A new type of hybrid synchronization between arbitrary hyperchaotic maps
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ORIGINAL ARTICLE
A new type of hybrid synchronization between arbitrary hyperchaotic maps Adel Ouannas1 • Ahmad Taher Azar2
•
Raghib Abu-Saris3
Received: 23 February 2016 / Accepted: 1 July 2016 Ó Springer-Verlag Berlin Heidelberg 2016
Abstract In this paper, a new approach is proposed to investigate new type of hybrid chaos synchronization in discrete-time hyperchaotic dynamical systems. We present, based on stability theory of linear discrete-time systems and Lyapunov stability theory, a general control scheme to study the co-existence of inverse projective synchronization, inverse generalized synchronization and Q-S synchronization between arbitrary 3D hyperchaotic maps. Numerical examples and computer simulations are used to validate the theoretical results derived in this paper. Keywords Hyperchaotic maps Inverse projective synchronization Inverse generalized synchronization QS synchronization Co-existence Lyapunov stability
& Ahmad Taher Azar [email protected]; [email protected] Adel Ouannas [email protected] Raghib Abu-Saris [email protected] 1
Laboratory of Mathematics, Informatics and Systems (LAMIS), University of Larbi Tebessi, 12002 Tebessa, Algeria
2
Faculty of Computers and Information, Benha University, Banha, Egypt
3
Department of Health Informatics, King Saud Bin Abdulaziz University for Health Science, Riyadh 11481, Saudi Arabia
1 Introduction In practice, discrete-time chaotic and hyperchaotic systems play a more important role than their continuous-parts. In fact, many mathematical models of physical processes [1], biological phenomena [2], chemical systems [3] and economic systems [4] were defined using discrete-time chaotic (hyperchaotic) systems. Some interesting 3D chaotic and hyperchaotic maps were presented in the past two decades such as Baier–Klain map [5], Hitzl–Zele map [6], Stefanski map [7], Wang system [8], discrete-time Ro¨ssler system [9] and Grassi–Miller map [10], etc. Historically, hyperchaos in discrete-time systems was firstly reported by Ro¨ssler [11]. A hyperchaotic system is usually defined as a chaotic system with more than one positive Lyapunov exponent. The occurrence of hyperchaotic behavior has been found in electronic circuits [12], semi-conductor Laze systems [13, 14] and in chemical reaction systems [15]. Hyperchaotic maps have been investigated extensively owing to their useful potential applications in encryption [16–20]. Thus it is a more important subject to study the synchronization of hyperchaotic maps. The idea of synchronization can be applied in the vast areas of intelligent systems [21, 22], information sciences [23, 24], and many other fields. Recently, more and more attention has been paid to the synchronization of hyperchaos in discrete-time dynamical systems, due to its potential applications in secure communication [25–27]. Different synchronization types have been proposed for chaotic (hyperchaotic) maps such as complete synchronization [28], anti-synchronization [29], projective synchronization [30], functio
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