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  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 & Continuous Dynamical Systems - B, 2020, 25 (9) : 3677-3714. doi: 10.3934/dcdsb.2020086
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[39]

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

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[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.   Google Scholar

[44]

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

[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.  Google Scholar

[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.  Google Scholar

[47]

W. Singer, Synchronization of cortical activity and its putative role in information processing and learning, Annu. Rev. Physiol., 55 (1993), 349-374.  doi: 10.1146/annurev.ph.55.030193.002025.  Google Scholar

[48]

Q. SongJ. Cao and F. Liu, Synchronization of complex dynamical networks with nonidentical nodes, Phys. Lett. A, 374 (2010), 544-551.   Google Scholar

[49]

F. Sorrentino and M. Porfiri, Analysis of parameter mismatches in the master stability function for network synchronization, EPL, 93 (2011), 50002. doi: 10.1209/0295-5075/93/50002.  Google Scholar

[50]

S. H. Strogatz and I. Stewart, Coupled oscillators and biological synchronization, Scientific American, 269 (1993), 102-109.  doi: 10.1038/scientificamerican1293-102.  Google Scholar

[51]

J. Sun, E. M. Bollt and T. Nishikawa, Master stability functions for coupled nearly identical dynamical systems, EPL, 85 (2009), 60011. doi: 10.1209/0295-5075/85/60011.  Google Scholar

[52]

J. A. K. SuykensP. F. Curran and L. O. Chua, Robust synthesis for master-slave synchronization of Lur'e systems, IEEE Trans. Circuits Syst. I, 46 (1999), 841-850.  doi: 10.1109/81.774230.  Google Scholar

[53]

D. TaylorP. S. Skardal and J. Sun, Synchronization of heterogeneous oscillators under network modifications: Perturbation and optimization of the synchrony alignment function, SIAM J. Appl. Math., 76 (2016), 1984-2008.  doi: 10.1137/16M1075181.  Google Scholar

[54]

J.-P. Tseng, A novel approach to synchronization of nonlinearly coupled network systems with delays, Phys. A, 452 (2016), 266-280.  doi: 10.1016/j.physa.2016.02.025.  Google Scholar

[55]

K. UriuY. Morishita and Y. Iwasa, Synchronized oscillation of the segmentation clock gene in vertebrate development, J. Math. Biol., 61 (2010), 207-229.  doi: 10.1007/s00285-009-0296-1.  Google Scholar

[56]

C. van Vreeswijk, Partial synchronization in populations of pulse-coupled oscillators, Phys. Rev. E, 54 (1996), 5522-5537.  doi: 10.1103/PhysRevE.54.5522.  Google Scholar

[57]

G. D. VanWiggeren and R. Roy, Communication with chaotic lasers, Science, 279 (1998), 1198-1200.  doi: 10.1126/science.279.5354.1198.  Google Scholar

[58]

J.-L. WangZ.-C. YangT. Huang and M. Xiao, Local and global exponential synchronization of complex delayed dynamical networks with general topology, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 393-408.  doi: 10.3934/dcdsb.2011.16.393.  Google Scholar

[59]

L. WangW. Qian and Q.-G. Wang, Bounded synchronization of a time-varying dynamical network with nonidentical nodes, Internat. J. Systems Sci., 46 (2015), 1234-1245.  doi: 10.1080/00207721.2013.815825.  Google Scholar

[60]

Y. Wang and J. Cao, Cluster synchronization in nonlinearly coupled delayed networks of non-identical dynamic systems, Nonlinear Anal. Real World Appl., 14 (2013), 842-851.  doi: 10.1016/j.nonrwa.2012.08.005.  Google Scholar

[61]

D. J. Watts and S. H. Strogatz, Collective dynamics of `small–world' networks, Nature, 393 (1998), 440-442.   Google Scholar

[62]

J. A. WhiteC. C. ChowJ. RitC. Soto-Treviño and N. Kopell, Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons, J. Comput. Neurosci., 5 (1998), 5-16.   Google Scholar

[63]

C. W. Wu, Synchronization in networks of nonlinear dynamical systems coupled via a directed graph, Nonlinearity, 18 (2005), 1057-1064.  doi: 10.1088/0951-7715/18/3/007.  Google Scholar

[64]

X. YangJ. Cao and Z. Yang, Synchronization of coupled reaction-diffusion neural networks with time-varying delays via pinning-impulsive controller, SIAM J. Control Optim., 51 (2013), 3486-3510.  doi: 10.1137/120897341.  Google Scholar

[65]

W. YuJ. Cao and J. Lü, Global synchronization of linearly hybrid coupled networks with time-varying delay, SIAM J. Appl. Dyn. Syst., 7 (2008), 108-133.  doi: 10.1137/070679090.  Google Scholar

[66]

J. ZhaoD. J. Hill and T. Liu, Global bounded synchronization of general dynamical networks with nonidentical nodes, IEEE Trans. Automat. Control, 57 (2012), 2656-2662.  doi: 10.1109/TAC.2012.2190206.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[39]

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

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[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.   Google Scholar

[44]

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

[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.  Google Scholar

[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.  Google Scholar

[47]

W. Singer, Synchronization of cortical activity and its putative role in information processing and learning, Annu. Rev. Physiol., 55 (1993), 349-374.  doi: 10.1146/annurev.ph.55.030193.002025.  Google Scholar

[48]

Q. SongJ. Cao and F. Liu, Synchronization of complex dynamical networks with nonidentical nodes, Phys. Lett. A, 374 (2010), 544-551.   Google Scholar

[49]

F. Sorrentino and M. Porfiri, Analysis of parameter mismatches in the master stability function for network synchronization, EPL, 93 (2011), 50002. doi: 10.1209/0295-5075/93/50002.  Google Scholar

[50]

S. H. Strogatz and I. Stewart, Coupled oscillators and biological synchronization, Scientific American, 269 (1993), 102-109.  doi: 10.1038/scientificamerican1293-102.  Google Scholar

[51]

J. Sun, E. M. Bollt and T. Nishikawa, Master stability functions for coupled nearly identical dynamical systems, EPL, 85 (2009), 60011. doi: 10.1209/0295-5075/85/60011.  Google Scholar

[52]

J. A. K. SuykensP. F. Curran and L. O. Chua, Robust synthesis for master-slave synchronization of Lur'e systems, IEEE Trans. Circuits Syst. I, 46 (1999), 841-850.  doi: 10.1109/81.774230.  Google Scholar

[53]

D. TaylorP. S. Skardal and J. Sun, Synchronization of heterogeneous oscillators under network modifications: Perturbation and optimization of the synchrony alignment function, SIAM J. Appl. Math., 76 (2016), 1984-2008.  doi: 10.1137/16M1075181.  Google Scholar

[54]

J.-P. Tseng, A novel approach to synchronization of nonlinearly coupled network systems with delays, Phys. A, 452 (2016), 266-280.  doi: 10.1016/j.physa.2016.02.025.  Google Scholar

[55]

K. UriuY. Morishita and Y. Iwasa, Synchronized oscillation of the segmentation clock gene in vertebrate development, J. Math. Biol., 61 (2010), 207-229.  doi: 10.1007/s00285-009-0296-1.  Google Scholar

[56]

C. van Vreeswijk, Partial synchronization in populations of pulse-coupled oscillators, Phys. Rev. E, 54 (1996), 5522-5537.  doi: 10.1103/PhysRevE.54.5522.  Google Scholar

[57]

G. D. VanWiggeren and R. Roy, Communication with chaotic lasers, Science, 279 (1998), 1198-1200.  doi: 10.1126/science.279.5354.1198.  Google Scholar

[58]

J.-L. WangZ.-C. YangT. Huang and M. Xiao, Local and global exponential synchronization of complex delayed dynamical networks with general topology, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 393-408.  doi: 10.3934/dcdsb.2011.16.393.  Google Scholar

[59]

L. WangW. Qian and Q.-G. Wang, Bounded synchronization of a time-varying dynamical network with nonidentical nodes, Internat. J. Systems Sci., 46 (2015), 1234-1245.  doi: 10.1080/00207721.2013.815825.  Google Scholar

[60]

Y. Wang and J. Cao, Cluster synchronization in nonlinearly coupled delayed networks of non-identical dynamic systems, Nonlinear Anal. Real World Appl., 14 (2013), 842-851.  doi: 10.1016/j.nonrwa.2012.08.005.  Google Scholar

[61]

D. J. Watts and S. H. Strogatz, Collective dynamics of `small–world' networks, Nature, 393 (1998), 440-442.   Google Scholar

[62]

J. A. WhiteC. C. ChowJ. RitC. Soto-Treviño and N. Kopell, Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons, J. Comput. Neurosci., 5 (1998), 5-16.   Google Scholar

[63]

C. W. Wu, Synchronization in networks of nonlinear dynamical systems coupled via a directed graph, Nonlinearity, 18 (2005), 1057-1064.  doi: 10.1088/0951-7715/18/3/007.  Google Scholar

[64]

X. YangJ. Cao and Z. Yang, Synchronization of coupled reaction-diffusion neural networks with time-varying delays via pinning-impulsive controller, SIAM J. Control Optim., 51 (2013), 3486-3510.  doi: 10.1137/120897341.  Google Scholar

[65]

W. YuJ. Cao and J. Lü, Global synchronization of linearly hybrid coupled networks with time-varying delay, SIAM J. Appl. Dyn. Syst., 7 (2008), 108-133.  doi: 10.1137/070679090.  Google Scholar

[66]

J. ZhaoD. J. Hill and T. Liu, Global bounded synchronization of general dynamical networks with nonidentical nodes, IEEE Trans. Automat. Control, 57 (2012), 2656-2662.  doi: 10.1109/TAC.2012.2190206.  Google Scholar

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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