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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


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:

Synchronization Methods for Chaotic Systems Involving Fractional Derivative with a Non-Singular Kernel

Journal of Applied Nonlinear Dynamics 11(2) (2022) 375--386 | DOI:10.5890/JAND.2022.06.008

Fatiha Mesdoui$^1$, Nabil Shawagfeh$^2$, Adel Ouannas$^3$

$^1$ University of Mohamed Seddik Benyahia, Jijel, Algeria

$^2$ Department of Mathematics, University of Jordan, Amman, Jordan

$^3$ Department of Mathematics and Computer Science, University of Larbi Ben M'hidi, Oum El Bouaghi, Algeria

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This study considers the problem of control-synchronization for chaotic systems involving fractional derivatives with a non-singular kernel. Using an extension of the Lyapunov Theorem for systems with Atangana-Baleanu-Caputo (ABC) derivative, a suitable control scheme is designed to achieve matrix projective synchronization (MP) between nonidentical ABC systems with different dimensions. The results are exemplified by the ABC version of the Lorenz system, Bloch system, and Liu system. To show the effectiveness of the proposed results, numerical simulations are performed based on the Adams-Bashforth-Mounlton numerical algorithm.


  1. [1]  Magin, R.L., (2006), Fractional calculus in bioengineering, Begell House Publishers.
  2. [2]  Carlson, G.E. and Halijak, C.A. (1964), Approximation of fractional capacitors (1/s)\^{ }(1/n) by a regular Newton process, IEEE Transactions on Circuit Theory, 11, pp 210-213.
  3. [3]  Pires, E.J.S., Machado, J.A.T., and de Moura, P.B. (2003), Fractional order dynamics in a GA planner, Signal Process, 83, 2377-2386.
  4. [4]  Kusnezov, D., Bulgac, A., and Dang, G.D. (1999), Quantum L{e}vy processes and fractional kinetics, Physical Review Letters, 82, 1136-1139.
  5. [5]  Kulich, V.V. and Lage, J.L. (2002), Application of fractional calculus to fluid mechanics, Journal of Fluids Engineering, 124, pp 803-806.
  6. [6]  Podunk, I. (1999), Fractional Differential Equations. Academic Press, New York.
  7. [7]  Caputo, M. and Fabrizio, M. (2015), A new definition of fractional derivative without singular kernel, Progr Fract Different Appl., 1, 73-85.
  8. [8]  Atangana, A. and Baleanu, D. (2016), New fractional derivatives with the nonlocal and non-singular kernel, Theory Appl. Heat Trans. Model. Therm. Sci., 20, 63-769.
  9. [9]  U\c{c}ar, S. (2020), Analysis of a basic SEIRA model with Atangana-Baleanu derivative, AIMS Mathematics, 5(2), pp. 1411.
  10. [10]  Saqib, M., Khan, I., and Shafie, S. (2018), Application of Atangana-Baleanu fractional derivative to MHD channel flow of CMC-based-CNT's nanofluid through a porous medium, Chaos, Solitons $&$ Fractals, 116, 79-85.
  11. [11]  Ghanbari, B., G\"{u}nerhan, H., and Srivastava, H.M. (2020), An application of the Atangana-Baleanu fractional derivative in mathematical biology: A three-species predator-prey model, Chaos, Solitons {$\&$ Fractals}, 138, 109910.
  12. [12]  Khan, M.A., Atangana, A., and Alzahrani, E. (2020), The dynamics of COVID-19 with quarantined and isolation, Advances in Difference Equations, 2020(1), pp 425.
  13. [13]  Koca, I. (2017), Analysis of rubella disease model with non-local and non-singular fractional derivatives, An International Journal of Optimization and Control. Theories {$\&$ Applications} (IJOCTA), 8(1), 17-25.
  14. [14]  Sweilam, N.H., Al-Mekhlafi, S.M., Assiri, T., and Atangana, A. (2020), Optimal control for cancer treatment mathematical model using Atangana-Baleanu-Caputo fractional derivative, Advances in Difference Equations, 2020(1), pp 334.
  15. [15]  U\c{c}ar, S., U\c{c}ar, E., \"{O}zdemir, N., and Hammouch, Z. (2019), Mathematical analysis and numerical simulation for a smoking model with A0tangana-Baleanu derivative, Chaos, Solitons {$\&$ Fractals}, 118(C), pp 300-306.
  16. [16]  Pecora, L.M. and Carrol, T.L. (1990), Synchronization in chaotic systems, Phys. Rev A., 64, pp. 8.
  17. [17]  Luo, A. (2009), A theory for synchronization of dynamical systems, Communication in Nonlinear Sciences and Numerical Simulation, 14(10), 1901-1951.
  18. [18]  Odibat, Z., Corson, N., Alaoui, M.A.A., and Bertelle, C. (2010), Synchronization of chaotic fractional-order systems via linear control, International Journal of Bifurcation and Chaos, 20, 81-97.
  19. [19]  Chen, X.R. and Liu, C.X. (2012), Chaos Synchronization of fractional-order unified chaotic system via nonlinear control, International Journal of Modern Physics B, 25, 407-415.
  20. [20]  Srivastava, M., Ansari, S.P., Agrawal, S.K., Das, S., Leung, A.Y.T. (2014), Antisynchronization between identical and non-identical fractional-order chaotic systems using active control method, Nonlinear Dynamics, 76, pp 905-914.
  21. [21]  Agrawal, S.K., Srivastava, M., and Das, S. (2012), Synchronization of fractional-order chaotic systems using active control method, Chaos, Solitons {$\&$ Fractals}, 45(6), 737-752.
  22. [22]  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(4), 2261-2272.
  23. [23]  Odibat, Z. (2010), Adaptive feedback control and synchronization of non-identical chaotic fractional-order systems, Nonlinear Dynamics, 60, 479-487.
  24. [24]  Agrawal, S.K. and Das, S. (2014), Function projective synchronization between four-dimensional chaotic systems with uncertain parameters using a modified adaptive control method, Journal of Process Control, 24(5), 517-530.
  25. [25]  Agrawal, S.K. and Das, S. (2013), Modified adaptive control method for synchronization of some fractional chaotic systems with unknown parameters, Nonlinear Dynamics, 73, 907-919.
  26. [26]  Razminia, A. and Baleanu, D. (2013), Complete synchronization of commensurate fractional-order chaotic systems using sliding mode control, Mechatronics, 23, 873-879.
  27. [27]  Al-Sawalha, M.M., Alomari, A.K., Goh, S.M., and Nooran, M.S.M. (2011), Active antisynchronization of two identical and different fractional-order chaotic systems, International Journal of Nonlinear Science, 11, 267-274.
  28. [28]  Si, G., Sun, Z., Zhang, Y., and Chen, W. (2012), Projective synchronization of different fractional-order chaotic systems with non-identical orders, Nonlinear Analytics: Real World Applications, 13, 1761-1771.
  29. [29]  Chai, Y., Chen, L., Wu, R., and Dai, J. (2013), Q-S synchronization of the fractional-order unified system, Pramana, 80, 449-461.
  30. [30]  Feng, H., Yang, Y., and Yang, S.P. (2013), A new method for full state hybrid projective synchronization of different fractional-order chaotic systems, Applied Mechanics and Materials, 385-38, 919-922.
  31. [31]  Zhang, X.D., Zhao, P.D., and Li, A.H. (2010), Construction of a new fractional chaotic system and generalized synchronization, Communications in Theoretical Physics, 53, 1105-1110.
  32. [32]  Ouannas, A. and Al-Sawalha, M.M. (2016), On $\Lambda -\phi $ generalized synchronization of chaotic dynamical systems in continuous-time, European Physical Journal Special Topics, 225, 187-196.
  33. [33]  Ouannas, A., Al-Sawalha, M.M., and Ziar, T. (2016), Fractional chaos synchronization schemes for different dimensional systems with non-identical fractional-orders via two scaling matrices, Optik, 127, 8410-8418.
  34. [34]  Ouannas, A. and Abu-Saris, R. (2016), On Matrix Projective Synchronization and Inverse Matrix Projective Synchronization for Different and Identical Dimensional Discrete-Time Chaotic Systems, Journal of Chaos, 2016, pp. 7.
  35. [35]  Yan, W. and Ding, Q. (2019), A New Matrix Projective Synchronization and Its Application in Secure Communication, IEEE Access, 7, 112977-112984.
  36. [36]  Taneco-Hern{a}ndez, M. and Vargas-De-Le{o}n, C. (2020), Stability and Lyapunov functions for systems with Atangana-Baleanu Caputo derivative: An HIV/AIDS epidemic model, Chaos, Solitons {$\&$ Fractals}, 132, 109586.
  37. [37]  Diethelm, K., Ford, N.J., and Freed, A.D. (2002), A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29(1-4). pp. 3-22.
  38. [38]  Atangana, A. and Koca, I. (2016), Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons & Fractals, 89, 447-454.
  39. [39]  G{o}mez-Aguilar, J.F. (2018), Chaos in a nonlinear Bloch system with Atangana-Baleanu fractional derivatives, Numer Methods Partial Differ Equations, 34, 1716-1738
  40. [40]  Han, Q., Liu, C.X., Sun, L. and Zhu, D.R. (2013), A fractional-order hyperchaotic system derived from a Liu system and its circuit realization, Chinese Physics B, 22, 6-020502.
  41. [41]  Ouannas, A., Odibat, Z., and Shawagfeh, N. (2016), A new Q-S synchronization results for discrete chaotic systems, Diff. Eq. Dyn. Syst., 27(4), 413-422.
  42. [42]  Ouannas, A., Odibat, Z., Shawagfeh, N., Alsaedi, A., and Ahmad, B. (2017), Universal chaos synchronization control laws for general quadratic discrete systems, Appl. Math. Model, 45, 636-641.
  43. [43]  Mesdoui, F., Ouannas, A., Shawagfeh, N., Grassi, G., and Pham, V.T. (2020), Synchronization Methods for the Degn-Harrison Reaction-Diffusion Systems, IEEE Access, 8, 91829-91836.
  44. [44]  Mesdoui, F., Shawagfeh, N., and Ouannas, A. (2020), Global synchronization of fractional-order and integer-order N component reaction-diffusion systems: Application to biochemical models, Math Meth Appl Sci, 1-10.