Fastest synchronized network and synchrony on the Julia set of complex-valued coupled map lattices

The purpose of this paper is to address synchronous chaos on the Julia set of complex-valued coupled map lattices (CCMLs). Our main results contain the following. First, we solve an inf min max problem for which its solution gives the fastest synchronized rate in a certain class of coupling matrices. Specifically, we show that for the class of real circulant matrices of size 4, the coupling weights, possible complex numbers, yielding the fastest synchronized rate can be exactly obtained. Second, for individual map of the form f_c(z) = z² + c with /c/ < 1/4 , we show that the corresponding CCMLs acquires global synchrony on its Julia set with the number of the oscillators being 3 or 4 for the diffusive coupling. For c = 0 and -2, the corresponding CCMLs obtain local synchronization if and only if the number of oscillators is less than or equal to 5. Global synchronization for the individual map of the form gc(z) = z³ + cz is also reported.