Module-phase synchronization in hyperchaotic complex Lorenz system after modified complex projection

•We propose the concept of modified complex projective synchronization.•We consider real parts, imaginary parts and module-phase synchronization.•State variables and projective factors which are plurals are taken into consideration.•More than one lyapunov exponents are more complex.

This paper studies the modified complex projective synchronization which has plural projective factors and is meaningful to complex systems, and applies this kind of synchronization in a hyperchaotic complex Lorenz system. Based on the proposed synchronization, we also study the hybrid synchronization which contains modified complex projection and module-phase synchronization because hybrid synchronization not only deal with the synchronization real part and imaginary part, respectively, but also take module and phase of complex system into consideration. Owing to it is difficult to design the controller directly, our work involve an intermediary system. The asymptotic convergence of the errors between the states is proven and the computer simulation results present the effectiveness of our method.