On Inverse Full State Hybrid Function Projective Synchronization For Continuous-time Chaotic Dynamical Systems with Arbi
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On Inverse Full State Hybrid Function Projective Synchronization For Continuous-time Chaotic Dynamical Systems with Arbitrary Dimensions Adel Ouannas1 · Ahmad Taher Azar2,3 · Toufik Ziar4
© Foundation for Scientific Research and Technological Innovation 2017
Abstract Referring to continuous-time chaotic dynamical systems, this paper investigates the inverse full state hybrid function projective synchronization (IFSHFPS) of non-identical systems characterized by different dimensions. By taking a master system of dimension n and a slave system of dimension m, the method enables each master system state to be synchronized with a linear combination of slave system states, where the scaling factor of the linear combination can be any arbitrary differentiable function. The approach, based on the Lyapunov stability theory and stability of linear continuous-time systems, presents some useful features: (i) it enables non-identical chaotic systems with different dimension n < m or n > m to be synchronized; (ii) it can be applied to a wide class of chaotic (hyperchaotic) systems for any differentiable scaling function; (iii) it is rigorous, being based on two theorems, one for the case n < m and the other for the case n > m. Two different numerical examples are reported. The examples clearly highlight the capability of the conceived approach in effectively achieving synchronized dynamics for any differentiable scaling function. Keywords Chaos · Continuous-time systems · Full state hybrid projective synchronization · Inverse problem · Different dimensions
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Ahmad Taher Azar [email protected]; [email protected] Adel Ouannas [email protected] Toufik Ziar [email protected]
1
Laboratory of Mathematics, Informatics and Systems (LAMIS), University of Larbi Tebessi, Tebessa 12002, Algeria
2
Faculty of computers and information, Benha University, Banha, Egypt
3
Nile University, Giza, Egypt
4
Department of Material Sciences, University of Tebessa, 12002 Tebessa, Algeria
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Differ Equ Dyn Syst
Introduction The topic of synchronization in chaotic dynamical systems has attracted great interest in nonlinear science [4,6–9,22,24,29,33–37,39,41]. The objective in chaos synchronization is to make the slave system variables synchronized in time with the corresponding chaotic master system variables [1,1,23,25,27,30–32,38,40,41]. Most of the methods that have been inspired by the Carroll and Pecora work [11] focuses on complete (identical) synchronization, where two identical chaotic systems asymptotically approach the same trajectory. Subsequently, different types of synchronization have been proposed in the literature [2,3,5,20,26,42,45]. Among these, full state hybrid projective synchronization (FSHPS) is one of the most noticeable types. It has been widly used in the synchronization of chaotic and hyperchaotic systems [14–19,44] . In this type of synchronization each slave system state achieves synchronization with linear combination of master system states. In [10], Cai et al. generalize the concept of (FS
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