Generalized Julia set in coupled map lattice and its coupling and synchronization

Abstract

This paper discusses first the basic theory and fractal characteristics of generalized Julia set in coupled map lattice (CML Julia set for short). Then, two different generalized CML Julia sets are coupled, which makes two different generalized CML Julia sets change to the same one. Using gradient control and optimal control, the synchronization of two different such generalized CML Julia sets is, respectively, also realized, which makes one CML Julia set change to another. The coupling and synchronization of two no-identical generalized CML Julia sets are accomplished via coupling and synchronizing their trajectories. To verify the feasibility of coupling and gradient and optimal control methods, we consider the generalized CML Julia sets, whose lattice length is 2, as examples to achieve their coupling and synchronization. Finally, digital simulations are carried out for α=3 and α=4 and the results verify the effectiveness of the coupling and synchronization.

Keywords:

• generalized Julia sets in coupled map lattice
• coupling
• gradient control
• optimal control
• synchronization

2010 AMS Subject Classifications:

• 37P40
• 34K35
• 34D06

Acknowledgements

The author gratefully acknowledges the suggestions of reviewers. The work was supported by the NNSF of China (Nos. 60874009 and 10971120).