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We propose a new fractional-order chaotic complex system and study its dynamical properties including symmetry, equilibria and their stability, and chaotic attractors. Chaotic behavior is verified with phase portraits, bifurcation diagrams, the histories, and the largest Lyapunov exponents. And we find that chaos exists in this system with orders less than 5 by numerical simulation. Additionally, antisynchronization of different fractional-order chaotic complex systems is considered based on the stability theory of fractional-order systems. This new system and the fractional-order complex Lorenz system can achieve antisynchronization. Corresponding numerical simulations show the effectiveness and feasibility of the scheme.

Chaotic behavior and synchronization of fractional-order dynamical systems have been extensively studied over the last decade. Many fractional-order systems can behave chaotically, such as the fractional-order Chua’s system [

However, most of the studies about fractional-order systems had been based on real variables, and complex systems are rarely involved. Complex systems provide an excellent instrument to describe a variety of physical phenomena, such as detuned laser systems, amplitudes of electromagnetic fields, and thermal convection of liquid flows [

Motivated by the above discussion, the aim of this paper is to investigate the chaotic phenomena in a newly proposed fractional-order complex Lü system, which may provide potential applications in secure communication. As will be shown below, this new system displays many interesting dynamical behaviors, such as fixed points, periodic motions, and chaotic motions. Besides, when the parameters of the system are fixed, the lowest order for chaos to exist is determined. Furthermore, antisynchronization between the new system and fractional-order complex Lorenz system is studied. More generally, we investigate antisynchronization of different fractional-order chaotic complex systems and give a usual scheme.

The remainder of this paper is organized as follows. In Section

There are many definitions of fractional derivatives [

In 2007, the complex Lü system was proposed by Mahmoud et al. [

Chaotic attractor of complex Lü system

In this paper, we modify the derivative operator in (

Separating the real and imaginary parts of system (

In 2002, Diethelm et al. proposed the Adams-Bashforth-Moulton predictor-corrector scheme [

where

Note that the symmetry of system (

The equilibria of system (

As to the equilibrium

Using the above discretization scheme (

Bifurcation diagram of system (

Time histories showing the rout to chaos via intermittency for system (

Phase portraits of system (

Bifurcation diagram of system (

Time histories showing the rout to chaos via intermittency for system (

Bifurcation diagram of system (

Phase portraits of system (

Phase portraits of system (

Time histories showing the rout to chaos via intermittency for system (

In this section, we give a general method to achieve antisynchronization of different fractional-order complex systems firstly. Consequently, antisynchronization between fractional-order complex Lü and Lorenz system can be achieved. Without loss of generality, we assume that the derivative order is

Consider the following fractional-order complex system:

Some fractional-order chaotic complex systems can be described by (

Now we give the stability results for linear fractional-order systems.

Autonomous linear system of the fractional-order

Antisynchronization between (

From the definition of antisynchronization, we obtain the error vector between (

If

In this section, the antisynchronization behavior between the fractional-order complex Lü and Lorenz systems is made. It is assumed that the fractional-order complex Lü system drives the fractional-order complex Lorenz system [

According to Theorem

The errors of antisynchronization converge asymptotically to zero in a quite short period as depicted in Figure

The time evolution of the synchronization errors between systems (

State variables of master system (

In this paper, a new fractional-order chaotic complex system is proposed. By means of phase portraits, bifurcation diagrams, the histories, and the largest Lyapunov exponents, we investigate chaotic behavior of this new system. Our results show that the new system displays many interesting dynamical behaviors, such as fixed points, periodic motions, and chaotic motions. Two typical routes to chaos—period doubling and intermittency—are found in this system. Besides, when the parameters of the system are fixed, the lowest order for chaos to exist is determined. Moreover, antisynchronization of different fractional-order chaotic complex systems has been studied. Meanwhile, the new system and the fractional-order complex Lorenz system can achieve antisynchronization.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The research is supported by the National Nature Science Foundation of China (nos. 61273088 and 10971120) and the Nature Science Foundation of Shandong province (no. ZR2010FM010). The authors would like to thank the editors and anonymous referees for their constructive comments and suggestions.