# American Institute of Mathematical Sciences

January  2016, 21(1): 173-184. doi: 10.3934/dcdsb.2016.21.173

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

 1 Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan, Taiwan, Taiwan

Received  August 2013 Revised  September 2015 Published  November 2015

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.
Citation: Yu-Hao Liang, Wan-Rou Wu, Jonq Juang. Fastest synchronized network and synchrony on the Julia set of complex-valued coupled map lattices. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 173-184. doi: 10.3934/dcdsb.2016.21.173
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##### References:
 [1] E. Ahmed, H. A. Abdusalam and E. S. Fahmy, On telegraph reaction diffusion and coupled map lattice in some biological systems,, Int. J. Mod. Phys. C, 12 (2001), 717. doi: 10.1142/S0129183101001936. Google Scholar [2] M. Barahona and L. M. Pecora, Synchronization in small-world systems,, Phys. Rev. Lett., 89 (2002). doi: 10.1103/PhysRevLett.89.054101. Google Scholar [3] Å. Brännström and D. J. T. Sumpter, Coupled map lattice approximations for spatially explicit individual-based models of ecology,, Bulletin Math. Biol., 67 (2005), 663. doi: 10.1016/j.bulm.2004.09.006. Google Scholar [4] P. J. Davis, Circulant Matrices,, Chelsea, (1994). Google Scholar [5] R. L. Devaney, An Introduction to Chaotic Dynamical Systems,, $2^{nd}$ edition, (1989). Google Scholar [6] K. S. Fink, G. Johnson, T. Carroll, D. Mar and L. Pecora, Three coupled oscillators as a universal probe of synchronization stability in coupled oscillator arrays,, Phys. Rev. Lett., 61 (2000), 5080. doi: 10.1103/PhysRevE.61.5080. Google Scholar [7] G. Hu, J. Yang and W. Liu, Instability and controllability of linearly coupled oscillators: Eigenvalue analysis,, Phys. Rev. E, 58 (1998), 4440. doi: 10.1103/PhysRevE.58.4440. Google Scholar [8] J. Jost and M. P. Joy, Spectral properties and synchronization in coupled map lattices,, Phys. Rev. E, 65 (2002). doi: 10.1103/PhysRevE.65.016201. Google Scholar [9] J. Juang and Y. H. Liang, Synchronous chaos in coupled map lattices with general connectivity topology,, SIAM J. Appl. Dyn. Syst., 7 (2008), 755. doi: 10.1137/070705179. Google Scholar [10] K. Kaneko, Overview of coupled map lattices,, Chaos, 2 (1992), 279. doi: 10.1063/1.165869. Google Scholar [11] X. Li and G. Chen, Synchronization and desynchronization of complex dynamical networks: An engineering viewpoint,, IEEE Trans. Circuits Syst. I, 50 (2003), 1381. doi: 10.1109/TCSI.2003.818611. Google Scholar [12] W. W. Lin and Y. Q. Wang, Chaotic synchronization in coupled map lattices with periodic boundary conditions,, SIAM J. Appl. Dyn. Syst., 1 (2002), 175. doi: 10.1137/S1111111101395410. Google Scholar [13] W. W. Lin and Y. Q. Wang, Proof of synchronized chaotic behaviors in coupled map lattices,, Int. J. Bifur. and Chaos, 21 (2011), 1493. doi: 10.1142/S0218127411029069. Google Scholar [14] L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems,, Phys. Rev. Lett., 80 (1998), 2109. doi: 10.1103/PhysRevLett.80.2109. Google Scholar [15] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,, $2^{nd}$ edition, (1999). Google Scholar [16] M. Zhan, G. Hu and J. Yang, Synchronization of chaos in coupled systems,, Phys. Rev. E, 62 (2000), 2963. doi: 10.1103/PhysRevE.62.2963. Google Scholar
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