International Journal of Differential Equations
Volume 2012 (2012), Article ID 623234, 16 pages
http://dx.doi.org/10.1155/2012/623234
Research Article

Chaos Control and Synchronization in Fractional-Order Lorenz-Like System

Department of Mathematics, Shivaji University, Vidyanagar, Kolhapur 416004, India

Received 6 May 2012; Accepted 22 June 2012

Copyright © 2012 Sachin Bhalekar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 [1013]. 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 [1618] 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 [2125], 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 system and observed that below a threshold order the chaos disappears [31]. Further, many systems such as Li and Peng [32], Lu [33], Li and Chen [34], Daftardar-Gejji and Bhalekar [35], and unified system [36] were investigated in this regard. Effect of delay on chaotic solutions in fractional order dynamical system is investigated by the present author [37]. It is demonstrated that the chaotic systems can be transformed into limit cycles or stable orbits with appropriate choice of delay parameter. Synchronization of fractional order chaotic systems was also studied by many researchers [3841].

In this paper, we propose fractional version of the Lorenz-like chaotic dynamical system [42]. We investigate minimum effective dimension of the system for chaos to exist. Then we control the chaos using simple linear feedback control. Further, we synchronize the proposed fractional order system using feedback control.

2. Preliminaries

2.1. Fractional Calculus

Few definitions of fractional derivatives are known [1618]. Probably the best known is the Riemann-Liouville formulation.

The Riemann-Liouville integral of order is given by An alternative definition was introduced by Caputo. Caputo's derivative is defined as where . The main advantage of the Caputo's formulation is that the Caputo derivative of a constant is equal to zero, that is not the case for the Riemann-Liouville derivative. Note that for ,

2.2. Numerical Method for Solving Fractional Differential Equations

Numerical methods used for solving ODEs have to be modified for solving fractional differential equations (FDEs). A modification of Adams-Bashforth-Moulton algorithm is proposed by Diethelm et al. in [4345] to solve FDEs.

Consider for the initial value problem (IVP) The IVP (2.4) is equivalent to the Volterra integral equation Consider the uniform grid for some integer and . Let be approximation to . Assume that we have already calculated approximations , and we want to obtain by means of the equation where The preliminary approximation is called predictor and is given by where Error in this method is where .

2.3. Asymptotic Stability of the Fractional-Ordered System

Consider the following fractional-ordered dynamical system: Let be an equilibrium point of the system (2.11) that is, and a small disturbance from a fixed point. Then System (2.12) can be written as where and Consider the linear autonomous system where is matrix and is asymptotically stable if and only if for all eigenvalues of . In this case, each component of solution decays towards like [29, 46].

This shows that if for all eigenvalues of then the solution of the system (2.13) tends to as . Thus, the equilibrium point of the system is asymptotically stable if , for all eigenvalues of , that is, if

3. Fractional Lorenz-Like System

In [42], Chongxin et al. proposed novel Lorenz-like chaotic system where and initial conditions ((2.2),(2.3), and (28)). In this paper, we study the corresponding fractional order system where . The equilibrium points of the system (3.1) and the eigenvalues of corresponding Jacobian matrix are given in Table 1. An equilibrium point of the system (3.1) is called as saddle point if the Jacobian matrix at has at least one eigenvalue with negative real part (stable) and one eigenvalue with nonnegative real part (unstable). A saddle point is said to have index one (/two) if there is exactly one (/two) unstable eigenvalue/s. It is established in the literature [4751] that scrolls are generated only around the saddle points of index two. Saddle points of index one are responsible only for connecting scrolls.

Table 1: Equilibrium points and corresponding eigenvalues.

It is clear from Table 1 that the equilibrium points and are saddle points of index two; hence, there exists a two-scroll attractor [47] in the system (3.2).

The system (3.2) shows regular behavior if it satisfies (2.16), that is, the system is stable if Thus, the system does not show chaotic behavior for . This result is supported by numerical experiments. Figure 1 shows phase portrait in -plane for . It is observed that the system shows chaotic behavior for . For   -phase portrait is shown in Figure 2. Figures 3 and 4 show - and -phase portraits, respectively, for . The phase portraits in - and -plane are drawn for in Figures 5 and 6 respectively. Thus, the minimum effective dimension of the system is .

Figure 1: phase portrait for .
Figure 2: -phase portrait for .
Figure 3: -phase portrait for .
Figure 4: -phase portrait for .
Figure 5: -phase portrait for .
Figure 6: -phase portrait for .

4. Control of Chaos

In this section, we control the chaos in system (3.2). Consider where is the linear feedback control term. We set , where is a parameter to be determined so that the system (4.1) is stable. Equilibrium points of system (4.1) are , and . The points and decide. The stability of the system. The Jacobian matrix is given by One eigenvalue of the matrix at point is The eigenvalue is having negative real part for all and hence stable for all . Other two complex-conjugate eigenvalues are given by The stability of eigenvalues depends on . We have plotted the curves and for in Figure 7. The intersection points of the curve with the curves , , and are , , and , respectively. Following stability condition (2.16), it is clear that the chaos in the system can be controlled if we take the value of greater than the corresponding intersection point. Figure 8 shows controlled time series for and . Similarly, Figures 9, 10, and 11 show controlled time series for , and , respectively.

Figure 7: Estimate for the control .
Figure 8: Controlled signal .
Figure 9: Controlled signal .
Figure 10: Controlled signal .
Figure 11: Controlled signal .

5. Synchronization

Present section deals with synchronization of proposed fractional-order system. Consider the master system and the slave system The unknown terms in (5.2) are control functions to be determined. Define the error functions as Equation (5.3) together with (5.1) and (5.2) yields the error system The control terms are chosen so that the system (5.4) becomes stable. There is not a unique choice for such functions. We choose With the choice of given by (5.5), the error system (5.4) becomes The eigenvalues of the coefficient matrix of linear system (5.6) are , and . Hence, the stability condition (2.16) is satisfied for and the errors tend to zero as . Thus, we achieve the required synchronization. The simulation results in case are summarized in Figures 1215. Synchronization is shown in Figure 12 (signals ), Figure 13 (signals ), and Figure 14 (signals ). Note that the master systems are shown by solid lines whereas slave systems are shown by dashed lines. The errors (solid line), (dashed line) and (dot-dashed line) in the synchronization are shown in Figure 15. We have studied other cases of namely, , and but the results are omitted.

Figure 12: Synchronized signals .
Figure 13: Synchronized signals .
Figure 14: Synchronized signals .
Figure 15: Synchronization errors.

6. Conclusion

In the present work, we demonstrate the fractional order Lorenz-like system. We have observed that the system is chaotic for the fractional order , that is, the minimum effective dimension of the system is 2.91. We have used simple linear feedback controller , and given sufficient condition on to control the chaos in the proposed system. Further, we have synchronized the system using feedback control.

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.