Chaotic synchronization between different fractional-order chaotic systems
Introduction
In 1990, Pecora and Carroll presented the concept of “chaotic synchronization” for the first time [1] and introduced a method to synchronize two identical chaotic systems with different initial conditions [2]. Chaos synchronization has become an active research subject in nonlinear science because of its potential applications in secure communication [3], [4], neuron systems [5], [6], [7], the study of laser dynamics [8] and control processing [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20].
In recent years, study on the dynamics of fractional-order differential systems has attracted interest of many researchers. It is demonstrated that some fractional-order differential systems behave chaotically or hyperchaotically, such as the fractional-order Lorenz system, the fractional-order Chua's system, the fractional Rossler system, the fractional modified Duffing system, the fractional order unified system and the Chen system [21], [22], [23], [24], [25]. On the other hand, chaos synchronization for the fractional order systems are just beginning to attract some attention due to its potential applications in secure communication and control processing [25], [26], [27], [28], [29], [30]. More recently, many authors begin to investigate the dynamics and synchronization of fractional-order chaotic systems [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34].
However, almost all existed synchronization methods for fractional-order chaotic systems mainly focus on that, which has the same fractional order q for drive fractional-order chaotic system and response fractional-order chaotic system. So far, there has been little information available in literature about the drive fractional-order chaotic system and response fractional-order chaotic system with different fractional order q, and little information available in literature about two different fractional-order chaotic systems with different fractional order q.
Motivated by all above works, in this paper, one synchronization method for fractional-order chaotic systems, which is different from the previous works, is given for the drive fractional-order chaotic system and response fractional-order chaotic system with different fractional order q, and for two different fractional-order chaotic systems with different fractional order q. The proposed technique is based on the idea of tracking control and stability theory of fractional order systems, which is simple and theoretically rigorous. Two groups of examples are considered and their numerical simulations are performed. The first example is the synchronization of fractional-order Chen chaotic system, in which the drive fractional-order chaotic system and response fractional-order chaotic system are the different fractional order q. The second example is the synchronization between fractional-order Lorenz chaotic system and fractional-order Chen chaotic system with different fractional order q. Numerical simulations coincide with the theoretical analysis.
Section snippets
Problem formulation
There are many definitions of fractional derivatives. In the following, we introduce the most common one of themwhere m is the first integer, which is not less than q, Dq is generally called “q-order Caputo differential operator” [30], f(m)(t) is the m-order derivative in the usual sense, and Jq(q>0) is the q-order Riemann–Liouville integral operator with expressionwhere Γ(.) is the gamma function.
Now, we consider the following n
Application of the above-mentioned scheme
To illustrate the effectiveness of the proposed synchronization scheme, two groups of examples are considered and their numerical simulations are performed.
First, consider the synchronization of fractional-order Chen chaotic system [36], and the drive fractional-order chaotic system and response fractional-order chaotic system with different fractional order.
The Chen system, introduced by Chen and Ueta in 1999 [36], is similar but not topologically equivalent to the Lorenz system and is
Conclusions
In this paper, we investigated the synchronization of fractional order chaotic systems based on the idea of tracking control and stability theory of fractional-order systems. The derived method in the present paper shows that the synchronization between drive system and response system with different fractional order q can be achieved, and the synchronization between different fractional-order chaotic systems with different fractional order q can be achieved. Numerical experiments show that the
Acknowledgements
This work is supported by Foundation of Science and Technology project of Chongqing Education Commission under Grant KJ110525.
References (38)
- et al.
Secure communication by chaotic synchronization: robustness under noisy conditions
Nonlinear Analysis
(2007) - et al.
A robust APD synchronization scheme and its application to secure communication
Journal of the Franklin Institute
(2009) - et al.
Synchronization of stochastic perturbed chaotic neural networks with mixed delays
Journal of the Franklin Institute
(2010) - et al.
Adaptive synchronization for delayed neural networks with stochastic perturbation
Journal of the Franklin Institute
(2008) - et al.
Synchronization effects using a piecewise linear map-based spiking–bursting neuron model
Neurocomputing
(2006) - et al.
Synchronization of mutually versus unidirectionally coupled chaotic semiconductor lasers
Optics Communications
(2006) Propagation of projective synchronization in a series connection of chaotic systems
Journal of the Franklin Institute
(2010)- et al.
Control and applications of chaos
Journal of the Franklin Institute
(1997) Generalized projective synchronization between Lorenz system and Chen's system
Chaos, Solitons & Fractals
(2007)- et al.
Chaos in a fractional order modified Duffing system
Chaos, Solitons & Fractals
(2007)
Chaos and hyperchaos in the fractional-order Rössler equations
Physica A
Generalized projective synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal
Physics Letters A
Generalized projective synchronization of fractional order chaotic systems
Physica A
The synchronization of fractional-order Rössler hyperchaotic systems
Physica A
Topology identification and adaptive synchronization of uncertain complex networks with non-derivative and derivative coupling
Journal of the Franklin Institute
Passive control on a unified chaotic system
Nonlinear Analysis: Real World Applications
Chaos control and chaos synchronization for ulti-scroll chaotic attractors generated using hyperbolic functions
Journal of Mathematical Analysis and Applications
Chaos in the fractional order unified system and its synchronization
Journal of the Franklin Institute
Chaos in the fractional order Chen system and its control
Chaos Solitons & Fractals
Cited by (21)
Fractal–fractional order stochastic chaotic model: A synchronization study
2023, Results in Control and OptimizationHybrid modified function projective synchronization of two different dimensional complex nonlinear systems with parameters identification
2013, Journal of the Franklin InstituteCitation Excerpt :Based on the wide scope of applications, the complex nonlinear systems have been receiving more attentions. After the seminal work on chaos synchronization proposed by Pecora and Carroll [13], in the past two decades, different synchronization schemes have been proposed such as complete synchronization (CS) [14,15], lag synchronization (LS) [16,17], generalized synchronization (GS) [18], projective synchronization (PS) [19], function projective synchronization (FPS) [20,21], etc. Recently, researchers proposed modified function projective synchronization (MFPS) as a generalization of FPS, in which the response system could be synchronized up to the drive system with a scaling function matrix [22–25].
Use of squared magnitude function in approximation and hardware implementation of siso fractional order system
2013, Journal of the Franklin InstituteCitation Excerpt :Fractional order systems (FOS) can be found in many fields [4–7]. Study of chaos in FOS and their control is a growing field of research [8,9]. Fractional order controllers have been used in many applications [10–19].
Robust stability and stabilization of fractional order interval systems with coupling relationships: The 0<α<1 case
2012, Journal of the Franklin InstituteCitation Excerpt :Chen et al. [13] presented some results for the global attractivity of solutions for nonlinear FDE with Riemann–Liouville and Caputo fractional calculus, respectively. Many researchers also have shown their interest in fractional chaotic systems [14–17]. Wu et al. [15] proposed the value of the lowest order of fractional chaotic system and investigated the synchronization of fractional chaotic system theoretically and numerically.
Projective synchronization of different fractional-order chaotic systems with non-identical orders
2012, Nonlinear Analysis: Real World ApplicationsCitation Excerpt :The synchronization between these two different 4-D FOHSs with non-identical orders is observed. From the available literature [29–35], it is known that the synchronization of chaotic systems with non-identical orders is somewhat more difficult to achieve than other synchronization cases when the drive system and response system are both with integer orders or both with identical fractional orders between corresponding states. In order to realize the synchronization between systems with non-identical orders, the controllers should be elaborately designed.
Synchronization of Fractional Stochastic Chaotic Systems via Mittag-Leffler Function
2022, Fractal and Fractional