Dynamics analysis of fractional order Yu-Wang system

Sachin Bhalekar 1
  • 1 Department of Mathematics, Shivaji University, Vidyanagar, Kolhapur, 416004, India

Abstract

Fractional order version of a dynamical system introduced by Yu and Wang (Engineering, Technology & Applied Science Research, 2, (2012) 209–215) is discussed in this article. The basic dynamical properties of the system are studied. Minimum effective dimension 0.942329 for the existence of chaos in the proposed system is obtained using the analytical result. For chaos detection, we have calculated maximum Lyapunov exponents for various values of fractional order. Feedback control method is then used to control chaos in the system. Further, the system is synchronized with itself and with fractional order financial system using active control technique. Modified Adams-Bashforth-Moulton algorithm is used for numerical simulations.

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  • [1] K. T. Alligood, T. D. Sauer, J. A. Yorke, Chaos: An Introduction to Dynamical Systems, (Springer, New York, 2008)

  • [2] G. Chen, X. Dong, IEEE T. Circ. Syst. 40, 591 (1993) http://dx.doi.org/10.1109/81.244908

  • [3] Q. Zongmin, C. Jiaxing, “A Novel Linear Feedback Control Approach of Lorenz Chaotic System,” Computational Intelligence for Modelling, Control and Automation, 2006 and International Conference on Intelligent Agents, Web Technologies and Internet Commerce, International Conference on, vol., no., pp.67, Nov. 28 2006–Dec. 1 2006.

  • [4] E. Ott, C. Grebogi, J. A. Yorke, Phys. Rev. Lett. 64, 1196 (1990) http://dx.doi.org/10.1103/PhysRevLett.64.1196

  • [5] J. Lu, S. Zhang, Phys. Lett. A 256, 148 (2001) http://dx.doi.org/10.1016/S0375-9601(01)00383-8

  • [6] E. N. Sanchez, J. P. Perez, M. Martinez, G. Chen, Latin Amer. Appl. Res: Int. J. 32, 111 (2002)

  • [7] J. Lu, J. Xie, J. Lü, S. Chen, Appl. Math. Mech. 24, 1309 (2003) http://dx.doi.org/10.1007/BF02439654

  • [8] Y. J. Cao, Phys. Lett. A 270, 171 (2000) http://dx.doi.org/10.1016/S0375-9601(00)00299-1

  • [9] C. C. Fuh, P. C. Tung, Phys. Rev. Lett. 75, 2952 (1995) http://dx.doi.org/10.1103/PhysRevLett.75.2952

  • [10] R. He, P. G. Vaidya, Phys. Rev. E 57(2), 1532 (1998) http://dx.doi.org/10.1103/PhysRevE.57.1532

  • [11] I. Podlubny, Fractional Differential Equations, (Academic Press, San Diego, 1999)

  • [12] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, (Gordon and Breach, Yverdon, 1993)

  • [13] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, (Elsevier, Amsterdam, 2006)

  • [14] M. Caputo, F. Mainardi, Pure Appl. Geophys. 91, 134 (1971) http://dx.doi.org/10.1007/BF00879562

  • [15] F. Mainardi, Y. Luchko, G. Pagnini, Frac. Calc. Appl. Anal. 4(2), 153 (2001)

  • [16] O P. Agrawal, Nonlinear Dyn. 29, 145 (2002) http://dx.doi.org/10.1023/A:1016539022492

  • [17] H. G. Sun, W. Chen, Y. Q. Chen, Physica A 388, 4586 (2009) http://dx.doi.org/10.1016/j.physa.2009.07.024

  • [18] I. S. Jesus, J. A. T. Machado, Nonlinear Dyn. 54(3), 263 (2008) http://dx.doi.org/10.1007/s11071-007-9322-2

  • [19] I. S. Jesus, J. A. T. Machado„ R. S. Barbosa, Comput. Math. Appl. 59(5), 1687 (2010) http://dx.doi.org/10.1016/j.camwa.2009.08.010

  • [20] J. Sabatier, S. Poullain, P. Latteux, J. Thomas, A. Oustaloup, Nonlinear Dyn. 38, 383 (2004) http://dx.doi.org/10.1007/s11071-004-3768-2

  • [21] T. J. Anastasio, Biol. Cybernet. 72, 69 (1994) http://dx.doi.org/10.1007/BF00206239

  • [22] M. D. Ortigueira, J. A. T. Machado, Signal Processing 86, 2503 (2006) http://dx.doi.org/10.1016/j.sigpro.2006.02.001

  • [23] R. L. Magin, Fractional Calculus in Bioengineering, (Begll House Publishers, USA, 2006)

  • [24] S. Bhalekar, V. Daftardar-Gejji, Commun. Nonlinear Sci. Numer. Simulat. 15(8), 2178 (2010) http://dx.doi.org/10.1016/j.cnsns.2009.08.015

  • [25] C. Li, G. Peng, Chaos Soliton. Fract. 22, 443 (2004) http://dx.doi.org/10.1016/j.chaos.2004.02.013

  • [26] C. Li, G. Chen, Phys. A: Stat. Mech. Appl. 341, 55 (2004) http://dx.doi.org/10.1016/j.physa.2004.04.113

  • [27] V. Daftardar-Gejji, S. Bhalekar, Comput. Math. Appl. 59(3), 1117 (2010) http://dx.doi.org/10.1016/j.camwa.2009.07.003

  • [28] S. Bhalekar, Int. J. Differential Equations 2012, Article ID 623234 (2012)

  • [29] S. Bhalekar, V. Daftardar-Gejji, Commun. Nonlinear Sci. Numer. Simulat. 15, 3536 (2010) http://dx.doi.org/10.1016/j.cnsns.2009.12.016

  • [30] W. H. Deng, C. P. Li, Physica A 353, 61 (2005) http://dx.doi.org/10.1016/j.physa.2005.01.021

  • [31] T. Zhou, C. P. Li, Physica D 212, 111 (2005) http://dx.doi.org/10.1016/j.physd.2005.09.012

  • [32] J. Wang, X. Xionga, Y. Zhang, Physica A 370, 279 (2006) http://dx.doi.org/10.1016/j.physa.2006.03.021

  • [33] F. Yu, C. Wang, Engineering, Technology & Applied Science Research 2, 209 (2012)

  • [34] D. Matignon, Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems and Application multiconference, IMACS, IEEE-SMC proceedings, Lille, France, July, Vol. 2; 1996. p. 963

  • [35] M. S. Tavazoei, M. Haeri, Nonlinear Dyn. 54, 213 (2008) http://dx.doi.org/10.1007/s11071-007-9323-1

  • [36] M. S. Tavazoei, M. Haeri, Phys. Lett. A 367, 102 (2007) http://dx.doi.org/10.1016/j.physleta.2007.05.081

  • [37] M. S. Tavazoei, M. Haeri, Physica D 237, 2628 (2008) http://dx.doi.org/10.1016/j.physd.2008.03.037

  • [38] K. Diethelm, N. J. Ford, A. D. Freed, Nonlinear Dyn. 29, 3 (2002) http://dx.doi.org/10.1023/A:1016592219341

  • [39] K. Diethelm, Elec. Trans. Numer. Anal. 5, 1 (1997)

  • [40] K. Diethelm, N. J. Ford, J. Math. Anal. Appl. 265, 229 (2002) http://dx.doi.org/10.1006/jmaa.2000.7194

  • [41] S. Kodba, M. Perc, M. Marhl, Eur. J. Phys. 26, 205 (2005) http://dx.doi.org/10.1088/0143-0807/26/1/021

  • [42] L. M. Pecora, T. L. Carroll, Phys. Rev. Lett. 64(8), 821 (1990) http://dx.doi.org/10.1103/PhysRevLett.64.821

  • [43] L. M. Pecora, T. L. Carroll, Phys. Rev. Lett. 44, 2374 (1991)

  • [44] G. Chen, X. Dong, From Chaos to Order, (World Scientific, Singapore, 1998)

  • [45] B. Blasius, A. Huppert, L. Stone, Nature 399, 354 (1999) http://dx.doi.org/10.1038/20676

  • [46] R. Hilfer, Applications of Fractional Calculus in Physics, (World Scientific, USA, 2001)

  • [47] S. H. Chen, J. Lü, Chaos Soliton. Fract. 14, 643 (2002) http://dx.doi.org/10.1016/S0960-0779(02)00006-1

  • [48] E. Bai, K. Lonngen, Phys. Rev. E 8, 51 (1997)

  • [49] E. Ott, C. Grebogi, J. A. Yorke, Phys. Rev. Lett. 64, 1196 (1990) http://dx.doi.org/10.1103/PhysRevLett.64.1196

  • [50] C. K. Ahn, Prog. T. Phys. 123, 421 (2010) http://dx.doi.org/10.1143/PTP.123.421

  • [51] C. K. Ahn, S. T. Jung, S. K. Kang, S. C. Joo, Commun. Nonlinear Sci. Numer. Simulat. 15, 2168 (2010) http://dx.doi.org/10.1016/j.cnsns.2009.08.009

  • [52] J. Cao, H. X. Li, D. W.C. Ho, Chaos Soliton. Fract. 23, 1285 (2005)

  • [53] C. K. Ahn, Nonlinear Analysis: Hybrid Systems 4, 16 (2010)

  • [54] C. K. Ahn, Nonlinear Dyn. 59, 535 (2010) http://dx.doi.org/10.1007/s11071-009-9560-6

  • [55] C. K. Ahn, Nonlinear Analysis: Hybrid Systems 9, 1 (2013)

  • [56] C. K. Ahn, Nonlinear Dyn. 60, 295 (2010) http://dx.doi.org/10.1007/s11071-009-9596-7

  • [57] C. K. Ahn, Nonlinear Dyn. 59, 319 (2010) http://dx.doi.org/10.1007/s11071-009-9541-9

  • [58] W. C. Chen, Chaos Soliton. Fract. 36, 1305 (2008) http://dx.doi.org/10.1016/j.chaos.2006.07.051

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