Since the pioneer research work of Ott et al. [

1], Pecora and Carroll [

2], the topic of chaos control and synchronization has attracted a lot of researchers in diverse areas including mathematics, physics, biology, medicine, engineering, and so on. Lots of research has been paid to study chaos control for real systems, and plenty of control methods have been put forward, such as feedback control [

3,

4], sliding mode control [

5,

6], backstepping method [

7], and so on. These control strategies can also be employed to realize various kinds of synchronization of real chaos. Further developments in this direction can be found in [

8,

9,

10,

11,

12,

13,

14].

The quoted literature above are only related to real chaotic systems and do not consider the chaotic systems which consist of complex variables. As is known to all, in the real world, many cases exist in the form of complex variables. For instance, Fowler et al. [

15] discovered the complex Lorenz system when they studied laser physics and baroclinic instability of the geophysical flows in 1982. Since then, the study on complex nonlinear systems has been paid a substantial amount of attentions and has become a hot topic due to its wide applications in chemical systems, optics and especially in secure communications [

16,

17,

18]. A considerable amount of complex dynamical systems exhibit chaotic motion, such as the complex Chen system [

19], the time-delay complex Lorenz system [

20], the complex generalised Lorenz hyperchaotic system [

21], just to name a few examples. Compared with real chaos, complex chaos has the diversity of synchronization types and results. On the one hand, a lot of authors extend some synchronization schemes of real chaos into complex space, for example, complete synchronization (CS) [

22], anti-synchronization (AS) [

23], lag synchronization (LS) [

24], combination synchronization [

25], etc. On the other hand, some new synchronization schemes have been proposed on the basis of the characteristics of complex systems, such as complex complete synchronization (CCS) [

26], complex lag synchronization (CLS) [

27], complex anti lag synchronization (CALS) [

28], combination complex synchronization [

29,

30], and so forth. However, the existing results on complex chaos have three disadvantages: Firstly, chaos control of the complex dynamical systems has gained little attention. Secondly, the existence of the synchronization problem, which is fundamental theoretical base, has not been considered so far. Finally, most of the current designed controllers eliminate the nonlinear term of the system, which are not only complicated but also difficult to realize in engineering. Therefore, control and synchronization in complex chaotic systems needs to be further and extensively studied.

Motivated by the aforementioned discussion, the current investigation concentrates on chaos control and synchronization of a novel complex dynamical system named as complex Rikitake system, which is proposed based on the Rikitake system. Following the idea of studying dynamics in chaotic systems, this paper investigates symmetry, dissipation, stability of equilibria, Lyapunov exponents, Poincaré-sections and bifurcation of the complex Rikitake system. Thus, along with the deeper understanding of feedback control method presented in [

9], we construct simple adaptive controllers to realize control and synchronization of the complex Rikitake system. Furthermore, we obtain a criterion to detect the existence of synchronization in the complex Rikitake system and further prove that there exist CS and the coexistence of CS and AS.