Control of a Fractional-Order Arneodo System

Abstract:

In this work, stability analysis of the Fractional-Order Arneodo system is studied by using the fractional Routh-Hurwitz criteria. Furthermore, the fractional Routh-Hurwitz conditions are used to control chaos in the proposed fractional-order system to its equilibria. Based on the fractional Routh-Hurwitz conditions and using specific choice of linear feedback controllers, it is shown that the Arneodo system is controlled to its equilibrium points. Numerical results show the effectiveness of the theoretical analysis.

• Periodical: Advanced Materials Research (Volumes 383-390)
• Pages: 4405-4412
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.383-390.4405
• Online since: November 2011
• Authors: Kun Zhang, Hua Wang, Hui Tao Wang
• Keywords: Chaos, Feedback Control, Fractional Routh-Hurwitz, Fractional-Order Arneodo System

References

[1] T.T. Hartley, C.F. Lorenzo and H.K. Qammer, Chaos in a fractional order Chua's system, IEEE Trans Circ Syst–I, vol. 42, p.485–490, August (1995).

DOI: 10.1109/81.404062

[2] I. Grigorenko and E. Grigorenko, Chaotic dynamics of the fractional Lorenz system, Phys Rev Lett, vol. 91, p.034101. 1–. 4, July (2003).

DOI: 10.1103/physrevlett.96.199902

[3] C.P. Li and G.J. Peng, Chaos in Chen's system with a fractional order, Chaos, Solitons & Fractals, vol. 22, p.443–450, Oct. (2004).

DOI: 10.1016/j.chaos.2004.02.013

[4] C.G. Li and G.R. Chen, Chaos and hyperchaos in the fractional-order Rössler equations, Physica A, vol. 341, p.55–61, Oct. (2004).

DOI: 10.1016/j.physa.2004.04.113

[5] X.Y. Wang and M.J. Wang, Dynamic analysis of the fractional-order Liu system and its synchronization, Chaos, vol. 17, 033106, (2007).

DOI: 10.1063/1.2755420

[6] LM. Pecora and TL. Carrol, Synchronization in chaotic systems, Phys Rev Lett, vol. 64, p.821–824, Feb. (1990).

[7] A.E. Matouk, Dynamic analysis feedback control and synchronization of Liu dynamical system, Nonlinear Anal Theor Meth Appl, vol. 69, p.3213–3224, (2008).

DOI: 10.1016/j.na.2007.09.029

[8] Arman Kiani-B, Kia Fallahi, Naser Pariz, and Henry Leung, A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter, Communications in Nonlinear Science and Numerical Simulation, vol. 14, p.863–879, March (2009).

DOI: 10.1016/j.cnsns.2007.11.011

[9] D. Matignon, Stability results for fractional differential equations with applications to control processing, Computational Engineering in Systems and Application. Multiconference, IMACS, IEEE-SMC, Lille, France, vol. 2, p.963–968, (1996).

[10] C.P. Li and J. P Yan, The synchronization of three fractional differential systems, Chaos, Solitons & Fractals, vol. 32, p.751–757, April (2007).

DOI: 10.1016/j.chaos.2005.11.020

[11] H. Zhu, S.B. Zhou and Z.S. He, Chaos synchronization of the fractional-order Chen's system, Chaos, Solitons & Fractals, vol. 41, p.2733–2740, Sept. (2009).

DOI: 10.1016/j.chaos.2008.10.005

[12] X.Y. Wang and J.M. Song, Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control, Commun Nonlinear Sci Numer Simulat, vol. 14, p.3351–3357, August (2009).

DOI: 10.1016/j.cnsns.2009.01.010

[13] X.Y. Wang, Y.J. He and M. J, Wang. Chaos control of a fractional order modified coupled dynamos system, Nonlinear Analysis, vol. 71, p.6126–6134, Dec. (2009).

DOI: 10.1016/j.na.2009.06.065

[14] Sara Dadras and Hamid Reza Moneni, Control of a fractional-order economical system via sliding mode, Physica A, vol. 389, p.24342442, June (2010).

[15] A.E. Matouk, Chaos, feedback control and synchronization of a fractional-order modified Autonomous Van der Pol-Duffing circuit, Commun Nonlinear Sci Numer Simulat, vol. 16, p.975–986, Feb. 2011, doi: 10. 1016/j. cnsns. 2010. 04. 027.

DOI: 10.1016/j.cnsns.2010.04.027

[16] A. Arneodo, P. Coullet, E. Spiegel and C. Tresser, Asymptotic chaos, Phys D, vol. 14, p.327–347, (1985).

DOI: 10.1016/0167-2789(85)90093-4

[17] S.M. Jiang, L.X. Tian and X.D. Wang, Control of Arneodo chaotic system, Journal of Jiangsu University, vol. 26, p.492–495, (2005).

[18] J.G. Lu, Chaotic dynamics and synchronization of fractional-order Arneodo's systems, Chaos, Solitons & Fractals, vol. 26, p.1125–1133, Nov. (2005).

DOI: 10.1016/j.chaos.2005.02.023

[19] M. Caputo, Linear models of dissipation whose Q is almost frequency independent, The Geophysical Journal of the Royal Astronomical Society, vol. 13, p.529–539, April (1967).

DOI: 10.1111/j.1365-246x.1967.tb02303.x

[20] E. Ahmed, A.M.A. El-Sayed and Hala A.A. El-Saka, On some Routh- Hurwitz conditions for fractional order differential equations and their applications in Lorenz, , Chua and Chen systems, Phys Let A, vol. 358, p.1–4, Oct. (2006).

DOI: 10.1016/j.physleta.2006.04.087

[21] E. Ahmed, A.M.A. El-Sayed and Hala A.A. El-Saka, Equilibrium points, stability and numerical solutions of fractional-order predator- prey and rabies models, Journal of Mathematical Analysis and Applications, vol. 325, p.542–553, Jan. (2007).

DOI: 10.1016/j.jmaa.2006.01.087

[22] K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Electron Trans Numer Anal, vol. 5, p.1–6, March (1997).

[23] K. Diethelm and Neville J. Ford, Analysis of fraction differential equations, J Math Anal Appl, vol. 265, p.229–248, Jan. (2002).

[24] K. Diethelm, Neville J. Ford and Alan D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn, vol. 29, p.3–22, (2002).

[25] K. Diethelm and Alan D. Freed, The FracPECE subroutine for the numerical solution of differential equations of fractional order, In: Heinzel S, Plesser T, editors. Forschung und wissenschaftliches Rechnen. Göttingen: Gesellschaft für wissenschaftliches Datenverarbeitung; p.57.