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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Chaos Control, Quad-Compound Anti-Synchronization, Analysis and Application on Novel Fractional Chaotic System

Discontinuity, Nonlinearity, and Complexity 11(3) (2022) 435--457 | DOI:10.5890/DNC.2022.09.007

Ayub Khan, Lone Seth Jahanzaib, Pushali Trikha

Department Of Mathematics, Jamia Millia Islamia, New Delhi, India Department of Applied Mathematics, School of Vocational Studies and Applied Sciences, Gautam Buddha University, Greater Noida-201308

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Abstract

In this paper chaos control and synchronization techniques are applied on the introduced novel fractional chaotic systems. The system is extensively studied for its dynamical properties using various tools such as Lyapunov spectrum, bifurcation diagrams, phase portraits, equilibrium points, dissipative character, uniqueness of solution and so on. Besides the effect of changing fractional order on the dynamics are also studied in detail. The chaotic behavior of the novel system is controlled about any randomly chosen point. The systems are then synchronized in quad compound combination anti-synchronization with eight chaotic systems in presence of disturbances and uncertainties. The achieved synchronization is illustrated in secure communication with help of an example.

Acknowledgments

L.S.Jahanzaib (MANF-2018-19-JAM-98362, U.G.C., India) and P. Trikha (09/466(0189)/2017-EMR-I, CSIR, India) thank the agencies for providing financial support as J. R. F. and S.R.F. respectively.

References

  1. [1]  Pecora, L.M. and Carroll, T.L. (1990), Synchronization in chaotic systems, Physical review letters, 64(8), 821.
  2. [2]  Zhang, B. and Deng, F. (2014), Double-compound synchronization of six memristor-based Lorenz systems, Nonlinear Dynamics, 77(4), 1519-1530.
  3. [3]  Dongmo, E.D., Ojo, K.S., Woafo, P., and Njah, A.N. (2018), Difference synchronization of identical \& non-identical chaotic \& hyper chaotic systems of different orders using active backstepping design, Journal of Computational and Nonlinear Dynamics, 13(5).
  4. [4]  Khan, A., Jahanzaib, L.S., Trikha, P., Khan, T., and others (2020), Compound difference anti-synchronization between hyper-chaotic systems of fractional order, Journal of Scientific Research, 12(2), 175-181.
  5. [5]  Trikha, P. and Jahanzaib, L.S. (2020), Combination difference synchronization between identical generalised Lotka-Volterra chaotic systems, Journal of Scientific Research, 12(2), 183-188.
  6. [6]  Mahmoud, E.E., Jahanzaib, L.S., Trikha, P., and Alkinani, M.H. (2020), Anti-synchronized quad-compound combination among parallel systems of fractional chaotic system with application, Alexandria Engineering Journal, Elsevier.
  7. [7]  Yadav, V.K., Prasad, G., Srivastava, M., and Das, S. (2019), Triple compound synchronization among eight chaotic systems with external disturbances via nonlinear approach, Differential Equations and Dynamical Systems, 1-24.
  8. [8]  Sun, J., Li, N., Wang, Y., and Wang, W. (2019), A novel chaotic system and its modified compound synchronization, Fundamenta Informaticae, Elsevier, 164(2-3), 259-275.
  9. [9]  Li, B., Zhou, X., and Wang, Y. (2019), Combination synchronization of three different fractional-order delayed chaotic systems, Complexity, 2019.
  10. [10]  Khan, A. and Trikha, P. (2019), Compound difference anti-synchronization between chaotic systems of integer and fractional order, SN Appl. Sci., 7(1), 757.
  11. [11]  Khan, A., Trikha, P., and Jahanzaib, L.S. (2020), Dislocated hybrid synchronization via. tracking control \& parameter estimation methods with application, International Journal of Modelling and Simulation, Taylor \& Francis, 1-11.
  12. [12]  Inan, B., Osman, M.S., Ak, T., and Baleanu, D. (2019), Analytical and numerical solutions of mathematical biology models: The Newell-Whitehead-Segel and Allen-Cahn equations, Mathematical Methods in the Applied Sciences, Wiley Online Library.
  13. [13]  Baleanu, D., Jajarmi, A., Mohammadi, H., and Rezapour, S. (2020), A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative, Chaos, Solitons $\&$ Fractals, 134, 109705.
  14. [14]  Wu, Y., Noonan, J.P., Yang, G., and Jin, H. (2012), Image encryption using the two-dimensional logistic chaotic map, Journal of Electronic Imaging, 21(1), 013, 014.
  15. [15]  Khan, A., Jahanzaib, L.S., Khan, T., and Trikha, P. (2020), Secure communication: Using fractional matrix projective combination synchronization, AIP Conference Proceedings, 2253(1), 020009.
  16. [16]  Trikha, P. and Jahanzaib, L.S. (2020), Dynamical analysis of a novel 5-d hyper-chaotic system with no equilibrium point and its application in secure communication, Differential Geometry-Dynamical Systems, 22.
  17. [17]  Trikha, P. and Jahanzaib, L.S. (2020), Secure communication: using double compound-combination hybrid synchronization, Proceedings of International Conference on Artificial Intelligence and Applications, Springer, 81-91.
  18. [18]  Khan, A., Jahanzaib, L.S., and Trikha, P.M. (2020), Fractional inverse matrix projective combination synchronization with application in secure communication, Proceedings of International Conference on Artificial Intelligence and Applications, Springer, 93-101.
  19. [19]  Khan, A., Jahanzaib, L.S., and Trikha, P. (2020), Secure communication: using parallel synchronization technique on novel fractional order chaotic system, IFAC-PapersOnLine, 53(1), 307-312.
  20. [20]  Khan, A., Trikha, P., and Lone, S.J. (2019), Secure communication: Using synchronization on a novel fractional order chaotic system, ICPECA-IEEE, 1-5.
  21. [21]  Wong, K., Man, K.P., Li, S., and Liao, X. (2005), A more secure chaotic cryptographic scheme based on the dynamic look-up table, Circuits, Systems and Signal Processing, 24(5), 571-584.
  22. [22]  Khan, A. and Trikha, P. (2014), Study of earth's changing polarity using compound difference synchronization, GEM-International Journal on Geomathematics, 11(1), 7.
  23. [23]  Vaidyanathan, S., Rajagopal, K., Volos, Ch.K., Kyprianidis, I.M., and Stouboulos, I.N. (2015), Analysis, adaptive control and synchronization of a seven-term novel 3-D chaotic system with three quadratic nonlinearities and its digital implementation in LabVIEW, J Eng Sci Technol Rev, 8(2), 130-141.
  24. [24]  Vaidyanathan, S., Volos, Ch.K., and Pham, V.T.(2015), Analysis, adaptive control and adaptive synchronization of a nine-term novel 3-D chaotic system with four quadratic nonlinearities and its circuit simulation, J Eng Sci Technol Rev A, 8(2), 181-191.
  25. [25]  Vaidyanathan, S. (2014), Analysis, control and synchronisation of a six-term novel chaotic system with three quadratic nonlinearities, International Journal of Modelling, Identification and Control, 22(1), 41-53.
  26. [26]  Vaidyanathan, S., Rajagopal, K., Volos, Ch.K., Kyprianidis, I.M., and Stouboulos, IN., Analysis, adaptive control and synchronization of a seven-term novel 3-D chaotic system with three quadratic nonlinearities and its digital implementation in LabVIEW, J Eng Sci Technol Rev., 8(2), 130-141.
  27. [27]  Sundarapandian, V. and Pehlivan, I. (2019), Analysis, control, synchronization, and circuit design of a novel chaotic system, Mathematical and Computer Modelling, 55(7-8), 1904-1915.
  28. [28]  Khan, A., Lone, S.J., and Trikha P., Analysis of a novel 3-D fractional order chaotic system, ICPECA, IEEE, 1-6.
  29. [29]  Broucke, M. (1987), One parameter bifurcation diagram for chua's circuit, IEEE transactions on circuits and systems, 34(2), 208-209.
  30. [30]  Zhang, S. and Zeng, Y. (2019), A simple Jerk-like system without equilibrium: Asymmetric coexisting hidden attractors, bursting oscillation and double full Feigenbaum remerging trees, Chaos, Solitons $\&$ Fractals, 120, 25-40.
  31. [31]  Wolf, A., Swift, J.B., Swinney, H.L., and Vastano, J.A. (1985), Determining lyapunov exponents from a time series, Physica D: Nonlinear Phenomena, 16(3), 285-317.
  32. [32]  Khan, A., Jahanzaib, L.S., Trikha, P., and Khan, T. (2020), Changing dynamics of the first, second \& third approximates of the exponential chaotic system \& their synchronization, Journal of Vibration Testing and System Dynamics, 4(4), 337-361.
  33. [33]  Geist, K., Parlitz, U., and Lauterborn, W. (1990), Comparison of different methods for computing lyapunov exponents, Progress of theoretical physics, 83(5), 875-893.
  34. [34]  Diethelm, K. and Ford, N.J. (2002), Journal of Mathematical Analysis and Applications, Elsevier, 265, 229-248.
  35. [35]  Tavazoei, M.S. and Haeri, M. (2007), A necessary condition for double scroll attractor existence in fractional-order systems, Physics Letters A, 367, 102-113.
  36. [36]  Matignon, D. (1996), Stability results for fractional differential equations with applications to control processing, Computational engineering in systems applications, IMACS, IEEE-SMC Lille, France, 2, 963-968.
  37. [37]  Vidyasagar, M. (2002), Nonlinear systems analysis, 42, Siam.
  38. [38]  Ahmad, I. and Shafiq, M. (2020), Robust adaptive anti-synchronization control of multiple uncertain chaotic systems of different orders, Automatika, 61(3), 396-414.
  39. [39]  Ahmad, I. and Shafiq, M. (2020), Oscillation free robust adaptive synchronization of chaotic systems with parametric uncertainties, Transactions of the Institute of Measurement and Control, SAGE.
  40. [40]  Ahmad, I. and Shafiq, M. (2020), A generalized analytical approach for the synchronization of multiple chaotic systems in the finite time, Arabian Journal for Science and Engineering, 45(3), 2297-2315.
  41. [41]  Shafiq, M. and Ahmad, I. (2019), Multi-switching combination anti-synchronization of unknown hyperchaotic systems, Arabian Journal for Science and Engineering, 44(8), 7335-7350.