September  2020, 25(9): 3677-3714. doi: 10.3934/dcdsb.2020086

From approximate synchronization to identical synchronization in coupled systems

1. 

Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan 300

2. 

Department of Mathematical Sciences, National Chengchi University, Taipei, Taiwan 116

* Corresponding author: Chih-Wen Shih

Received  March 2019 Revised  August 2019 Published  September 2020 Early access  April 2020

Fund Project: The authors are supported in part by the Ministry of Science and Technology of Taiwan.

We establish a framework to investigate approximate synchronization of coupled systems under general coupling schemes. The units comprising the coupled systems may be nonidentical and the coupling functions are nonlinear with delays. Both delay-dependent and delay-independent criteria for approximate synchronization are derived, based on an approach termed sequential contracting. It is explored and elucidated that the synchronization error, the distance between the asymptotic state and the synchronous set, decreases with decreasing difference between subsystems, difference between the row sums of connection matrix, and difference of coupling time delays between different units. This error vanishes when these factors decay to zero, and approximate synchronization becomes identical synchronization for the coupled system comprising identical subsystems and connection matrix with identical row sums, and with identical coupling delays. The application of the present theory to nonlinearly coupled heterogeneous FitzHugh-Nagumo neurons is illustrated. We extend the analysis to study approximate synchronization and asymptotic synchronization for coupled Lorenz systems and show that for some coupling schemes, the synchronization error decreases as the coupling strength increases, whereas in another case, the error remains at a substantial level for large coupling strength.

Citation: Chih-Wen Shih, Jui-Pin Tseng. From approximate synchronization to identical synchronization in coupled systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3677-3714. doi: 10.3934/dcdsb.2020086
References:
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A. AyJ. HollandA. SperleaG. S. DevakanmalaiS. KniererS. SangervasiA. Stevenson and E. M. Özbudak, Spatial gradients of protein-level time delays set the pace of the traveling segmentation clock waves, Development, 141 (2014), 4158-4167.  doi: 10.1242/dev.111930.

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show all references

References:
[1]

S. Acharyya and R. E. Amritkar, Synchronization of coupled nonidentical dynamical systems, EPL, 99 (2012), 40005. doi: 10.1209/0295-5075/99/40005.

[2]

S. Acharyya and R. E. Amritkar, Synchronization of nearly identical dynamical systems: Size instability, Phys. Rev. E, 92 (2015), 052902, 10 pp. doi: 10.1103/PhysRevE.92.052902.

[3]

A. AyJ. HollandA. SperleaG. S. DevakanmalaiS. KniererS. SangervasiA. Stevenson and E. M. Özbudak, Spatial gradients of protein-level time delays set the pace of the traveling segmentation clock waves, Development, 141 (2014), 4158-4167.  doi: 10.1242/dev.111930.

[4]

A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512.  doi: 10.1126/science.286.5439.509.

[5]

S. BoccalettiV. LatoraY. MorenoM. Chavez and D.-U. Hwang, Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), 175-308.  doi: 10.1016/j.physrep.2005.10.009.

[6]

N. Burić and D. Todorović, Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling, Phys. Rev. E, 67 (2003), 066222, 15 pp. doi: 10.1103/PhysRevE.67.066222.

[7]

N. Burić, K. Todorović and N. Vasović, Synchronization of bursting neurons with delayed chemical synapses, Phys. Rev. E, 78 (2008), 036211.

[8]

S. A. Campbell, Time delays in neural systems, in Handbook of Brain Connectivity (eds. A. McIntosh and V. K. Jirsa), Springer, Berlin, Heidelberg, (2007), 65–90. doi: 10.1007/978-3-540-71512-2_2.

[9]

J. CaoZ. Wang and Y. Sun, Synchronization in an array of linearly stochastically coupled networks with time delays, Phys. A, 385 (2007), 718-728.  doi: 10.1016/j.physa.2007.06.043.

[10]

K.-W. ChenK.-L. Liao and C.-W. Shih, The kinetics in mathematical models on segmentation clock genes in zebrafish, J. Math. Biol., 76 (2018), 97-150.  doi: 10.1007/s00285-017-1138-1.

[11]

S. M. CrookG. B. ErmentroutM. C. Vanier and J. M. Bower, The role of axonal delay in the synchronization of networks of coupled cortical oscillators, J. Comput. Neurosci., 4 (1997), 161-172. 

[12]

K. M. Cuomo and A. V. Oppenheim, Circuit implementation of synchronized chaos with applications to communications, Phys. Rev. Lett., 71(1996), 65-68. doi: 10.1103/PhysRevLett.71.65.

[13]

Z. Duan and G. Chen, Global robust stability and synchronization of networks with Lorenz-type nodes, IEEE Trans. Circuits Syst. II, 56 (2009), 679-683. 

[14]

G. B. Ermentrout and N. Kopell, Fine structure of neural spiking and synchronization in the presence of conduction delays, Proc. Natl. Acad. Sci., 95 (1998), 1259-1264.  doi: 10.1073/pnas.95.3.1259.

[15]

R. Femat and G. Solís-Perales, On the chaos synchronization phenomena, Phys. Lett. A, 262 (1999), 50-60.  doi: 10.1016/S0375-9601(99)00667-2.

[16]

S.-Y. HaS. E. Noh and J. Park, Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics, Netw. Heterog. Media, 10 (2015), 787-807.  doi: 10.3934/nhm.2015.10.787.

[17]

J. K. Hale, Diffusive coupling, dissipation, and synchronization, J. Dynam. Differential Equations, 9 (1997), 1-52.  doi: 10.1007/BF02219051.

[18]

W. HeW. DuF. Qian and J. Cao, Synchronization analysis of heterogeneous dynamical networks, Neurocomputing, 104 (2013), 146-154.  doi: 10.1016/j.neucom.2012.10.008.

[19]

W. HeF. QianJ. Cao and Q.-L. Han, Impulsive synchronization of two nonidentical chaotic systems with time-varying delay, Phys. Lett. A, 375 (2011), 498-504.  doi: 10.1016/j.physleta.2010.11.052.

[20]

W. HeF. QianQ.-L. Han and J. Cao, Synchronization error estimation and controller design for delayed Lur'e systems with parameter mismatches, IEEE Trans. Neural Netw. Learn. Syst., 23 (2012), 1551-1563. 

[21]

F. C. Hoppensteadt and E. M. Izhikevich, Pattern recognition via synchronization in phase-locked loop neural networks, IEEE Trans. Neural Netw., 11 (2000), 734-738.  doi: 10.1109/72.846744.

[22]

C. H. HsiaC. Y. Jung and B. Kwon, On the global convergence of frequency synchronization for Kuramoto and Winfree oscillators, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3319-3334.  doi: 10.3934/dcdsb.2018322.

[23]

L. Huang, Q. Chen, Y.-C. Lai and L. M. Pecora, Generic behavior of master-stability functions in coupled nonlinear dynamical systems, Phys. Rev. E, 80 (2009), 036204. doi: 10.1103/PhysRevE.80.036204.

[24]

T. Huang, C. Li and X. Liao, Synchronization of a class of coupled chaotic delayed systems with parameter mismatch, Chaos, 17 (2007), 033121, 5 pp. doi: 10.1063/1.2776668.

[25]

T. HuangC. LiW. Yu and G. Chen, Synchronization of delayed chaotic systems with parameter mismatches by using intermittent linear state feedback, Nonlinearity, 22 (2009), 569-584.  doi: 10.1088/0951-7715/22/3/004.

[26]

X. Huang and J. Cao, Generalized synchronization for delayed chaotic neural networks: A novel coupling scheme, Nonlinearity, 19 (2006), 2797-2811.  doi: 10.1088/0951-7715/19/12/004.

[27]

J. K. Kim and D. B. Forger, A mechanism for robust circadian timekeeping via stoichiometric balance, Mol. Syst. Biol., 8 (2012), 1-14.  doi: 10.1038/msb.2012.62.

[28]

V. I. KrinskyV. N. Biktashev and I. R. Efimov, Autowave principles for parallel image processing, Phys. D, 49 (1991), 247-253.  doi: 10.1016/0167-2789(91)90213-S.

[29]

R. LevyW. D. HutchisonA. M. Lozano and J. O. Dostrovsky, High-frequency synchronization of neuronal activity in the subthalamic nucleus of Parkinsonian patients with limb tremor, J. Neurosci., 20 (2000), 7766-7775.  doi: 10.1523/JNEUROSCI.20-20-07766.2000.

[30]

C.-H. Li and S.-Y. Yang, Eventual dissipativeness and synchronization of nonlinearly coupled dynamical network of Hindmarsh-Rose neurons, Appl. Math. Model., 39 (2015), 6631-6644.  doi: 10.1016/j.apm.2015.02.017.

[31]

K.-L. LiaoC.-W. Shih and J.-P. Tseng, Synchronized oscillations in a mathematical model of segmentation in zebrafish, Nonlinearity, 25 (2012), 869-904.  doi: 10.1088/0951-7715/25/4/869.

[32]

E. N. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sci., 20 (1963), 130-141.  doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.

[33]

W. LuT. Chen and G. Chen, Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay, Phys. D, 221 (2006), 118-134.  doi: 10.1016/j.physd.2006.07.020.

[34]

A. Margheri and R. Martins, Generalized synchronization in linearly coupled time periodic systems, J. Differential Equations, 249 (2010), 3215-3232.  doi: 10.1016/j.jde.2010.09.005.

[35]

G. S. Medvedev, Electrical coupling promotes fidelity of responses in the networks of model neurons, Neural Comput., 21 (2009), 3057-3078.  doi: 10.1162/neco.2009.07-08-813.

[36]

J. M. MontenbruckM. Bürger and F. Allgöwer, Practical synchronization with diffusive couplings, Automatica J. IFAC, 53 (2015), 235-243.  doi: 10.1016/j.automatica.2014.12.024.

[37]

M. E. J. Newman, The structure and function of complex networks, SIAM Rev., 45 (2003), 167-256.  doi: 10.1137/S003614450342480.

[38]

E. Panteley, A. Loría and L. Conteville, On practical synchronization of heterogeneous networks of nonlinear systems: Application to chaotic systems, 2015 American Control Conference (ACC), (2015), 5359–5364. doi: 10.1109/ACC.2015.7172177.

[39]

L. M. Pecora and T. L. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett., 80 (1998), 2109-2112. 

[40]

C. PradeepY. CaoR. Murugesu and R. Rakkiyappan, An event-triggered synchronization of semi-Markov jump neural networks with time-varying delays based on generalized free-weighting-matrix approach, Math. Comput. Simulation, 155 (2019), 41-56.  doi: 10.1016/j.matcom.2017.11.001.

[41]

H. M. Rodrigues, Abstract methods for synchronization and applications, Appl. Anal., 62 (1996), 263-296.  doi: 10.1080/00036819608840483.

[42]

H. M. RodriguesL. F. C. Alberto and N. G. Bretas, Uniform invariance principle and synchronization. Robustness with respect to parameter variation, J. Differential Equations, 169 (2001), 228-254.  doi: 10.1006/jdeq.2000.3902.

[43]

M. G. RosenblumA. S. Pikovsky and J. Kurths, From phase to lag synchronization in coupled chaotic oscillators, Phys. Rev. Lett., 78 (1997), 4193-4196. 

[44]

M. G. RosenblumA. S. Pikovsky and J. Kurths, Phase synchronization of chaotic oscillators, Phys. Rev. Lett., 76 (1996), 1804-1807. 

[45]

C.-W. Shih and J.-P. Tseng, A general approach to synchronization of coupled cells, SIAM J. Appl. Dyn. Syst., 12 (2013), 1354-1393.  doi: 10.1137/130907720.

[46]

C.-W. Shih and J.-P. Tseng, Global synchronization and asymptotic phases for a ring of identical cells with delayed coupling, SIAM J. Math. Anal., 43 (2011), 1667-1697.  doi: 10.1137/10080885X.

[47]

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Figure 1.  Plot of $ (u_1(t),v_1(t)) $ for the solution $ (u_i(t),v_i(t))_{i = 1, 2, 3} $ of system (69), starting from $ (1.3,-1.2,1.5,-1.5,1.4,-1.3) $ at $ t_0 = 0 $, and evolution of $ Err(t): = \max_{1\leq i \leq2} \{ \max\{|u_{i}(t)-u_{i+1}(t)|, |v_{i}(t)-v_{i+1}(t)|\}\} $, for various $ (\delta^\ast_{F},\delta^\ast_\kappa,\delta^\ast_\tau) $ in Example 1
Figure 2.  Evolutions of $ Err(t): = \max_{1\leq k \leq 3}\{|x_{1,k}(t)-x_{2,k}(t)|\} $ for the solutions $ (x_{i,j}(t)) $ of system (99) with $ \vartheta = 1 $, and (a) $ c = 64 $, (b) $ c = 640 $, (c) $ c = 6400 $, in Example 2, starting from $ (14,12,-23,15,13,-22) $ at $ t_0 = 0 $
Figure 3.  Evolutions of $ Err(t): = \max_{1\leq k \leq 3}\{|x_{1,k}(t)-x_{2,k}(t)|\} $ for the solutions $ (x_{i,j}(t)) $ of system (114) with $ \vartheta = 1 $ and (a) $ c = 3014 $, (b) $ c = 6028 $, and (c) $ c = 12056 $, starting from $ (14,12,-23,15,13,-22) $ at $ t_0 = 0 $, in Example 3
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