Combination-Combination Synchronization of Four Nonlinear Complex Chaotic Systems

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 953265, 14 pages http://dx.doi.org/10.1155/2014/953265 Research Article Combination-Combination Synchronization of Four Nonlinear Complex Chaotic Systems Xiaobing Zhou,1 Lianglin Xiong,2 and Xiaomei Cai3 1 School of Information Science and Engineering, Yunnan University, Kunming 650091, China School of Mathematics and Computer Science, Yunnan University of Nationalities, Kunming 650031, China 3 Bureau of Asset Management, Yunnan University, Kunming 650091, China 2 Correspondence should be addressed to Xiaobing Zhou; Received 13 August 2013; Revised 25 October 2013; Accepted 30 October 2013; Published 3 February 2014 Academic Editor: Narcisa C. Apreutesei Copyright © 2014 Xiaobing Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper investigates the combination-combination synchronization of four nonlinear complex chaotic systems. Based on the Lyapunov stability theory, corresponding controllers to achieve combination-combination synchronization among four different nonlinear complex chaotic systems are derived. The special cases, such as combination synchronization and projective synchronization, are studied as well. Numerical simulations are given to illustrate the theoretical analysis. 1. Introduction In 1982, Fowler et al. [1] generalized the real Lorenz model to a complex Lorenz model, which can be used to describe and simulate the physics of a detuned laser and the thermal convection of liquid flows [2, 3]. After that, many new chaotic and hyperchaotic complex systems have been reported and intensively studied, including the complex Van der Pol oscillators [4], the complex Chen and complex Lü systems [5], complex detuned laser system [6], complex hyperchaotic Lorenz system [7], complex modified hyperchaotic Lü system [8], and a novel hyperchaotic complex-variable system [9] which generates 2-, 3-, and 4-scroll attractors. Since Pecora and Carroll [10] first proposed the driveresponse concept for constructing synchronization of coupled chaotic systems, synchronization in chaotic systems has been extensively investigated due to their potential applications in the fields of secure communications; optical, chemical, physical, and biological systems; neural networks; and so forth [11–13]. When applying the complex systems in communications, the complex variables will double the number of variables and can increase the content and security of the transmitted information. Based on the Lyapunov stability theory, linear feedback controller was derived to achieve hybrid projective synchronization in a chaotic complex nonlinear system [14]. The authors [15] achieved adaptive antisynchronization of a class of chaotic complex nonlinear systems described by a united mathematical expression with fully uncertain parameters. In [16], the author investigated the modified projective phase synchronization of chaotic complex nonlinear systems. Based on the passive theory, the authors studied the projective synchronization of hyperchaotic complex nonlinear systems and its application in secure communications [17]. In [18], the authors achieved fast synchronization of a novel hyperchaotic complex system based on finite-time stability theory. However, most of the existing synchronization schemes are based on the usual drive-response synchronization mode, which has one drive system and one response system. In [19], Luo et al. proposed the combination synchronization scheme, which has two drive systems and one response system. Zhou et al. investigated combination synchronization of three nonlinear complex hyperchaotic systems in [20]. Sun et al. [21] extended the combination synchronization scheme to the combination-combination synchronization scheme, where synchronization is achieved between two drive systems and two response systems. This synchronization scheme has advantages over the other synchronization schemes, such that it can provide greater security in secure communication. 2 Abstract and Applied Analysis For the nonlinear complex chaotic or hyperchaotic systems, there are no work on combination-combination synchronization for them. This paper aims to study the combination-combination synchronization of four nonlinear complex chaotic systems. The rest of this paper is organized as follows. Section 2 introduces the scheme of combinationcombination synchronization. In Section 3, we investigate combination-combination synchronization of four complex nonlinear chaotic systems. Numerical simulations are conducted in Section 4. Finally, conclusions are given in Section 5. The first drive system [22] is given by ̇ = 𝛼1 (𝑥12 − 𝑥11 ) + 𝑥12 𝑥13 , 𝑥11 ̇ = 𝛾1 𝑥11 − 𝑥12 − 𝑥11 𝑥13 , 𝑥12 (6) 1 ̇ = −𝛽1 𝑥13 + (𝑥11 𝑥12 + 𝑥11 𝑥12 ) , 𝑥13 2 and the second drive system [23] is described as follows: ̇ = 𝑎1 𝑥21 + 𝑏1 𝑥22 𝑥23 , 𝑥21 ̇ = 𝑎2 𝑥22 + 𝑏2 𝑥21 𝑥23 , 𝑥22 2. The Scheme of Combination-Combination Synchronization ̇ = 𝑎3 𝑥23 + 𝑥23 In the scheme of combination-combination synchronization, there are four nonlinear dynamical systems, two drive systems, and two response system. The two drive systems are, respectively, given by 𝑥1̇ = 𝑓1 (𝑥1 ) , (1) 𝑥2̇ = 𝑓2 (𝑥2 ) . (2) 𝑏3 (𝑥 𝑥 + 𝑥21 𝑥22 ) . 2 21 22 The first response system [6] takes the following form: ̇ = 𝜎3 𝑦12 − 𝜎3 (1 − 𝑖𝛿3 ) 𝑦11 + 𝜑1 + 𝑖𝜑2 , 𝑦11 ̇ = (𝛼3 − 𝑦13 ) 𝑦11 − (1 + 𝑖𝛿3 ) 𝑦12 + 𝜑3 + 𝑖𝜑4 , 𝑦12 and the second response [9] is given by ̇ = 𝑦22 − 𝛼4 𝑦21 + 𝛽4 𝑦22 𝑦23 + 𝜑1∗ + 𝑖𝜑2∗ , 𝑦21 𝑦1̇ = 𝑔1 (𝑦1 ) + 𝜑, (3) ̇ = 𝛾4 𝑦22 − 𝑦21 𝑦23 + 𝑦23 + 𝜑3∗ + 𝑖𝜑4∗ , 𝑦22 𝑦2̇ = 𝑔2 (𝑦2 ) + 𝜑∗ , (4) ̇ = 𝑦23 where 𝑥1 = (𝑥11 , 𝑥12 , . . . , 𝑥1𝑛 )𝑇 , 𝑥2 = (𝑥21 , 𝑥22 , . . . , 𝑥2𝑛 )𝑇 , 𝑦1 = (𝑦11 , 𝑦12 , . . . , 𝑦1𝑛 )𝑇 , and 𝑦2 = (𝑦21 , 𝑦22 , . . . , 𝑦2𝑛 )𝑇 are the state vectors of the systems (1), (2), (3), and (4), respectively; 𝑓1 (⋅), 𝑓2 (⋅), 𝑔1 (⋅), 𝑔2 (⋅) : 𝑅𝑛 → 𝑅𝑛 are four continuous vector functions and 𝜑, 𝜑∗ : 𝑅𝑛 ×𝑅𝑛 ×𝑅𝑛 ×𝑅𝑛 → 𝑅𝑛 are two controller vectors which will be designed. Definition 1 (see [21]). If there exist four constant matrices 𝐴, 𝐵, 𝐶, and 𝐷 ∈ 𝑅𝑛 and 𝐶 ≠ 0 or 𝐷 ≠ 0 such that 󵄩 󵄩 lim 󵄩󵄩𝐴𝑥1 + 𝐵𝑥2 − 𝐶𝑦1 − 𝐷𝑦2 󵄩󵄩󵄩 = 0, (8) 1 ̇ = −𝛽3 𝑦13 + (𝑦11 𝑦12 + 𝑦11 𝑦12 ) + 𝜑5 , 𝑦13 2 The two response systems are, respectively, described by 𝑡 → +∞ 󵄩 (7) (5) the drive systems (1) and (2) are realized combination-combination synchronization with the response systems (3) and (4), where ‖ ⋅ ‖ represents the matrix norm. Remark 2. The combination-combination synchronization can be reduced to combination synchronization, projective synchronization, and even control problem, if we choose specific values of 𝐴, 𝐵, 𝐶, and 𝐷. 3. Combination-Combination Synchronization of Four Nonlinear Complex Chaotic Systems In this section, we investigate the combination-combination synchr (...truncated)