Abstract
The article aims to study the reduced-order anti-synchronization between projections of fractional order hyperchaotic and chaotic systems using active control method. The technique is successfully applied for the pair of systems viz., fractional order hyperchaotic Lorenz system and fractional order chaotic Genesio-Tesi system. The sufficient conditions for achieving anti-synchronization between these two systems are derived via the Laplace transformation theory. The fractional derivative is described in Caputo sense. Applying the fractional calculus theory and computer simulation technique, it is found that hyperchaos and chaos exists in the fractional order Lorenz system and fractional order Genesio-Tesi system with order less than 4 and 3 respectively. The lowest fractional orders of hyperchaotic Lorenz system and chaotic Genesio-Tesi system are 3.92 and 2.79 respectively. Numerical simulation results which are carried out using Adams-Bashforth-Moulton method, shows that the method is reliable and effective for reduced order anti-synchronization.
Similar content being viewed by others
References
R. Hifer, Applications of Fractional Calculus in Physics (World Scientific, New Jersey, 2001)
I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999)
R.C. Koeller, J. Appl. Mech. 51, 299 (1984)
H. H. Sun, A. A. Abdelwahed, B. Onaral, IEEE T. Autom. Control 29, 441 (1984)
M. Ichise, Y. Nagayanagi, T. Kojima, J. Electroanal. Chem. 33, 253 (1971)
O. Heaviside, Electromagnetic theory (Chelsea, New York, 1971)
N. Laskin, Physica A 287, 482 (2000)
D. Kunsezov, A. Bulagc, G. D. Dang, Phys. Rev. Lett. 82, 1136 (1999)
T.T. Hartley, C.F. Lorenzo, Nonlinear Dyn. 29, 201 (2002)
S.G. Samko, A.A. Kilbas, O.I. Maricev, Fractional Integrals and Derivatives, Theory and Applications (Gordon and Breach, 1993)
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Amsterdam, Elsevier Science, 2006)
D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos (World Scientific 2012)
L. M. Pecora, T. L. Carroll, Phys. Rev. Lett. 64, 821 (1990)
B. Blasius, A. Huppert, L. Stone, Nature 399, 354 (1999)
M. Lakshmanan, K. Murali, Chaos in Nonlinear Oscillators: Controlling and Synchronization (World Scientific, Singapore, 1996)
S. K. Han, C. Kerrer, Y. Kuramoto, Phys. Rev. Lett. 75, 3190 (1995)
K. Murali, M. Lakshmanan, Appld. Math. Mech. 11, 1309 (2003)
A. Razminia, D. Baleanu, J. Comput. Nonlin. Dyn. 8, 31012, (2013)
J. W. Shuai, K. W. Wong, Phys. Rev. E 57, 7002 (1998)
R. Roy, K. S. Thornburg, Phys. Rev. Lett. 72, 2009 (1994)
M. Srivastava, S. K. Agrawal, S. Das, Int. J. Nonlinear Sci. 13, 482 (2012)
M. T. Yassen, Chaos Soliton. Fract. 23, 1527 (2005)
X. Wu, J. Lü, Chaos Soliton. Fract. 18, 721 (2003)
H. Delavari, D.M. Senejohnny, D. Baleanu, Cent. Eur. J. Phys. 10, 1095 (2012)
S. K. Agrawal, M. Srivastava, S. Das, Chaos Soliton. Fract. 45, 737 (2012)
Y. Zhang, J. Sun, Phys. Lett. A 330, 442 (2004)
M. G. Rosenblum, A. S. Pikovsky, J. Kurths, Phys. Rev. Lett. 78, 4193 (1997)
G. Si, Z. Sun, Y. Zhang, W. Chen, Nonlinear Anal. Real World Appl. 13, 1761 (2012)
P. Zhoua, W. Zhu, Nonlinear Anal. Real World Appl. 12, 811 (2011)
J. P. Yan, C. P. Li, Chaos Soliton. Fract. 32, 725 (2007)
G. H. Erjaee, H. Taghvafard, Commun. Nonlinear Sci. Numer. Simulat. 16, 4079 (2011)
X. Y. Wang, J. M. Song, Commun Nonlinear Sci. Numer. Simulat. 14, 3351 (2009)
S. K. Agrawal, M. Srivastava, S. Das, Nonlinear Dyn. 69, 2277 (2012)
R. Femat, G. Perales, Phys. Rev. E 65, 036226 (2002)
M. Ho, Y. Hung, Z. Liua, I. Jiang, Phys. Lett. A 348, 251 (2006)
S. Bowong, Phys. Lett. A 326, 102 (2004)
M. M. Al-sawalha, M. S. M. Noorani, Commun. Nonlinear Sci. Numer. Simulat. 15, 3022 (2010)
M. M. Al-sawalha, M. S. M. Noorani, Commun. Nonlinear Sci. Numer. Simulat. 17, 1908 (2012)
S. A. Lazzouni, S. Bowong, F. M. M. Kakmeni, B. Cherki, Commun. Nonlinear Sci. Numer. Simulat. 12, 568 (2007)
X. Wang, M. Wang, Physica A 387, 3751 (2008)
R. Genesio, A. Tesi, Automatica 28, 531 (1992)
M. R. Faieghia, H. Delavari, Commun. Nonlinear Sci. Numer. Simulat. 17, 731 (2012)
H.J. Haubold, A.M. Mathai, R.K. Saxena, Journal of Applied Mathematics 2011, 1, doi:10.1155/2011/298628
K. Diethelm, J. Ford, A. Freed, Numer. Algorithms 36, 31 (2004)
K. Diethelm, J. Ford, Appl. Math. Comput. 154, 621 (2004)
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Srivastava, M., Agrawal, S.K. & Das, S. Reduced-order anti-synchronization of the projections of the fractional order hyperchaotic and chaotic systems. centr.eur.j.phys. 11, 1504–1513 (2013). https://doi.org/10.2478/s11534-013-0310-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.2478/s11534-013-0310-5