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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Stability Analysis, Control of Simple Chaotic System and its Hybrid Projective Synchronization with Fractional Lu System

Journal of Applied Nonlinear Dynamics 9(1) (2020) 93--107 | DOI:10.5890/JAND.2020.03.008

Vijay K. Yadav, Vijay K. Shukla, Mayank Srivastava, Subir Das

Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi-221005, India

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Abstract

In this article, the stability analysis and chaos control of Simple chaotic system have been discussed. A new lemma based on Caputo derivative is used during the stability analysis of the fractional order Simple chaotic system through Lyapunov stability theory. During hybrid projective synchronization the Simple chaotic system is considered as drive system and Lu chaotic system is taken as response system. Nonlinear control method has been used to analyse the hybrid projective synchronization of fractional systems. For numerical simulations Adams-Bashforth-Moulton method has been used and results obtained are presented graphically.

Acknowledgments

The authors are extending their heartfelt thanks to the revered reviewers for their valuable comments to upgrade the present manuscript.

References

  1. [1]  Lorenz, E.N. (1963), Deterministic non-periodic flow, Journal of Atmosoheric Sciences, 20, 130-141.
  2. [2]  Jafari, H., Kadem, A., Baleanu, D., and Yilmaz, T. (2012), Solutions of the fractional davey-stewartson equations with variational iteration method, Romanian Reports in Physics, 64, 337-346.
  3. [3]  Jafari, H. and Daftardar-Gejji, V. (2006), Solving a system of nonlinear fractional differential equations using Adomian decomposition, Journal of Computational and Applied Mathematics, 196, 644-651.
  4. [4]  Das, S. and Yadav, V.K. (2016), Chaos control and function projective synchronization of fractional-order systems through the backstepping method, Theoretical and Mathematical Physics, 189, 1430-1439.
  5. [5]  Yadav, V.K., Srikanth, N., and Das, S. (2016), Dual function projective synchronization of fractional order complex chaotic systems, Optik, 127, 10527-10538.
  6. [6]  Koeller, R.C. (1984), Application of fractional calculus to the theory of viscoelasticity, Journal of Applied Mechanics, 51, 299-307.
  7. [7]  Sun, H.H., Abdelwahed, A.A., and Onaral, B. (1984), Linear approximation for transfer function with a pole of fractional order, IEEE Transactions on Automatic Control, 29, 441-444.
  8. [8]  Ichise, M., Nagayanagi, Y., and Kojima, T. (1971), An analog simulation of noninteger order transfer functions for analysis of electrode process, Journal of Electroanalytical Chemistry , 33, 253-265.
  9. [9]  Pecora, L.M. and Carroll, T.L. (1990), Synchronization in chaotic systems, Physical Review Journals, 64,821-824.
  10. [10]  Blasius, B., Huppert, A., and Stone, L. (1999), Complex dynamics and phase synchronization in Spatially extended ecological system, Nature, 39, 354-359.
  11. [11]  Lakshmanan, M. and Murali, K. (1996), Chaos in Nonlinear Oscillators: Controlling and Synchronization, World Scientific, Singapore, 340.
  12. [12]  Han, S.K., Kerrer, C., and Kuramoto, Y. (1995), Dephasing and bursting in coupled neural oscillators, Physical Review Letters, 75, 3190-3193.
  13. [13]  Murali, K. and Lakshmanan, M. (2003), Secure communication using a compound signal using sampled-data feedback, Applied Mathematics and Mechanics, 11, 1309-1315.
  14. [14]  Barajas-Ramirez, J.G., Chen, G., and Shieh, L.S. (2003), Hybrid chaos synchronization, International Journal of Bifurcation and Chaos, 13, 1197-1216.
  15. [15]  Xie, Q. and Chen, G. (2002), Hybrid chaos synchronization and its application in information processing, Mathematical and Computer Modelling, 35, 145-163.
  16. [16]  Sudheer, K.S. and Sabir, M. (2009), Hybrid synchronization of hyperchaotic Lu system, Journal of Physics, 73, 781-786.
  17. [17]  Das, S., Srivastava, M., and Leung, A.Y.T. (2013), Hybrid phase synchronization between identical and nonidentical three-dimensional chaotic systems using the active control method, Nonlinear dynamics, 73, 2261-2272.
  18. [18]  Si, G., Sun, Z., Zhang, Y., and Chen, W. (2012), Projective synchronization of different fractional-order chaotic systems with non-identical orders, Nonlinear Analysis, 13, 1761-1771.
  19. [19]  Jia, N. and Wang, T. (2011), Chaos control and hybrid projective synchronization for a class of new chaotic systems, Computers and Mathematics with Applications, 62, 4783-4795.
  20. [20]  Tripathi, P., Aneja, N., and Sharma, B.K. (2018),Stability of dynamical behavior of a new hyper chaotic system in certain range and its hybrid projective synchronization behavior, International Journal of Dynamics and Control, 1-10.
  21. [21]  Boroujeni, E.A., Yazdanpanah, M.J., and Momeni, H.R. (2012), Controller design for fractional order chaotic Lu system, American Control Conference, 997-10002.
  22. [22]  Zhou, P., Ding, R., and Cao, Y. (2012), Hybrid projective synchronization for two identical fractional-order chaotic systems, Discrete Dynamics in Nature and Society, 1-11.
  23. [23]  Zhang, L. and Liu, T. (2016), Full state hybrid projective synchronization of variable-order fractional chaotic/hyperchaotic systems with nonlinear external disturbances and unknown parameters, Journal of Nonlinear Sciences and Applications, 9, 1064-1076.
  24. [24]  Zayernouri, M. and Matzavinos, A. (2016), Fractional Adams-Bashforth-Moulton methods: An application to the fractional Keller-Segel chemotaxis system, Journal of Computational Physics, 317, 1-14.
  25. [25]  Kilbas, A., Srivastava, H.M., and Trujillo, J.(2006), Theory and applications of fractional differential equations, Elsevier, Amsterdam.
  26. [26]  Aguila-Camacho, N., Duarte-Mermoud, M.A., and Gallegos, J.A.(2014), Lyapunov functions for fractional order systems, Communications in Nonlinear Science and Numerical Simulation, 19, 2951-2957.
  27. [27]  Faieghi, M.R. and Delavari, H. (2012), Chaos in fractional-order Genesio–Tesi system and its synchronization, Commun Nonlinear Sci Numer Simulat, 17, 731-741.
  28. [28]  Matignon, D. (1996), Stability results for fractional differential equations with applications to control processing, in: Computational Engineering in Systems and Application Multi Conference, IMACS, IEEE-SMC Proceedings, 2, 963-968.
  29. [29]  Novikov, M.D. and Pavlov, B.M. (2000), On one nonlinear model with complex dynamics, Vestnik MSU, 2, 3-7.
  30. [30]  Lu, J.G. (2006), Chaotic dynamics of the fractional-order Lü system and its synchronization, Physics Letters A, 354, 305-311.
  31. [31]  Diethelm, K., Ford, J., and Freed, A. (2004), Detailed error analysis for a fractional Adams method, Numerical Algorithms, 36, 31-52.
  32. [32]  Diethelm, K. and Ford, J.(2004), Multi-order fractional differential equations and their numerical solution, Applied Mathematics and Computation, 154, 621-640.

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