The present paper deals with fractional-order version of a dynamical system introduced by Chongxin et al. (2006). The chaotic behavior of the system is studied using analytic and numerical methods. The minimum effective dimension is identified for chaos to exist. The chaos in the proposed system is controlled using simple linear feedback controller. We design a controller to place the eigenvalues of the system Jacobian in a stable region. The effectiveness of the controller in eliminating the chaotic behavior from the state trajectories is also demonstrated using numerical simulations. Furthermore, we synchronize the system using nonlinear feedback. 1. Introduction A variety of problems in engineering and natural sciences are modeled using chaotic dynamical systems.A chaotic system is a nonlinear deterministic system possessing complex dynamical behaviors such as being extremely sensitive to tiny variations of initial conditions, unpredictability, and having bounded trajectories in the phase space [1]. Controlling the chaotic behavior in the dynamical systems using some form of control mechanism has recently been the focus of much attention. So many approaches are proposed for chaos control namely, OGY method [2], backstepping design method [3], differential geometric method [4], inverse optimal control [5], sampled-data feedback control [6], adaptive control [7], and so on. One simple approach is the linear feedback control [8]. Linear feedback controllers are easy to implement, they can perform the job automatically, and stabilize the overall control system efficiently [9]. The controllers can also be used to synchronize two identical or distinct chaotic systems [10–13]. Synchronization of chaos refers to a process wherein two chaotic systems adjust a given property of their motion to a common behavior due to a coupling. Synchronization has many applications in secure communications of analog and digital signals [14] and for developing safe and reliable cryptographic systems [15]. Fractional calculus deals with derivatives and integration of arbitrary order [16–18] and has deep and natural connections with many fields of applied mathematics, engineering, and physics. Fractional calculus has wide range of applications in control theory [19], viscoelasticity [20], diffusion [21–25], turbulence, electromagnetism, signal processing [26, 27], and bioengineering [28]. Study of chaos in fractional order dynamical systems and related phenomena is receiving growing attention [29, 30]. I. Grigorenko and E. Grigorenko investigated fractional ordered Lorenz
References
[1]
K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer, New York, NY, USA, 2008.
[2]
E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Physical Review Letters, vol. 64, no. 11, pp. 1196–1199, 1990.
[3]
J. Lü and S. Zhang, “Controlling Chen's chaotic attractor using backstepping design based on parameters identification,” Physics Letters, Section A, vol. 286, no. 2-3, pp. 148–152, 2001.
[4]
C. C. Fuh and P. C. Tung, “Controlling chaos using differential geometric method,” Physical Review Letters, vol. 75, no. 16, pp. 2952–2955, 1995.
[5]
E. N. Sanchez, J. P. Perez, M. Martinez, and G. Chen, “Chaos stabilization: an inverse optimal control approach,” Latin American Applied Research, vol. 32, no. 1, pp. 111–114, 2002.
[6]
J. A. Lu, J. Xie, J. H. Lu, and S. H. Chen, “Control chaos in transition system using sampled-data feedback,” Applied Mathematics and Mechanics, vol. 24, no. 11, pp. 1309–1315, 2003.
[7]
Y. J. Cao, “A nonlinear adaptive approach to controlling chaotic oscillators,” Physics Letters, Section A, vol. 270, no. 3-4, pp. 171–176, 2000.
[8]
G. Chen and X. Dong, “On feedback control of chaotic continuous-time systems,” IEEE Transactions on Circuits and Systems I, vol. 40, no. 9, pp. 591–601, 1993.
[9]
Q. Zongmin and C. Jiaxing, “A novel linear feedback control approach of Lorenz chaotic system,” in Proceedings of the International Conference on Computational Intelligence for Modelling, Control and Automation (CIMCA '06), Jointly with International Conference on Intelligent Agents Web Technologies and International Commerce (IAWTIC '06), p. 67, December 2006.
[10]
L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, 1990.
[11]
L. M. Pecora and T. L. Carroll, “Driving systems with chaotic signals,” Physical Review A, vol. 44, no. 4, pp. 2374–2383, 1991.
[12]
L. Kocarev and U. Parlitz, “General approach for chaotic synchronization with applications to communication,” Physical Review Letters, vol. 74, no. 25, pp. 5028–5031, 1995.
[13]
T. L. Carroll, J. F. Heagy, and L. M. Pecora, “Transforming signals with chaotic synchronization,” Physical Review E, vol. 54, no. 5, pp. 4676–4680, 1996.
[14]
R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2001.
[15]
R. He and P. G. Vaidya, “Implementation of chaotic cryptography with chaotic synchronization,” Physical Review E, vol. 57, no. 2, pp. 1532–1535, 1998.
[16]
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
[17]
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Detivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
[18]
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
[19]
J. Sabatier, S. Poullain, P. Latteux, J. L. Thomas, and A. Oustaloup, “Robust speed control of a low damped electromechanical system based on CRONE control: application to a four mass experimental test bench,” Nonlinear Dynamics, vol. 38, no. 1–4, pp. 383–400, 2004.
[20]
M. Caputo and F. Mainardi, “A new dissipation model based on memory mechanism,” Pure and Applied Geophysics, vol. 91, no. 1, pp. 134–147, 1971.
[21]
F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the spacetime fractional diffusion equation,” Fractional Calculus and Applied Analysis, vol. 4, no. 2, pp. 153–192, 2001.
[22]
O. P. Agrawal, “Solution for a fractional diffusion-wave equation defined in a bounded domain,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 145–155, 2002.
[23]
H. Sun, W. Chen, and Y. Chen, “Variable-order fractional differential operators in anomalous diffusion modeling,” Physica A, vol. 388, no. 21, pp. 4586–4592, 2009.
[24]
I. S. Jesus and J. A. Tenreiro MacHado, “Fractional control of heat diffusion systems,” Nonlinear Dynamics, vol. 54, no. 3, pp. 263–282, 2008.
[25]
I. S. Jesus, J. A. T. Machado, and R. S. Barbosa, “Control of a heat diffusion system through a fractional order nonlinear algorithm,” Computers and Mathematics with Applications, vol. 59, no. 5, pp. 1687–1694, 2010.
[26]
T. J. Anastasio, “The fractional-order dynamics of brainstem vestibulo-oculomotor neurons,” Biological Cybernetics, vol. 72, no. 1, pp. 69–79, 1994.
[27]
M. D. Ortigueira and J. A. T. Machado, “Fractional calculus applications in signals and systems,” Signal Processing, vol. 86, no. 10, pp. 2503–2504, 2006.
[28]
R. L. Magin, Fractional Calculus in Bioengineering, Begll House Publishers, Redding, Conn, USA, 2006.
[29]
D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Proceedings of the Computational Engineering in Systems and Application Multiconference, vol. 2, pp. 963–968, Lille, France, July 1996.
[30]
A. Kiani-B, K. Fallahi, N. Pariz, and H. 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, no. 3, pp. 863–879, 2009.
[31]
I. Grigorenko and E. Grigorenko, “Chaotic dynamics of the fractional Lorenz system,” Physical Review Letters, vol. 91, no. 3, Article ID 034101, 4 pages, 2003.
[32]
C. Li and G. Peng, “Chaos in Chen's system with a fractional order,” Chaos, Solitons and Fractals, vol. 22, no. 2, pp. 443–450, 2004.
[33]
J. G. Lu, “Chaotic dynamics of the fractional-order Lü system and its synchronization,” Physics Letters, Section A, vol. 354, no. 4, pp. 305–311, 2006.
[34]
C. G. Li and G. Chen, “Chaos and hyperchaos in the fractional order Rossler equations,” Physica A, vol. 341, pp. 55–61, 2004.
[35]
V. Daftardar-Gejji and S. Bhalekar, “Chaos in fractional ordered Liu system,” Computers and Mathematics with Applications, vol. 59, no. 3, pp. 1117–1127, 2010.
[36]
X. Wu, J. Li, and G. Chen, “Chaos in the fractional order unified system and its synchronization,” Journal of the Franklin Institute, vol. 345, no. 4, pp. 392–401, 2008.
[37]
S. Bhalekar and V. Daftardar-Gejji, “Fractional ordered Liu system with time-delay,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 8, pp. 2178–2191, 2010.
[38]
W. H. Deng and C. P. Li, “Chaos synchronization of the fractional Lü system,” Physica A, vol. 353, no. 1–4, pp. 61–72, 2005.
[39]
T. Zhou and C. Li, “Synchronization in fractional-order differential systems,” Physica D, vol. 212, no. 1-2, pp. 111–125, 2005.
[40]
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.
[41]
S. Bhalekar and V. Daftardar-Gejji, “Synchronization of different fractional order chaotic systems using active control,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 11, pp. 3536–3546, 2010.
[42]
L. Chongxin, L. Ling, L. Tao, and L. Peng, “A new butterfly-shaped attractor of Lorenz-like system,” Chaos, Solitons and Fractals, vol. 28, no. 5, pp. 1196–1203, 2006.
[43]
K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 3–22, 2002.
[44]
K. Diethelm, “An algorithm for the numerical solution of differential equations of fractional order,” Electronic Transactions on Numerical Analysis, vol. 5, pp. 1–6, 1997.
[45]
K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, pp. 229–248, 2002.
[46]
M. S. Tavazoei and M. Haeri, “Regular oscillations or chaos in a fractional order system with any effective dimension,” Nonlinear Dynamics, vol. 54, no. 3, pp. 213–222, 2008.
[47]
M. S. Tavazoei and M. Haeri, “A necessary condition for double scroll attractor existence in fractional-order systems,” Physics Letters, Section A, vol. 367, no. 1-2, pp. 102–113, 2007.
[48]
L. O. Chua, M. Komuro, and T. Matsumoto, “The double-scroll family,” IEEE Transactions on Circuits and Systems, vol. 33, no. 11, pp. 1072–1118, 1986.
[49]
C. P. Silva, “Shil'nikov's theorem-a tutorial,” IEEE Transactions on Circuits and Systems I, vol. 40, pp. 675–682, 1993.
[50]
D. Cafagna and G. Grassi, “New 3D-scroll attractors in hyperchaotic Chua's circuits forming a ring,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 13, no. 10, pp. 2889–2903, 2003.
[51]
J. Lü, G. Chen, X. Yu, and H. Leung, “Design and analysis of multiscroll chaotic attractors from saturated function series,” IEEE Transactions on Circuits and Systems I, vol. 51, no. 12, pp. 2476–2490, 2004.
