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Chaos control of a fractional-order financial system. (English) Zbl 1195.91185
Summary: The fractional-order financial system introduced by {\it W.-C. Chen} [“Nonlinear dynamics and chaos in a fractional-order financial system”, Chaos Solitons Fractals 36, No. 5, 1305--1314 (2008)] displays chaotic motions at order less than 3. In this paper we extend the nonlinear feedback control in ODE systems to fractional-order systems, in order to eliminate the chaotic behavior. The results are proved analytically by applying the Lyapunov linearization method and stability condition for fractional system. Moreover, numerical simulations are shown to verify the effectiveness of the proposed control scheme.

MSC:
91G80Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
34H10Chaos control (ODE)
37N40Dynamical systems in optimization and economics
37D45Strange attractors, chaotic dynamics
WorldCat.org
Full Text: DOI EuDML
References:
[1] 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. I,” Applied Mathematics and Mechanics, vol. 22, no. 11, pp. 1240-1251, 2001. · Zbl 1001.91501 · doi:10.1023/A:1016313804297
[2] 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, no. 12, pp. 1375-1382, 2001. · Zbl 1143.91341 · doi:10.1023/A:1022806003937
[3] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their solution and Some of Their Application, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[4] R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000. · Zbl 0998.26002
[5] A. Oustaloup, J. Sabatier, and P. Lanusse, “From fractal robustness to the CRONE control,” Fractional Calculus & Applied Analysis, vol. 2, no. 1, pp. 1-30, 1999. · Zbl 1111.93310
[6] A. Oustaloup, F. Levron, B. Mathieu, and F. M. Nanot, “Frequency-band complex noninteger differentiator: characterization and synthesis,” IEEE Transactions on Circuits and Systems I, vol. 47, no. 1, pp. 25-39, 2000. · doi:10.1109/81.817385
[7] T. T. Hartley and C. F. Lorenzo, “Dynamics and control of initialized fractional-order systems,” Nonlinear Dynamics, vol. 29, no. 1-4, pp. 201-233, 2002. · Zbl 1021.93019 · doi:10.1023/A:1016534921583
[8] J. Wang, X. Xiong, and Y. Zhang, “Extending synchronization scheme to chaotic fractional-order Chen systems,” Physica A, vol. 370, no. 2, pp. 279-285, 2006. · doi:10.1016/j.physa.2006.03.021
[9] Z. M. Odibat, N. Corson, M. A. Aziz-Alaoui, and C. Bertelle, “Synchronization of chaotic fractional-order systems via linear control,” International Journal of Bifurcation and Chaos, vol. 20, no. 1, pp. 81-97, 2010. · Zbl 1183.34095 · doi:10.1142/S0218127410025429
[10] X.-y. Wang, Y.-j. He, and M.-j. Wang, “Chaos control of a fractional order modified coupled dynamos system,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. 6126-6134, 2009. · Zbl 1187.34080 · doi:10.1016/j.na.2009.06.065
[11] W.-C. Chen, “Nonlinear dynamics and chaos in a fractional-order financial system,” Chaos, Solitons and Fractals, vol. 36, no. 5, pp. 1305-1314, 2008. · doi:10.1016/j.chaos.2006.07.051
[12] A. M. Lyapunov, The General Problem of the Stability of Motion, Taylor & Francis, London, UK, 1992. · Zbl 0786.70001
[13] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Translation edited and with a preface by A. Jeffrey and D. Zwillinger, Academic Press, San Diego, Calif, USA, 6th edition, 2000. · Zbl 0981.65001
[14] A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D, vol. 16, no. 3, pp. 285-317, 1985. · Zbl 0585.58037 · doi:10.1016/0167-2789(85)90011-9
[15] M. Caputo, “Linear models of dissipation whose Q is almost frequency independent-II,” Geophysical Journal of the Royal Astronomical Society, vol. 13, no. 5, pp. 529-539, 1967.
[16] K. Diethelm and A. D. Freed, “The FracPECE subroutine for the numerical solution of differential equations of fractional order,” in Forschung und wissenschaftliches Rechnen, S. Heinzel and T. Plesser, Eds., pp. 57-71, Gesellschaft für Wisseschaftliche Datenverarbeitung, Gottingen, Germany, 1998.
[17] D. Matignon, “Stability properties for generalized fractional differential systems,” in Systèmes différentiels fractionnaires, vol. 5 of ESAIM Proc., pp. 145-158, Soc. Math. Appl. Indust., Paris, France, 1998. · Zbl 0920.34010 · doi:10.1051/proc:1998004 · http://www.edpsciences.org/articles/proc/Vol.5/contents.htm
[18] M. Moze and J. Sabatier, “LMI tools for stability analysis of fractional systems,” in Proceedings of ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, pp. 1-9, Long Beach, Calif, USA, September 2005.
[19] D. Auerbach, P. Cvitanović, J.-P. Eckmann, G. Gunaratne, and I. Procaccia, “Exploring chaotic motion through periodic orbits,” Physical Review Letters, vol. 58, no. 23, pp. 2387-2389, 1987. · doi:10.1103/PhysRevLett.58.2387