Cluster Synchronization of Networks of Chaotic Systems: A Comprehensive Review of Theory and Applications

References

• E. N. Lorenz, “Deterministic non-periodic flow,” J. Atmos. Sci. , Vol. 20, no. 2, pp. 130–141, 1963.
   Web of Science ®Google Scholar
• N. K. Hayles, Chaos and Order: Complex Dynamics in Literature and Science. University of Chicago Press, 1991.
   Google Scholar
• O. Bohigas and H. A. Weidenmuller, “Aspects of chaos in nuclear physics,” Annu. Rev. Nucl. Part. Sci., Vol. 38, no. 1, pp. 421–453, 1988.
   Google Scholar
• Y. Chen and A. Y. Leung, Bifurcation and Chaos in Engineering. London: Springer, 1998, pp. 176–229.
   Google Scholar
• J. N. Weiss, A. Garfinkel, M. L. Spano, and W. L. Ditto, “Chaos and chaos control in biology,” J. Clin. Invest., Vol. 93, no. 4, pp. 1355–1360, 1994.
   PubMed Web of Science ®Google Scholar
• D. Guegan, “Chaos in economics and finance,” Annu. Rev. Control, Vol. 33, no. 1, pp. 89–93, 2009.
   Web of Science ®Google Scholar
• L. Glass, “Introduction to controversial topics in nonlinear science: is the normal heart rate chaotic?,” Chaos, Vol. 19, no. 2, p. 028501, 2009.
   PubMed Web of Science ®Google Scholar
• T. Matsumoto, “Chaos in electronic circuits,” Proc. IEEE, Vol. 75, no. 8, pp. 1033–1057, 1987.
   Web of Science ®Google Scholar
• M. B. Kennel, R. Brown, and H. D. Abarbanel, “Determining embedding dimension for phase-space reconstruction using a geometrical construction,” Phys. Rev. A, Vol. 45, no. 6, pp. 3403–3411.1992.
   PubMed Web of Science ®Google Scholar
• J. B. Dingwell, “Lyapunov exponents,” Wiley Encyclopedia of Biomedical Engineering, Wiley, Apr. 2006. DOI: 10.1002/9780471740360.ebs0702.
   Google Scholar
• S. Chatterjee and M. R. Yilmaz, “Chaos, fractals and statistics,” Stat. Sci., Vol. 7, no. 1, pp. 49–68, 1992.
   Google Scholar
• K. W. Chung, and H. M. Ma, “Automatic generation of aesthetic patterns on fractal tilings by means of dynamical systems”, Chaos, Solit. Fract., Vol. 24, no. 4, pp. 1145–1158, 2005. DOI: 10.1016/j.chaos.2004.09.115.
   Web of Science ®Google Scholar
• T. Yang and L. O. Chua, “Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication,” IEEE Trans. Circ. Syst. I: Fundam. Theory Appl., Vol. 44, no. 10, pp. 976–988, 1997.
   Google Scholar
• Y. Pourasad, R. Ranjbarzadeh, and A. Mardani, “A new algorithm for digital image encryption based on chaos theory,” Entropy, Vol. 23, no. 3, p. 341, 2021.
   PubMed Web of Science ®Google Scholar
• L. M. Pecora, T. L. Carroll, G. A. Johnson, D. J. Mar, and J. F. Heagy, “Fundamentals of synchronization in chaotic systems, concepts, and applications,” Chaos, Vol. 7, no. 4, pp. 520–543, 1997.
   PubMed Web of Science ®Google Scholar
• A. Volkovskii and N. Rulkov, “Experimental study of bifurcations at the threshold for stochastic locking,” Sov. Tech. Phys. Lett., Vol. 15, p. 249, 1989.
   Google Scholar
• I. Aranson and N. Rul'kov, “Nontrivial structure of synchronization zones in multi-dimensional systems,” Phys. Lett. A, Vol. 139, no. 8, pp. 375–378, 1989.
   Web of Science ®Google Scholar
• H. Fujisaka and T. Yamada, “Stability theory of synchronized motion in coupled-oscillator systems,” Prog. Theor. Phys., Vol. 69, no. 1, pp. 32–47, 1983.
   Web of Science ®Google Scholar
• M. Haris, M. Shafiq, I. Ahmad, A. Ibrahim, and M. Misiran, “A nonlinear adaptive controller for the synchronization of unknown identical chaotic systems,” Arab. J. Sci. Eng., no. 10, pp. 10097–10112, Feb. 2021. DOI: 10.1007/s13369-020-05222-x.
   Web of Science ®Google Scholar
• S. Zhai, X. Wang, and Y. Zheng, “Cluster synchronization of a nonlinear network with fixed and switching topologies,” IEEE Syst. J., Vol. 17, no. 3, pp. 3752–3761, Sep. 2023. DOI: 10.1109/jsyst.2023.3246735.
   Web of Science ®Google Scholar
• I. Ahmad and M. Shafiq, “Robust adaptive anti-synchronization control of multiple uncertain chaotic systems of different orders,” Automatika, Vol. 61, no. 3, pp. 396–414, 2020.
   Web of Science ®Google Scholar
• X. Zhu, Z. Tang, J. Feng, and D. Ding, “Aperiodically intermittent pinning cluster synchronization of complex networks with hybrid delays: A region-division event-triggered protocol,” J. Franklin. Inst., Vol. 360, no. 15, pp. 11094–11113, 2023.
   Web of Science ®Google Scholar
• I. Ahmad, M. Shafiq, and M. Shahzad, “Global finite-time multi-switching synchronization of externally perturbed chaotic oscillators,” Circuits Syst. Signal Process., Vol. 37, pp. 5253–5278, 2018.
   Web of Science ®Google Scholar
• A. Fowler, J. Gibbon, and M. McGuinness, “The complex Lorenz equations,” Phys. D: Nonlinear Phenom., Vol. 4, no. 2, pp. 139–163, 1982.
   Google Scholar
• J. Gibbon and M. McGuinness, “The real and complex Lorenz equations in rotating fluids and lasers,” Phys. D: Nonlinear Phenom., Vol. 5, no. 1, pp. 108–122, 1982.
   Google Scholar
• H. Zeghlache and P. Mandel, “Influence of detuning on the properties of laser equations,” JOSA B, Vol. 2, no. 1, pp. 18–22, 1985.
   Web of Science ®Google Scholar
• C.-z. Ning and H. Haken, “Detuned lasers and the complex Lorenz equations: subcritical and supercritical Hopf bifurcations,” Phys. Rev. A, Vol. 41, no. 7, pp. 3826–3837.1990.
   PubMed Web of Science ®Google Scholar
• T. Živković and K. Rypdal, “Experimental evidence of low-dimensional chaotic convection dynamics in a toroidal magnetized plasma,” Phys. Rev. E, Vol. 77, no. 3, p. 037401, 2008.
   Web of Science ®Google Scholar
• A. E. Bell, “The Horologium oscillatorium of christian huygens,” Nature, Vol. 148, no. 3748, pp. 245–248, Aug. 1941. DOI: 10.1038/148245a0.
   Google Scholar
• C. M. Gray, “Synchronous oscillations in neuronal systems: mechanisms and functions,” J. Comput. Neurosci., Vol. 1, no. 1-2, pp. 11–38, 1994.
   PubMedGoogle Scholar
• S. H. Strogatz and I. Stewart, “Coupled oscillators and biological synchronization,” Sci. Am., Vol. 269, no. 6, pp. 102–109, 1993.
   PubMed Web of Science ®Google Scholar
• L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett., Vol. 64, no. 8, pp. 821–824.1990.
   PubMed Web of Science ®Google Scholar
• G. D. Vanwiggeren and R. Roy, “Communication with chaotic lasers,” Science, Vol. 279, no. 5354, pp. 1198–1200, 1998.
   PubMed Web of Science ®Google Scholar
• A. K. Jaiswal and B. B. Sharma, “Synchronisation scheme for cluster-based interconnected network of nonlinear systems,” Int. J. Syst., Control Commun., Vol. 15, no. 1, pp. 1–16, 2024.
   Google Scholar
• L. Kuhnert, K. Agladze, and V. Krinsky, “Image processing using light-sensitive chemical waves,” Nature, Vol. 337, no. 6204, pp. 244–247, 1989.
   Web of Science ®Google Scholar
• A. Pogromsky, T. Glad, and H. Nijmeijer, “On diffusion driven oscillations in coupled dynamical systems,” Int. J. Bifurc. Chaos, Vol. 9, no. 04, pp. 629–644, 1999.
   Google Scholar
• S. Boccaletti, J. Kurths, G. Osipov, D. Valladares, and C. Zhou, “The synchronization of chaotic systems,” Phys. Rep., Vol. 366, no. 1-2, pp. 1–101, 2002.
   Web of Science ®Google Scholar
• B. Sharma and I. Kar, “Observer-based synchronization scheme for a class of chaotic systems using contraction theory,” Nonlinear Dyn., Vol. 63, pp. 429–445, 2011.
   Web of Science ®Google Scholar
• M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “From phase to lag synchronization in coupled chaotic oscillators,” Phys. Rev. Lett., Vol. 78, no. 22, pp. 4193–4196.1997.
   Web of Science ®Google Scholar
• A. K. Jaiswal, Y. Chauhan, and B. B. Sharma, “Cluster synchronization conditions for coupled network of a class of nonlinear systems using partial contraction,” 2023, pp. 1–6.
   Google Scholar
• J. Zhou, T. Chen, and L. Xiang, “Chaotic lag synchronization of coupled delayed neural networks and its applications in secure communication,” Circuits Syst. Signal Process., Vol. 24, pp. 599–613, 2005.
   Web of Science ®Google Scholar
• B. B. Sharma and I. N. Kar, “Chaotic synchronization and secure communication using contraction theory,” in Pattern Recognition and Machine Intelligence, Berlin, Heidelberg: Springer, 2009, pp. 549–554.
   Google Scholar
• Z. Duan, G. Chen, and L. Huang, “Complex network synchronizability: analysis and control,” Phys. Rev. E, Vol. 76, no. 5, p. 056103, 2007.
   Web of Science ®Google Scholar
• S. Zhai and W. X. Zheng, “Stability conditions for cluster synchronization in directed networks of diffusively coupled nonlinear systems,” IEEE Trans. Circuits Syst. I: Regul. Pap., Vol. 70, no. 1, pp. 413–423, 2022.
   Web of Science ®Google Scholar
• F. Sorrentino and E. Ott, “Network synchronization of groups,” Phys. Rev. E, Vol. 76, no. 5, p. 056114, 2007.
   Web of Science ®Google Scholar
• L. Chen and J. Lu, “Cluster synchronization in a complex dynamical network with two nonidentical clusters,” J. Syst. Sci. Complex., Vol. 21, no. 1, pp. 20–33, 2008.
   Web of Science ®Google Scholar
• Y. Wang and J. Cao, “Cluster synchronization in nonlinearly coupled delayed networks of non-identical dynamic systems,” Nonlinear Anal. Real World Appl., Vol. 14, no. 1, pp. 842–851, 2013.
   Web of Science ®Google Scholar
• J.-S. Wu, L.-C. Jiao, and G.-R. Chen, “Cluster synchronization in a network of non-identical dynamic systems,” Chin. Phys. B, Vol. 20, no. 6, p. 060503, 2011.
   Web of Science ®Google Scholar
• I. Ahmad, A. B. Saaban, A. B. Ibrahim, and M. Shahzad, “Global chaos synchronization of new chaotic system using linear active control,” Complexity, Vol. 21, no. 1, pp. 379–386, 2015.
   Web of Science ®Google Scholar
• L. Liu and Q. Liu, “Cluster synchronization in a complex dynamical network of non-identical nodes with delayed and non-delayed coupling via pinning control,” Phys. Scr., Vol. 94, no. 4, p. 045204, 2019.
   Web of Science ®Google Scholar
• X. Zhu, D. Ding, and Z. Tang, “Cluster synchronization of nonlinearly coupled lur'e networks with non-identical nodes under adaptive pinning control,” in 2022 41st Chinese Control Conference (CCC). IEEE, Jun. 2022, pp. 67–72. DOI: 10.23919/ccc55666.2022.9901864.
   Google Scholar
• M. Derakhshannia and S. S. Moosapour, “Disturbance observer-based sliding mode control for consensus tracking of chaotic nonlinear multi-agent systems,” Math. Comput. Simul., Vol. 194, pp. 610–628, 2022.
   Web of Science ®Google Scholar
• P. Gaspard, “Rössler systems,” Encycl. Nonlinear Sci., Vol. 231, pp. 808–811, 2005.
   Google Scholar
• L. O. Chua, “Chua's circuit: an overview ten years later,” J. Circuits Syst. Comput., Vol. 4, no. 02, pp. 117–159, 1994.
   Google Scholar
• A. G. Radwan, “On some generalized discrete logistic maps,” J. Adv. Res., Vol. 4, no. 2, pp. 163–171, 2013.
   PubMedGoogle Scholar
• Y. Peng, K. Sun, and S. He, “A discrete memristor model and its application in hénon map,” Chaos Solit. Fractals, Vol. 137, p. 109873, 2020.
   Web of Science ®Google Scholar
• A. Ouannas, A.-A. Khennaoui, Z. Odibat, V.-T. Pham, and G. Grassi, “On the dynamics, control and synchronization of fractional-order Ikeda map,” Chaos Solit. Fractals, Vol. 123, pp. 108–115, 2019.
   Web of Science ®Google Scholar
• J. K. Moser, “Lectures on Hamiltonian systems,” in Hamiltonian Dynamical Systems. CRC Press, Aug. 2020, pp. 77–136. DOI: 10.1201/9781003069515-7.
   Google Scholar
• I. Podlubny, “Fractional-order systems and fractional-order controllers,” Inst. Exp. Phys. Slovak Acad. Sci. Kosice, Vol. 12, no. 3, pp. 1–18, 1994.
   Google Scholar
• J. A. Suykens, P. F. Curran, and L. O. Chua, “Robust synthesis for master-slave synchronization of lur'e systems,” IEEE Trans. Circuits Syst. I: Fundam. Theory Appl., Vol. 46, no. 7, pp. 841–850, 1999.
   Google Scholar
• C. Li, W. Sun, and J. Kurths, “Synchronization between two coupled complex networks,” Phys. Rev. E, Vol. 76, no. 4, p. 046204, 2007.
   Web of Science ®Google Scholar
• L. Zhou, F. Tan, X. Li, and L. Zhou, “A fixed-time synchronization-based secure communication scheme for two-layer hybrid coupled networks,” Neurocomputing, Vol. 433, pp. 131–141, 2021.
   Web of Science ®Google Scholar
• L. Zhou and F. Tan, “A chaotic secure communication scheme based on synchronization of double-layered and multiple complex networks,” Nonlinear Dyn., Vol. 96, pp. 869–883, 2019.
   Web of Science ®Google Scholar
• S. Vaidyanathan and S. Rasappan, “Global chaos synchronization of n-scroll Chua circuit and lur'e system using backstepping control design with recursive feedback,” Arab. J. Sci. Eng., Vol. 39, pp. 3351–3364, 2014.
   Web of Science ®Google Scholar
• F. A. Rodrigues, T. K. D. Peron, P. Ji, and J. Kurths, “The Kuramoto model in complex networks,” Phys. Rep., Vol. 610, pp. 1–98, 2016.
   Web of Science ®Google Scholar
• M. Willis, “Proportional-integral-derivative control,” Dept. of Chemical and Process Engineering University of Newcastle, 1999.
   Google Scholar
• M. M. Zirkohi and S. Khorashadizadeh, “Chaos synchronization using higher-order adaptive PID controller,” AEU – Int. J. Electron. Commun., Vol. 94, pp. 157–167, 2018.
   Web of Science ®Google Scholar
• B. Wittenmark, “Adaptive dual control methods: An overview,” in Adaptive Systems in Control and Signal Processing 1995, Elsevier, 1995, pp. 67–72. DOI: 10.1016/b978-0-08-042375-3.50010-x.
   Google Scholar
• Y. Shtessel, C. Edwards, L. Fridman, and A. Levant, Sliding Mode Control and Observation, Vol. 10, New York: Springer, 2014. DOI: 10.1007/978-0-8176-4893-0.
   Google Scholar
• H. Hou, Q. Zhang, and M. Zheng, “Cluster synchronization in nonlinear complex networks under sliding mode control,” Nonlinear Dyn., Vol. 83, pp. 739–749, 2016.
   Web of Science ®Google Scholar
• C. Yi and Y. Yang, “Cluster synchronization in a heterogeneous network with mixed coupling via event-triggered and optimizing pinning control,” Neural Process. Lett., Vol. 55, no. 7, pp. 8785–8808, Feb. 2023. DOI: 10.1007/s11063-023-11177-5.
   Web of Science ®Google Scholar
• X. Wang, J.-A. Wang, Q. Sun, X. Wen, and J. Zhang, “Synchronization of nonlinear coupled complex network via event-dependent intermittent pinning control,” Eur. J. Control, Vol. 69, p. 100757, 2023.
   Web of Science ®Google Scholar
• W. Li, J. Zhou, J. Li, T. Xie, and J.-A. Lu, “Cluster synchronization of two-layer networks via aperiodically intermittent pinning control,” IEEE Trans. Circuits Syst. II Express Briefs, Vol. 68, no. 4, pp. 1338–1342, 2020.
   Web of Science ®Google Scholar
• W. Wu, W. Zhou, and T. Chen, “Cluster synchronization of linearly coupled complex networks under pinning control,” IEEE Trans. Circuits Syst. I: Regul. Pap., Vol. 56, no. 4, pp. 829–839, 2008.
   Web of Science ®Google Scholar
• Q. Song, J. Cao, and W. Yu, “Second-order leader-following consensus of nonlinear multi-agent systems via pinning control,” Syst. Control Lett., Vol. 59, no. 9, pp. 553–562, 2010.
   Web of Science ®Google Scholar
• S. Changder and B. B. Sharma, “Synchronisation among multiple chaotic systems connected in chain and star configuration using backstepping strategy,” Int. J. Syst. Control Commun., Vol. 14, no. 1, pp. 60–80, 2023.
   Google Scholar
• J. Lu, J. Cao, and D. W. Ho, “Adaptive stabilization and synchronization for chaotic lur'e systems with time-varying delay,” IEEE Trans. Circuits Syst. I: Regul. Pap., Vol. 55, no. 5, pp. 1347–1356, 2008.
   Web of Science ®Google Scholar
• P. Anand and B. B. Sharma, “Generalized finite-time synchronization scheme for a class of nonlinear systems using backstepping like control strategy,” Int. J. Dyn. Contr., Vol. 11, no. 1, pp. 258–270, 2023.
   Google Scholar
• R. K. Ranjan and B. B. Sharma, “Reduced-order observer-based synchronization and output tracking in chain network of a class of nonlinear systems using contraction framework,” Int. J. Dyn. Contr., pp. 1–15, 2023.
   Google Scholar
• Y. Yu and H.-X. Li, “Adaptive hybrid projective synchronization of uncertain chaotic systems based on backstepping design,” Nonlinear Anal. Real World Appl., Vol. 12, no. 1, pp. 388–393, 2011.
   Web of Science ®Google Scholar
• C. P. Chen, G.-X. Wen, Y.-J. Liu, and Z. Liu, “Observer-based adaptive backstepping consensus tracking control for high-order nonlinear semi-strict-feedback multiagent systems,” IEEE Trans. Cybern., Vol. 46, no. 7, pp. 1591–1601, 2015.
   PubMed Web of Science ®Google Scholar
• B. Hou, F. Sun, H. Li, Y. Chen, and G. Liu, “Observer-based cluster consensus control of high-order multi-agent systems,” Neurocomputing, Vol. 168, pp. 979–982, 2015.
   Web of Science ®Google Scholar
• Z. Li and G. Chen, “Global synchronization and asymptotic stability of complex dynamical networks,” IEEE Trans. Circuits Syst. II Express Briefs, Vol. 53, no. 1, pp. 28–33, 2006.
   Web of Science ®Google Scholar
• V. Agrawal and B. B. Sharma, “An observer based approach for multi-scroll chaotic system synchronization and secure communication with multi-shift ciphering,” in 2014 International Conference on Advances in Engineering & Technology Research (ICAETR-2014). IEEE, Apr. 2014, pp. 1–6. DOI: 10.1109/icaetr.2014.7012905.
   Google Scholar
• M. K. Vaishnav and B. B. Sharma, “Observer design & synchronization of fourth order Lorenz Stenflo chaotic system,” in 2013 Students Conference on Engineering and Systems (SCES). IEEE, Apr. 2013, pp. 1–6. DOI: 10.1109/sces.2013.6547550.
   Google Scholar
• J. Wang, Q. Ma, and L. Zeng, “Observer-based synchronization in fractional-order leader–follower complex networks,” Nonlinear Dyn., Vol. 73, pp. 921–929, 2013.
   Web of Science ®Google Scholar
• T.-L. Liao and N.-S. Huang, “An observer-based approach for chaotic synchronization with applications to secure communications,” IEEE Trans. Circuits Syst. I: Fundam. Theory Appl., Vol. 46, no. 9, pp. 1144–1150, 1999.
   Google Scholar
• J. Liu, S. Vazquez, L. Wu, A. Marquez, H. Gao, and L. G. Franquelo, “Extended state observer-based sliding-mode control for three-phase power converters,” IEEE Trans. Ind. Electron., Vol. 64, no. 1, pp. 22–31, 2016.
   Web of Science ®Google Scholar
• J. Jouffroy and J.-J. Slotine, “Methodological remarks on contraction theory,” in 2004 43rd IEEE Conference on Decision and Control (CDC)(IEEE Cat. No. 04CH37601), Vol. 3. IEEE, 2004, pp. 2537–2543. DOI: 10.1109/cdc.2004.1428824.
   Google Scholar
• B. B. Sharma and I. N. Kar, “Contraction theory based adaptive synchronization of chaotic systems,” Chaos Solit. Fractals, Vol. 41, no. 5, pp. 2437–2447, 2009.
   Web of Science ®Google Scholar
• Y. Chauhan and B. B. Sharma, “Synchronisation conditions for a class of non-linearly coupled networks using contraction based strategy,” 2021.
   Google Scholar
• Z. Aminzare, B. Dey, E. N. Davison, and N. E. Leonard, “Cluster synchronization of diffusively coupled nonlinear systems: A contraction-based approach,” J. Nonlinear Sci., Vol. 30, pp. 2235–2257, 2020.
   Web of Science ®Google Scholar
• Y. Chauhan and B. B. Sharma, “Design of coupling function for synchronization of linear & nonlinear systems in chain formation using contraction based approach,” in 2020 IEEE 5th International Conference on Computing Communication and Automation (ICCCA). IEEE, Oct. 2020, pp. 203–209. DOI: 10.1109/iccca49541.2020.9250827.
   Google Scholar
• W. Wang and J.-J. E. Slotine, “On partial contraction analysis for coupled nonlinear oscillators,” Biol. Cybern., Vol. 92, no. 1, pp. 38–53, 2005.
   PubMed Web of Science ®Google Scholar
• G. Wen, W. Yu, G. Hu, J. Cao, and X. Yu, “Pinning synchronization of directed networks with switching topologies: A multiple Lyapunov functions approach,” IEEE Trans. Neural Netw. Learn. Syst., Vol. 26, no. 12, pp. 3239–3250, 2015.
   PubMed Web of Science ®Google Scholar
• L. Zhou, C. Wang, and L. Zhou, “Cluster synchronization on multiple sub-networks of complex networks with nonidentical nodes via pinning control,” Nonlinear Dyn., Vol. 83, pp. 1079–1100, 2016.
   Web of Science ®Google Scholar
• K. Wang, X. Fu, and K. Li, “Cluster synchronization in community networks with nonidentical nodes,” Chaos, Vol. 19, no. 2, May 2009. DOI: 10.1063/1.3125714.
   Web of Science ®Google Scholar
• S. C. Manrubia and A. S. Mikhailov, “Mutual synchronization and clustering in randomly coupled chaotic dynamical networks,” Phys. Rev. E, Vol. 60, no. 2, pp. 1579–1589.1999.
   Web of Science ®Google Scholar
• L. Zhang, W. Pan, L. Yan, B. Luo, X. Zou, and M. Xu, “Cluster synchronization of coupled semiconductor lasers network with complex topology,” IEEE J. Sel. Top. Quantum Electron., Vol. 25, no. 6, pp. 1–7, 2019.
   Web of Science ®Google Scholar
• L. Huang, Y.-C. Lai, and R. A. Gatenby, “Optimization of synchronization in complex clustered networks,” Chaos, Vol. 18, no. 1, Jan. 2008. DOI: 10.1063/1.2826289.
   Web of Science ®Google Scholar
• A. C. Ferreira, M. A. Vilaça, L. B. Oliveira, E. Habib, H. C. Wong, and A. A. Loureiro, “On the security of cluster-based communication protocols for wireless sensor networks,” in Networking-ICN 2005: 4th International Conference on Networking, Reunion Island, France, April –21, 2005, Proceedings, Part I 4. Berlin, Heidelberg: Springer, 2005, pp. 449–458. DOI: 10.1007/978-3-540-31956-6_53.
   Google Scholar
• X. Li, L. Zhou, and F. Tan, “An image encryption scheme based on finite-time cluster synchronization of two-layer complex dynamic networks,” Soft Comput., Vol. 26, no. 2, pp. 511–525, 2022.
   Web of Science ®Google Scholar
• J.-Y. Li, Z. Wang, R. Lu, and Y. Xu, “Cluster synchronization control for discrete-time complex dynamical networks: when data transmission meets constrained bit rate,” IEEE Trans. Neural. Netw. Learn. Syst., Vol. 34, no. 5, pp. 2554–2568, 2021.
   Web of Science ®Google Scholar
• L. Zhou, X. Li, F. Tan, Y. Huang, and W. Ma, “A two-layer networks-based audio encryption/decryption scheme via fixed-time cluster synchronization,” Soft Comput., Vol. 26, no. 19, pp. 9761–9774, 2022.
   Web of Science ®Google Scholar
• H. Wang, X. Yang, Z. Xiang, R. Tang, and Q. Ning, “Synchronization of switched neural networks via attacked mode-dependent event-triggered control and its application in image encryption,” IEEE Trans. Cybern., Vol. 53, no. 9, pp. 5994–6003, Sep. 2023. DOI: 10.1109/tcyb.2022.3227021.
   PubMed Web of Science ®Google Scholar
• D. Sun, C. Wang, W. Shang, and G. Feng, “A synchronization approach to trajectory tracking of multiple mobile robots while maintaining time-varying formations,” IEEE Trans. Robot., Vol. 25, no. 5, pp. 1074–1086, 2009.
   Web of Science ®Google Scholar