Fastest synchronized network and synchrony on the Julia set of complex-valued coupled map lattices
Yu-Hao Liang Wan-Rou Wu Jonq Juang
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 $f_c(z)= z^2+ c$ with $|c|< \frac{1}{4}$, 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 $g_c(z)= z^3+ cz$ is also reported.
keywords: fastest synchronized network. Julia set Complex-valued coupled map lattices

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