Performance of synchronization in networks of chaotic systems under different PID coupling schemes

The conventional PID control method is utilized for synchronizing a network of chaotic systems, where the coupling is defined on the proportions, integrals and derivatives of the state variables. The master stability function approach is taken, with a special technique to bypass the difficulties in calculations of integral and derivative couplings by using the hyperjerk systems. For demonstration, six different hyperjerk systems are examined, and the effects of different coupling schemes are studied and compared. It is found that the most efficient coupling for achieving network synchronization is the proportional–integral coupling. It is also verified that the same networked systems are not synchronizable under derivative, integral, integral–derivative, and proportional–derivative coupling configurations. The results obtained from master stability function method are verified by numerical simulations.