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
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.

[2]

M. Barahona and L. M. Pecora, Synchronization in small-world systems,, Phys. Rev. Lett., 89 (2002). 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. doi: 10.1016/j.bulm.2004.09.006.

[4]

P. J. Davis, Circulant Matrices,, Chelsea, (1994).

[5]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems,, $2^{nd}$ edition, (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. 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. 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). 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. doi: 10.1137/070705179.

[10]

K. Kaneko, Overview of coupled map lattices,, Chaos, 2 (1992), 279. 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. 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. 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. 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. doi: 10.1103/PhysRevLett.80.2109.

[15]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,, $2^{nd}$ edition, (1999).

[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.

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. doi: 10.1142/S0129183101001936.

[2]

M. Barahona and L. M. Pecora, Synchronization in small-world systems,, Phys. Rev. Lett., 89 (2002). 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. doi: 10.1016/j.bulm.2004.09.006.

[4]

P. J. Davis, Circulant Matrices,, Chelsea, (1994).

[5]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems,, $2^{nd}$ edition, (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. 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. 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). 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. doi: 10.1137/070705179.

[10]

K. Kaneko, Overview of coupled map lattices,, Chaos, 2 (1992), 279. 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. 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. 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. 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. doi: 10.1103/PhysRevLett.80.2109.

[15]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos,, $2^{nd}$ edition, (1999).

[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.

[1]

Xin Li, Wenxian Shen, Chunyou Sun. Invariant measures for complex-valued dissipative dynamical systems and applications. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2427-2446. doi: 10.3934/dcdsb.2017124

[2]

Jerry L. Bona, Stéphane Vento, Fred B. Weissler. Singularity formation and blowup of complex-valued solutions of the modified KdV equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11/12) : 4811-4840. doi: 10.3934/dcds.2013.33.4811

[3]

Gerhard Keller, Carlangelo Liverani. Coupled map lattices without cluster expansion. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2/3) : 325-335. doi: 10.3934/dcds.2004.11.325

[4]

Lu Gan Liu, ming Wei. Multiple complex-valued solutions for the nonlinear Schrödinger equations involving magnetic potentials. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1957-1975. doi: 10.3934/cpaa.2017096

[5]

Chenxi Guo, Guillaume Bal. Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields. Inverse Problems & Imaging, 2014, 8 (4) : 1033-1051. doi: 10.3934/ipi.2014.8.1033

[6]

B. Fernandez, P. Guiraud. Route to chaotic synchronisation in coupled map lattices: Rigorous results. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 435-456. doi: 10.3934/dcdsb.2004.4.435

[7]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519

[8]

Luke G. Rogers, Alexander Teplyaev. Laplacians on the basilica Julia set. Communications on Pure & Applied Analysis, 2010, 9 (1) : 211-231. doi: 10.3934/cpaa.2010.9.211

[9]

Michael Braun. On lattices, binary codes, and network codes. Advances in Mathematics of Communications, 2011, 5 (2) : 225-232. doi: 10.3934/amc.2011.5.225

[10]

Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751

[11]

James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667

[12]

David J. Aldous. A stochastic complex network model. Electronic Research Announcements, 2003, 9: 152-161.

[13]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461

[14]

Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115

[15]

Koh Katagata. On a certain kind of polynomials of degree 4 with disconnected Julia set. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 975-987. doi: 10.3934/dcds.2008.20.975

[16]

Volodymyr Nekrashevych. The Julia set of a post-critically finite endomorphism of $\mathbb{PC}^2$. Journal of Modern Dynamics, 2012, 6 (3) : 327-375. doi: 10.3934/jmd.2012.6.327

[17]

Rich Stankewitz. Density of repelling fixed points in the Julia set of a rational or entire semigroup, II. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2583-2589. doi: 10.3934/dcds.2012.32.2583

[18]

Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327

[19]

Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457

[20]

Zengjing Chen, Yuting Lan, Gaofeng Zong. Strong law of large numbers for upper set-valued and fuzzy-set valued probability. Mathematical Control & Related Fields, 2015, 5 (3) : 435-452. doi: 10.3934/mcrf.2015.5.435

2016 Impact Factor: 0.994

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]