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

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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

Yu-Hao Liang ^1, , Wan-Rou Wu ^1, and Jonq Juang ^1,

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

Received August 2013 Revised September 2015 Published November 2015

Abstract
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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.

Mathematics Subject Classification: 37F10, 37E05, 37N9.

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 and Continuous Dynamical Systems - B, 2016, 21 (1) : 173-184. doi: 10.3934/dcdsb.2016.21.173

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), 717-726. doi: 10.1142/S0129183101001936.
[2]	M. Barahona and L. M. Pecora, Synchronization in small-world systems, Phys. Rev. Lett., 89 (2002), 054101. doi: 10.1103/PhysRevLett.89.054101.
[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), 663-682. doi: 10.1016/j.bulm.2004.09.006.
[4]	P. J. Davis, Circulant Matrices, Chelsea, New York, 1994.
[5]	R. L. Devaney, An Introduction to Chaotic Dynamical Systems, $2^{nd}$ edition, Addison-Wesley, California, 1989.
[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), 5080-5090. doi: 10.1103/PhysRevE.61.5080.
[7]	G. Hu, J. Yang and W. Liu, Instability and controllability of linearly coupled oscillators: Eigenvalue analysis, Phys. Rev. E, 58 (1998), 4440-4453. doi: 10.1103/PhysRevE.58.4440.
[8]	J. Jost and M. P. Joy, Spectral properties and synchronization in coupled map lattices, Phys. Rev. E, 65 (2002), 016201, 9pp. doi: 10.1103/PhysRevE.65.016201.
[9]	J. Juang and Y. H. Liang, Synchronous chaos in coupled map lattices with general connectivity topology, SIAM J. Appl. Dyn. Syst., 7 (2008), 755-765. doi: 10.1137/070705179.
[10]	K. Kaneko, Overview of coupled map lattices, Chaos, 2 (1992), 279-282. doi: 10.1063/1.165869.
[11]	X. Li and G. Chen, Synchronization and desynchronization of complex dynamical networks: An engineering viewpoint, IEEE Trans. Circuits Syst. I, 50 (2003), 1381-1390. doi: 10.1109/TCSI.2003.818611.
[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), 175-189. doi: 10.1137/S1111111101395410.
[13]	W. W. Lin and Y. Q. Wang, Proof of synchronized chaotic behaviors in coupled map lattices, Int. J. Bifur. and Chaos, 21 (2011), 1493-1500. doi: 10.1142/S0218127411029069.
[14]	L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett., 80 (1998), 2109-2112. doi: 10.1103/PhysRevLett.80.2109.
[15]	C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, $2^{nd}$ edition, CRC Press, Florida, 1999.
[16]	M. Zhan, G. Hu and J. Yang, Synchronization of chaos in coupled systems, Phys. Rev. E, 62 (2000), 2963-2966. doi: 10.1103/PhysRevE.62.2963.

show all references

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), 717-726. doi: 10.1142/S0129183101001936.
[2]	M. Barahona and L. M. Pecora, Synchronization in small-world systems, Phys. Rev. Lett., 89 (2002), 054101. doi: 10.1103/PhysRevLett.89.054101.
[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), 663-682. doi: 10.1016/j.bulm.2004.09.006.
[4]	P. J. Davis, Circulant Matrices, Chelsea, New York, 1994.
[5]	R. L. Devaney, An Introduction to Chaotic Dynamical Systems, $2^{nd}$ edition, Addison-Wesley, California, 1989.
[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), 5080-5090. doi: 10.1103/PhysRevE.61.5080.
[7]	G. Hu, J. Yang and W. Liu, Instability and controllability of linearly coupled oscillators: Eigenvalue analysis, Phys. Rev. E, 58 (1998), 4440-4453. doi: 10.1103/PhysRevE.58.4440.
[8]	J. Jost and M. P. Joy, Spectral properties and synchronization in coupled map lattices, Phys. Rev. E, 65 (2002), 016201, 9pp. doi: 10.1103/PhysRevE.65.016201.
[9]	J. Juang and Y. H. Liang, Synchronous chaos in coupled map lattices with general connectivity topology, SIAM J. Appl. Dyn. Syst., 7 (2008), 755-765. doi: 10.1137/070705179.
[10]	K. Kaneko, Overview of coupled map lattices, Chaos, 2 (1992), 279-282. doi: 10.1063/1.165869.
[11]	X. Li and G. Chen, Synchronization and desynchronization of complex dynamical networks: An engineering viewpoint, IEEE Trans. Circuits Syst. I, 50 (2003), 1381-1390. doi: 10.1109/TCSI.2003.818611.
[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), 175-189. doi: 10.1137/S1111111101395410.
[13]	W. W. Lin and Y. Q. Wang, Proof of synchronized chaotic behaviors in coupled map lattices, Int. J. Bifur. and Chaos, 21 (2011), 1493-1500. doi: 10.1142/S0218127411029069.
[14]	L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett., 80 (1998), 2109-2112. doi: 10.1103/PhysRevLett.80.2109.
[15]	C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, $2^{nd}$ edition, CRC Press, Florida, 1999.
[16]	M. Zhan, G. Hu and J. Yang, Synchronization of chaos in coupled systems, Phys. Rev. E, 62 (2000), 2963-2966. doi: 10.1103/PhysRevE.62.2963.

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