Abstract
We show that time-delayed feedback methods, which have successfully been used to control unstable steady states or periodic orbits, provide a tool to control Hopf bifurcation for a small-world network model with nonlinear interactions and time delays. We choose the interaction strength parameter as a bifurcation parameter. Without control, bifurcation will occur early; meanwhile, the model can maintain a stationary total influenced volume only in a certain domain of the interaction strength parameter. However, outside of this domain the model still possesses a stable total influenced volume that can be guaranteed by delayed feedback perturbation, and the onset of the Hopf bifurcation is postponed. The feedback perturbation vanishes if the stabilization is successful and thus the domain of stability can be extended under only small control force. We present an analytical investigation of the feedback scheme using characteristic equation and discuss effects of both a low-pass filter included in the control loop and nonzero latency times associated with generation and injection of the feedback signal.
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The research of M. Xiao was jointly supported by the project of Jiangsu Province of China, and the Fundamental Discipline Construction Foundation of Nanjing Xiaozhuang University.
The research of D.W.C. Ho was supported by CityU SRG 7002355.
The research of J. Cao was jointly supported by the National Natural Science Foundation of China under Grant No. 60874088, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20070286003, and 333 project of Jiangsu Province of China.
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Xiao, M., Ho, D.W.C. & Cao, J. Time-delayed feedback control of dynamical small-world networks at Hopf bifurcation. Nonlinear Dyn 58, 319–344 (2009). https://doi.org/10.1007/s11071-009-9485-0
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DOI: https://doi.org/10.1007/s11071-009-9485-0