Abstract
In this paper, we demonstrate that the adaptive hybrid synchronization behavior can coexist in two identical and different hyperchaotic systems with terms of parametric uncertainty. By using rigorous mathematical theory, the controller is designed based on Lyapunov stability theory and an analytic expression of the controller with its adaptive laws of parameters is shown. The adaptive hybrid synchronization between two identical systems (hyperchaotic Lü system) and different systems (hyperchaotic Lorenz and hyperchaotic Chen systems) are taken as two illustrative examples to show the effectiveness of the proposed method. Theoretical analysis and numerical simulations are shown to verify the results.
Similar content being viewed by others
References
Luo ACJ (2009) A theory for synchronization of dynamical systems. Commun Nonlinear Sci Numer Simul 4:1901–1951
Dou F, Sun J, Duan W, Lu K (2009) Controlling hyperchaos in the new hyperchaotic system. Commun Nonlinear Sci Numer Simul 14:552–559
Rafikov M, Balthazar J (2008) On control and synchronization in chaotic and hyperchaotic systems via linear feedback control. Commun Nonlinear Sci Numer Simul 13:1246–1255
Ho MC, Hung YC, Chou CH (2002) Phase and anti-phase synchronization of two chaotic systems by using active control. Phys Lett A 296:43–48
Li GH, Zhou SP (2007) Anti-synchronization in different chaotic systems. Chaos, Solitons Fractals 32:516–520
Wang Z (2008) Anti-synchronization in two non-identical hyperchaotic systems with known or unknown parameters. Commun Nonlinear Sci Numer Simul 14:2366–2372
Al-Sawalha MM, Noorani MSM (2010) Adaptive anti-synchronization of chaotic systems with fully unknown parameters. Comput Math Appl 59:3234–3244
Al-Sawalha MM, Noorani MSM (2010) Adaptive anti-synchronization of two identical and different hyperchaotic systems with uncertain parameters. Commun Nonlinear Sci Numer 15:1036–1047
Al-Sawalha MM, Noorani MSM (2009) On anti-synchronization of chaotic systems via nonlinear control. Chaos, Solitons Fractals 42:170–179
Al-Sawalha MM, Noorani MSM (2009) Anti-synchronization of two hyperchaotic systems via nonlinear control. Commun Nonlinear Sci Numer Simul 14:3402–3411
Feng J, Chen S, Wang C (2005) Adaptive synchronization of uncertain hyperchaotic systems based on parameter identification. Chaos, Solitons Fractals 26:1163–1169
Chen S, Hua J, Wang C, Lu J (2004) Adaptive synchronization of uncertain Rössler hyperchaotic system based on parameter identification. Phys Lett A 321:50–55
Wua X, Guana Z, Wua Z (2008) Adaptive synchronization between two different hyperchaotic systems. Nonlinear Anal 68:1346–1351
Gao T, Chen Z, Yuan Z, Yu D (2007) Adaptive synchronization of a new hyperchaotic system with uncertain parameters. Chaos, Solitons Fractals 33:922–928
Wang B, Wen G (2008) On the synchronization of a hyperchaotic system based on adaptive method. Phys Lett A 372:3015–3020
Jia Q (2007) Adaptive control and synchronization of a new hyperchaotic system with unknown parameters. Phys Lett A 362:424–429
Li RH (2008) A special full-state hybrid-synchronization in symmetrical chaotic system. Appl Math Comput 200:319–321
Xie Q, Chen G, Bollt EM (2002) Hybrid chaos synchronization and its application in information processing. Math Comput Model 35:145–163
Zhang Q, Lu J (2008) Full state hybrid lag projective synchronization in chaotic (hyperchaotic) systems. Phys Lett A 372:1416–1421
Chen J, Chen H, Lin Y (2009) Synchronization and anti-synchronization coexist in ChenLee chaotic systems. Chaos, Solitons Fractals 39:707–716
Sun W, Chen Z, Lu Y, Chen S (2010) An intriguing hybrid synchronization phenomenon of two coupled complex networks. Appl Math Comput 216:2301–2309
Vaidyanathan S, Rasappan S (2011) Hybrid chaos synchronization of 4D hyperchaotic Qi and Jia systems by active nonlinear control. Int J Distrib Parallel Syst (IJDPS) 2:83–94
Chen A, Lu J, Lü J, Yu S (2006) Generating hyperchaotic Lü attractor via state feedback control. Physica A 364:103110
Yuxia L, Wallace K, Chen G (2005) Generating hyperchaos via state feedback control. Int J Bifurcat Chaos 15:3367–3375
Park J (2005) Adaptive synchronization of hyperchaotic Chen system with uncertain parameters. Chaos, Solitons Fractals 26:959–964
Jia Q (2007) Hyperchaos generated from the Lorenz chaotic system and its control. Phys Lett A 366:217–222
Jia Q (2007) Projective synchronization of a new hyperchaotic Lorenz system. Phys Lett A 370:40–45
Lasalle J, Lefschtg S (1961) Stability by Lyapunov’s direct method with application. Academic Press, New York
Acknowledgments
This work is financially supported by the Malaysian Ministry of Higher Education Grant: UKM–ST–06–FRGS0008–2008.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Al-sawalha, M.M., Shoaib, M. Adaptive modified synchronization of hyperchaotic systems with fully unknown parameters. Int. J. Dynam. Control 4, 23–30 (2016). https://doi.org/10.1007/s40435-014-0104-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40435-014-0104-7