Abstract
A novel criterion for achieving synchronisation in fractional-order chaotic and hyperchaotic systems is presented. Here, it is proved that the existence of a Lyapunov function in the integer-order differential system implies local stability of the steady state in its fractional-order counterpart. So, our criterion is based on computations of suitable linear feedback controllers of the fractional-order systems according to an appropriate choice of Lyapunov function. Furthermore, a new fractional-order hyperchaotic system is introduced here. The case of hyperchaos in the proposed system is verified by computing its greatest two Lyapunov exponents which are shown to be positive. The new synchronisation criterion is successfully applied to the fractional Liu system, the fractional Samardzija–Greller system, the fractional financial system and a novel fractional-order hyperchaotic system. Numerical results are used to verify the analytical results.
Similar content being viewed by others
References
A M A El-Sayed, H M Nour, A Elsaid, A E Matouk and A Elsonbaty, Appl. Math. Model. 40, 3516 (2016)
W S Sayed, H A H Fahmy, A A Rezk and A G Radwan, Int. J. Bifurc. Chaos 27, 22 (2017), Article ID 1730004
A D Arulsamy, Pramana – J. Phys. 85, 161 (2015)
M Mirzazadeh, Pramana – J. Phys. 86, 957 (2016)
O S Iyiola and F D Zaman, Pramana – J. Phys. 87: 51 (2016)
J Banerjee, U Ghosh, S Sarkar and S Das, Pramana – J. Phys. 88: 70 (2017)
E Ahmed, A M A El-Sayed and H A A El-Saka, J. Math. Anal. Appl. 325, 542 (2007)
A E Matouk, A A Elsadany, E Ahmed and H N Agiza, Commun. Nonlinear Sci. Numer. Simul. 27, 153 (2015)
P Song, H Zhao and X Zhang, Theory Biosci. 135, 1 (2016)
N Laskin, Physica A 287, 482 (2000)
J Zhao, S Wang, Y Chang and X Li, Nonlinear Dyn. 80, 1721 (2015)
A M A El-Sayed, A Elsonbaty, A A Elsadany and A E Matouk, Int. J. Bifurc. Chaos 26, 35 (2016), Article ID 1650222
A E M El-Misiery and E Ahmed, Appl. Math. Comput. 178, 207 (2006)
P Balasubramaniam, P Muthukumar and K Ratnavelu, Nonlinear Dyn. 80, 249 (2015)
M P Aghababa, Nonlinear Dyn. 78, 2129 (2014)
N Laskin, Chaos Solitons Fractals 102, 16 (2017)
M Caputo, J. R. Astron. Soc. 13, 529 (1967)
A E Matouk, Commun. Nonlinear Sci. Numer. Simul. 16, 975 (2011)
S P Ansari, S Agrawal and S Das, Pramana – J. Phys. 84, 23 (2015)
Y Lin, C Wang, H He and L L Zhou, Pramana – J. Phys. 86, 801 (2016)
A Deshpande and V Daftardar-Gejji, Pramana – J. Phys. 87: 49 (2016)
M Bagyalakshmi and S Gangadharan, Int. J. Bifurc. Chaos 27, 19 (2017), Article ID 1750168
Z Wei, K Rajagopal, W Zhang, S T Kingni and A Akgül, Pramana – J. Phys. 90: 50 (2018)
V K Tamba, S T Kingni, G F Kuiate, H B Fotsin and P K Talla, Pramana – J. Phys. 91: 12 (2018)
A E Matouk, Math. Probl. Eng. 2009, 11 (2009), Article ID 572724
A S Hegazi, E Ahmed and A E Matouk, J. Fract. Calc. Appl. 1, 1 (2011)
A S Hegazi and A E Matouk, Appl. Math. Lett. 24, 1938 (2011)
L Chen, Y He, X Lv and R Wu, Pramana – J. Phys. 85, 91 (2015)
K Vishal, S Agrawal and S Das, Pramana – J. Phys. 86, 59 (2016)
A Nourian and S Balochian, Pramana – J. Phys. 86, 1401 (2016)
A E Matouk, Complexity 21, 116 (2016)
L M Wang, Pramana – J. Phys. 89: 38 (2017)
K Rabah, S Ladaci and M Lashab, Pramana – J. Phys. 89: 46 (2017)
A Khan, D Khattar and N Prajapati, Pramana – J. Phys. 89: 90 (2017)
A Karthikeyan and K Rajagopal, Pramana – J. Phys. 90: 14 (2018)
N Noghredani, A Riahi, N Pariz and A Karimpour, Pramana – J. Phys. 90: 26 (2018)
M A Bhat and A Khan, Pramana – J. Phys. 90: 73 (2018)
P Muthukumar, P Balasubramaniam and K Ratnavelu, Chaos 24, 033105 (2014)
P Muthukumar, P Balasubramaniam and K Ratnavelu, Nonlinear Dyn. 80, 1883 (2015)
P Muthukumar, P Balasubramaniam and K Ratnavelu, Nonlinear Dyn. 86, 751 (2016)
P Muthukumar, P Balasubramaniam and K Ratnavelu, Multimed. Tools Appl. 76, 23517 (2017)
P Muthukumar, P Balasubramaniam and K Ratnavelu, ISA Trans. 82, 51 (2018), https://doi.org/10.1016/j.isatra.2017.07.007
A S Hegazi, E Ahmed and A E Matouk, Commun. Nonlinear Sci. Numer. Simul. 18, 1193 (2013)
A E Matouk and A A Elsadany, Nonlinear Dyn. 85, 1597 (2016)
W C Chen, Chaos Solitons Fractals 36, 1305 (2008)
K Diethelm and N J Ford, J. Math. Anal. Appl. 265, 229 (2002)
K Diethelm, N J Ford and A D Freed, Nonlinear Dyn. 29, 3 (2002)
D Matignon, Proccedings of IMACS, IEEE-SMC (Lille, 1996) Vol. 2, p. 963
E Ahmed and A S Elgazzar, Physica A 379, 607 (2007)
A E Matouk, Phys. Lett. A 373, 2166 (2009)
R Bhatia, Positive definite matrices (Princeton University Press, Princeton, NJ, 2007)
A E Matouk, Int. J. Dyn. Control (2018) (in press), https://doi.org/10.1007/s40435-018-0439-6
Acknowledgements
This work is supported by the Deanship of Scientific Research at King Saud University, Research Group No. RG-1438-046. The authors appreciate the valuable comments given by the anonymous reviewers that help to improve the presentation of our work.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Al-Khedhairi, A., Matouk, A.E. & Askar, S.S. Computations of synchronisation conditions in some fractional-order chaotic and hyperchaotic systems. Pramana - J Phys 92, 72 (2019). https://doi.org/10.1007/s12043-019-1747-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12043-019-1747-x
Keywords
- Fractional differential equations
- Lyapunov stability
- chaos
- hyperchaos
- new feedback synchronisation scheme