References
- Kocarev L. Chaos-based cryptography: a brief overview. IEEE Circuits Syst Mag. 2001;1(3):6–21. doi: 10.1109/7384.963463
- Dachselt F, Schwarz W. Chaos and cryptography. IEEE Trans Circuits Syst I: Fund Theory Appl. 2001;48(12):1498–1509. doi: 10.1109/TCSI.2001.972857
- Masuda N, Aihara K. Cryptosystems with discretized chaotic maps. IEEE Trans Circuits Syst I: Fund Theory Appl. 2002;49(1):28–40. doi: 10.1109/81.974872
- Masuda N, Jakimoski G, Aihara K, et al. Chaotic block ciphers: from theory to practical algorithms. IEEE Trans Circuits Syst I: Reg Pap. 2006;53:1341–1352. doi: 10.1109/TCSI.2006.874182
- Lawande QV, Ivan BR, Dhodapkar SD. Chaos based cryptography: a new approach to secure communications. Bombay: BARC Newsletter; 2005.
- Yamada T, Fujisaca H. Stability theory of synchronized motion in coupled-oscillator. Syst II Prog Theor Phys. 1983;70:1240–1248. doi: 10.1143/PTP.70.1240
- Yamada T, Fujisaca H. Stability theory of synchronized motion in coupled-oscillator. Syst III Prog Theor Phys. 1984;72:885–894. doi: 10.1143/PTP.72.885
- Afraimovich VS, Verochev NN, Robinovich MI. Stochastic synchronization of oscillations in dissipative systems, Radio. Phys Quant Electron. 1983;29:795–803. doi: 10.1007/BF01034476
- Pecora LM, Carrol TL. Synchronization in chaotic systems. Phys Rev A. 1990;64:821–824.
- Martinez-Guerra R, Mata-Machuca JL. Fractional generalized synchronization in a class of nonlinear fractional order systems. Nonlinear Dyn. 2014;77:1237–1244. doi: 10.1007/s11071-014-1373-6
- Mahmoud GM, Abed-Elhameed TM, Ahmed ME. Generalization of combination–combination synchronization of chaotic n-dimensional fractional-order dynamical systems. Nonlinear Dyn. 2016;83(4):1885–1893. doi: 10.1007/s11071-015-2453-y
- Maheri M, Arifin N. Synchronization of two different fractional-order chaotic systems with unknown parameters using a robust adaptive nonlinear controller. Nonlinear Dyn. 2016;85(2):825–838. doi: 10.1007/s11071-016-2726-0
- Mathiyalagan K, Park JH, Sakthivel R. Exponential synchronization for fractional-order chaotic systems with mixed uncertainties. Complexity. 2015;21(1):114–125. doi: 10.1002/cplx.21547
- Goodrich C, Peterson AC. Discrete fractional calculus. Berlin: Springer; 2015.
- Wu GC, Baleanu D, Luo WH. Lyapunov functions for Riemann–Liouville-like fractional difference equations. Appl Math Comput. 2017;314:228–236.
- Wu GC, Baleanu D, Xie HP, et al. Chaos synchronization of fractional chaotic maps based on stability results. Phys A. 2016;460:374–383. doi: 10.1016/j.physa.2016.05.045
- Cross MC, Hohenberg PC. Pattern formation outside of equilibrium. Rev Modern Phys. 1993;65(3):851–1112. doi: 10.1103/RevModPhys.65.851
- Lai YC, Winslow RL. Extreme sensitive dependence on parameters and initial conditions in spatio-temporal chaotic dynamical systems. Phys D: Nonlinear Phenomena. 1994;74(3–4):353–371. doi: 10.1016/0167-2789(94)90200-3
- Parekh N, Kumar VR, Kulkarni BD. Control of spatiotemporal chaos: a study with an autocatalytic reaction-diffusion system. Pramana J Phys. 1997;48(1):303–323. doi: 10.1007/BF02845637
- Zelik SV. Spatial and dynamical chaos generated by reaction–diffusion systems in unbounded domains. J Dyn Differ Equ. 2007;19(1):1–74. doi: 10.1007/s10884-006-9007-4
- Wang Y, Cao J. Synchronization of a class of delayed neural networks with reaction–diffusion terms. Phys Lett A. 2007;369:201–211. doi: 10.1016/j.physleta.2007.04.079
- Yu F, Jiang H. Global exponential synchronization of fuzzy cellular neural networks with delays and reaction–diffusion terms. Neurocomputing. 2011;74:509–515. doi: 10.1016/j.neucom.2010.08.017
- Yang X, Cao J, Yang Z. Synchronization of coupled reaction–diffusion neural networks with time-varying delays via pinning impulsive control. SIAM J Cont Optim. 2013;51(5):3486–3510. doi: 10.1137/120897341
- Bao H, Park JH, Cao J. Synchronization of fractional-order complex-valued neural networks with time delay. Neural Networks. 2016;81:16–28. doi: 10.1016/j.neunet.2016.05.003
- Bao H, Park JH, Cao J. Adaptive synchronization of fractional-order memristor-based neural networks with time delay. Nonlinear Dyn. 2015;82(3):1343–1354. doi: 10.1007/s11071-015-2242-7
- Hu G, Li X, Wang Y. Pattern formation and spatiotemporal chaos in a reaction–diffusion predator–prey system. Nonlinear Dyn. 2015;81(1):265–275. doi: 10.1007/s11071-015-1988-2
- Zaitseva MF, Magnitskii NA, Poburinnaya NB. Control of space-time Chaos in a system of equations of the FitzHugh–Nagumo type. Differ Equ. 2016;52(12):1585–1593. doi: 10.1134/S0012266116120065
- Zaitseva MF, Magnitskii NA. Space–time Chaos in a system of reaction–diffusion equations. Differ Equ. 2017;53(11):1519–1523. doi: 10.1134/S0012266117110155
- Leipnik RB, Newton TA. Double strange attractors in rigid body motion with linear feedback control. Phys Lett A. 1981;86:63–67. doi: 10.1016/0375-9601(81)90165-1
- Wolf A, Swift J, Swinney H, et al. Determining Lyapunov exponents from a time series. Phys D. 1985;16:285–317. doi: 10.1016/0167-2789(85)90011-9
- Qiang J. Chaos control and synchronization of the Newton–Leipnik chaotic system. Chaos Solitons Fractals. 2008;35:814–824. doi: 10.1016/j.chaos.2006.05.069
- Jovic B. Synchronization techniques for Chaotic communication systems. Berlin: Springer-Verlag; 2011.
- Bendoukha S. On the existence of chaos and complete synchronization of the fractional-order Newton–Leipnik chaotic system. to appear.
- Bendoukha S, Abdelmalek S. Complete synchronization of the Newton–Leipnik reaction diffusion chaotic system. to appear.
- Kanga Y, Lina KT, Chenb JH, et al. Parametric analysis of a fractional-order Newton-Leipnik system. J Phys: Conf Ser. 2008;96:012140.
- Sheu LJ, Chen HK, Chen JH, et al. Chaos in the Newton–Leipnik system with fractional order. Chaos Solitons Fract. 2008;36:98–103. doi: 10.1016/j.chaos.2006.06.013
- Khan A, Tyagi A. Fractional order disturbance observer based adaptive sliding mode hybrid projective synchronization of fractional order Newton–Leipnik chaotic system. Int J Dyn Control. 2017;6:1–14.
- Kilbas A, Srivastava H, Trujillo J. Theory and applications of fractional differential equations. Elsevier; 2006.
- Li Y, Chen YQ, Podlubny I. Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag–Leffler stability. Comput Math Appl. 2010;59(5):1810–1821. doi: 10.1016/j.camwa.2009.08.019
- Aguila–Camacho N, Duarte–Mermoud MA, Gallegos JA. Lyapunov functions for fractional order systems. Commun Nonlinear Sci Numer Simul. 2014;19:2951–2957. doi: 10.1016/j.cnsns.2014.01.022
- Douaifia R, Abdelmalek S, Bendoukha S. Asymptotic stability conditions for autonomous time-fractional reaction–diffusion systems, to appear.
- Mansouri D, Abdelmalek S, Bendoukha S. On the asymptotic stability of the time-fractional Lengyel–Epstein system. Comput Math Appl. 2019. DOI:10.1016/j.camwa.2019.04.015