Synchronization of dynamical systems of different orders and different dimensions

Ayub Khan; Mridula Budhraja; Aysha Ibraheem

doi:10.26637/MJM0602/0009

Abstract

A method of tracking control is proposed to achieve synchronization between the systems of fractional order and integer order. This article presents two cases of synchronization, in first case synchronization for three dimensional integer order Cai system and a four dimensional fractional order hyperchaotic Gao system is achieved and in second case synchronization for three dimensional fractional order Newton-Leipnik system and four dimensional hyperchaotic Pang-Liu system is achieved by tracking control method. Computational results shows that the controllers designed are useful to synchronize the considered master and slave systems in both the cases. In order to execute numerical results Matlab software is used.

Keywords:

Synchronization, fractional order system, integer order system, tracking Control

Mathematics Subject Classification:

Mathematics

Authors Imprint References Funding Related Articles Downloads

Ayub Khan Department of Mathematics, Jamia Millia Islamia, New Delhi, 110025, India.
Mridula Budhraja Department of Mathematics, Shivaji College, New Delhi, 110027, India.
Aysha Ibraheem Department of Mathematics, University of Delhi, Delhi, 110007, India.

Pages: 354-361
Date Published: 01-04-2018
Vol. 6 No. 02 (2018): Malaya Journal of Matematik (MJM)

E.N. Lorenz, Deterministic nonperiodic flow. J. Atmos. Sci. 20 (2) (1963), 130-141.

I. Grigorenko, E. Grigorenko, Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett. 91(3) (2003), 034101-4.

C. Li, G. Chen, Chaos and hyperchaos in the fractionalorder Rossler equations. Physica A, 341 (2004), 55-61.

X.Y. Wang, M.J. Wang, Dynamic analysis of the fractional-order Liu system and its synchronization. Chaos, 17(3) (2007), 033106.

C. Li, G. Peng, Chaos in Chen's system with a fractional order. Chaos Solitons Fractals 22(2) (2004), 443-450.

I. Petras, Chaos in the fractional-order Volta's system: modeling and simulation. Nonlinear Dyn. 57(1-2) (2009), $157-170$.

I. Podlubny, Factional differential equations, New York, Academic Press, (1999).

I. Petras, Fractional-order nonlinear systems, modelling, analysis and simulation, Spinger (2011).

A. Anguraja, M. Kasthurib, P. Karthikeyan, Existence results of boundary value problem for implicit impulsive fractional differential equations. Malaya J. Mat. $5(3)(2017), 550-555$

A. Benaissa Cherif, F. Z. Ladranib, Asymptotic behavior of solution for a fractional Riemann-Liouville differential equations on time scales. Malaya J. Mat. 5(3)(2017), 561568

J.A. Nanwarea, N.B. Jadhav, D.B. Dhaigude, Initial value problems for fractional differential equations involving Riemann-Liouville derivative. Malaya J. Mat. 5(2)(2017), $337-345$

I.S. Jesus, J.A. Machado, Fractional control of heat difusion systems. Nonlinear Dyn. 54(3)(2008), 263-282.

R.L. Magin, Fractional calculus in bioengineering. USA, Begll House Publishers, (2006).

W.G. Glockle, T. Mattfeld, T.F. Nonnenmacher, E.R. Weibel, Fractals in biology and medicine. Basel, Birkhauser, (1998).

L.M. Peccora, T.L. Carroll. Synchronization in chaotic systems. Phy. Rev. Lett. 64(8) (1990), 821-824.

S. Wang, Y. Yu, M. Diao, Hybrid projective synchronization of chaotic fractional order systems with different dimensions. Phvsica A. 389(21) (2010), 4981-4988.

H. Taghvafard, G.H. Erjaee, Phase and anti-phase synchronization of fractional order chaotic systems via active control. Commun. Nonlinear Sci. Numer. Simul. 16(10)(2011), 4079-4088

P. Zhou, W. Zhu, Function projective synchronization for fractional-order chaotic systems, Nonlinear Anal. Real World Appl. 12(2) (2011), 811-816.

W. Kinzel, A. Englert, I. Kanter, On chaos synchronization and secure communication. Philos Trans A Math Phys Eng Sci, 368 (2010), 379-389.

X. Wu, H. Wang, H. Lu, Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication. Nonlinear Anal.: Real World Appl. 13(3)(2012), 1441-1450.

R.Roy, K. S. Thornburg, Experimental synchronization of chaotic lasers, Phys. Rev. Lett. 72(13)(1994), 2009-2012.

S. Bhalekar, V. Daftardar-Gejji, Synchronization of different fractional order chaotic systems using active control, Commun. Nonlinear Sci. Numer. Simul. 15 (11)(2010), 3536-3546.

R. X. Zhang, S. P. Yang, Adaptive synchronization of fractional-order chaotic systems via a single driving variable. Nonlinear Dyn. 66(4) (2011), 831-837.

J. H. Park, O.M. Kwon, A novel criterion for delayed feedback control of time-delay chaotic systems. Chaos Solitons Fractals 23(2)(2005), 495-501.

L. Pan, W. Zhou, J. Fang, D. Li, A novel active pinning control for synchronization and anti-synchronization of new uncertain unified chaotic systems, Nonlinear Dyn. $62(1-2)(2010), 417-425$.

B. Wang, Y. Zhou, J. Xue, D. Zhu, Active sliding mode for synchronization of a wide class of four-dimensional fractional-order chaotic systems. ISRN Applied Mathematics, 2014 (2014), Article ID 472371.

D. Chen, W. Zhao, J.C. Sprott, X. Ma, Application of Takagi-Sugeno fuzzy model to a class of chaotic synchronization and anti-synchronization, Nonlinear Dyn. 73(3)(2013), 1495-1505.

Z. Ping , C.Y. Ming, K. Fei, Synchronization between fractional-order chaotic systems and integer order chaotic systems, Chin. Phy. B 19 (9) (2010), 090503.

D. Y. Chen, R. F. Zhang, X. Y. Ma, J. Wang, Synchronization between a novel class of fractional-order and integer-order chaotic systems via a sliding mode controller. Chin. Phy. B, 21(12)(2012), 120507.

G. Cai, Z. Tan, Chaos synchronization of a new chaotic system via nonlinear control. J. Uncertain Systems 1 (3) (2007) 235-240.

Y.Gao, C. Liang, Q. Wu, H. Yuan, A new fractionalorder hyperchaotic system and its modified projective synchronization. Chaos Solitons Fractals 76, (2015) 190204.

L.J Sheu ,L.M. Tam, S.K. Lao, Y. Kang, K.T. Lin, J.H. Chen, H.K. Chen, Parametric analysis and impulsive synchronization of fractional order Newton Leipnik systems. Int. J. Nonlinear Sci. Numer. Simul.(10) (1)(2009), 33-44.

S. Pang, Y. Liu, A new hyperchaotic system from the Lu system and its control, J. Comput. Appl. Math $235(8)(2011), 2775-2789$.