying parameter, Applied Mathematics and Computation, 217, 1, 355-362, (2010) · Zbl 1206.65255
[29] Danca, M. F.; Romera, M.; Pastor, G.; Montoya, F., Finding attractors of continuous-time systems by parameter switching, Nonlinear Dynamics, 67, 4, 2317-2342, (2012) · Zbl 1242.37056
[30] Sanders, J. A.; Verhulst, F., Averaging methods in nonlinear dynamical systems, (1985), Springer Berlin · Zbl 0586.34040
[31] Petráš, I., Fractional-order nonlinear systems, (2011), Springer Berlin
[32] R. Garrappa, Predictor-corrector PECE method for fractional differential equations, MATLAB Central File Exchange, File ID: 32918, 2011.
[33] Garrappa, R., On linear stability of predictor-corrector algorithms for fractional differential equations, International Journal of Computer Mathematics, 87, 10, 2281-2290, (2010) · Zbl 1206.65197
[34] Falconer, K. J., Fractal geometry: mathematical foundations and applications, (2003), Wiley New York · Zbl 1060.28005
[35] Z. Danziger, Hausdorff distance, MATLAB Central File Exchange, File ID: 26738, 2009-2012, 2009.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.