Complex Modified Hybrid Projective Synchronization of Different Dimensional Fractional-Order Complex Chaos and Real Hyper-Chaos

Abstract

This paper introduces a type of modified hybrid projective synchronization with complex transformation matrix (CMHPS) for different dimensional fractional-order complex chaos and fractional-order real hyper-chaos. The transformation matrix in this type of chaotic synchronization is a non-square matrix, and its elements are complex numbers. Based on the stability theory of fractional-order systems, by employing the feedback control technique, necessary and sufficient criteria on CMHPS are derived. Furthermore, CMHPS between fractional-order real hyper-chaotic Rössler system and other two different dimensional fractional-order complex Lorenz-like chaotic systems is provided as two examples to discuss reduced order and increased order synchronization, respectively.

Projective Synchronization for a Class of Fractional-Order Chaotic Systems with Fractional-Order in the (1, 2) Interval

“…However, many previous synchronization methods [6][7][8][9][13][14][15][16] for fractional-order chaotic systems only focused on the fractional-order 1 0 < < q , when in fact, there are many fractional-order systems with fractional-order 1 < q < 2 in the real world. For example, the time fractional heat conduction equation [17], the fractional telegraph equation [18], the time fractional reaction-diffusion systems [19], the fractional diffusion-wave equation [20], the space-time fractional diffusion equation [21], the super-diffusion systems [22], etc., but the chaos phenomenon was not considered in [17][18][19][20][21][22].…”

“…Rössler chaotic system [8,9], the fractional-order Chen chaotic system [6][7][8], the fractional-order memristor chaotic system [10], and so on.…”

Adaptive generalized combination complex synchronization of uncertain real and complex nonlinear systems

“…When the AGCCS controller is activated, the response system (27) changes its original behavior according to the drive systems (25) and (26) with respect to the given complex mapping vectors (28). The AGCCS errors between the drive systems (25), (26) and response system (27) (25), (26) and (27) are drawn in Figs. 3, 4, and 5, respectively, which show that the estimated values of unknown parameters adaptively converge to their true values in a second.…”

“…The true values of unknown parameters are set as A = (10, 28, 8/3) T , B = (35, 28, 3) T and C = (45, 25, 6, 5) T , respectively, which can ensure that the drive systems (25) and (26) operate chaotically, and the response system (27) operates in hyperchaotic state without control. The initial conditions of the systems (25), (26) and (27) are arbitrarily chosen as x(0) = (−10, 0, 37) T , y(0) = (5, −14, −16) T and z(0) = (3.6 − 0.6 j, 0.9 − j, 13, 15) T , separately. The initial values of estimated parameters are randomly selected asÂ(0) = (10, 10, 10) T ,B(0) = (10, 10, 10) T andĈ(0) = (10, 10, 10, 10) T , respectively.…”

“…23 As introduced in Refs. [24][25][26][27][28], synchronization between chaotic real and complex systems can offer more options of chaotic generators for secure communication and digital cryptography, which can increase the difficulty to decrypt and decipher the encrypted signals and information. Hence, it is meaningful and challenging to explore various synchronization schemes of nonlinear real and complex systems.…”