Adaptive control of bifurcation and chaos in a time-delayed system

In this paper, the stabilization of a continuous time-delayed system is considered. To control the bifurcation and chaos in a time-delayed system, a parameter perturbation control and a hybrid control are proposed. Then, to ensure the asymptotic stability of the system in the presence of unexpected system parameter changes, the adaptive control idea is introduced, i.e., the perturbation control parameter and the hybrid control parameter are automatically tuned according to the adaptation laws, respectively. The adaptation algorithms are constructed based on the Lyapunov-Krasovskii stability theorem. The adaptive parameter perturbation control and the adaptive hybrid control methods improve the corresponding constant control methods. They have the advantages of increased stability, adaptability to the changes of the system parameters, control cost saving, and simplicity. Numerical simulations for a well-known chaotic time-delayed system are performed to demonstrate the feasibility and superiority of the proposed control methods. Besides, comparison of the two adaptive control methods are made in an experimental study.     

Bifurcations and chaos control in discrete small-world networks

An impulsive delayed feedback control strategy to control period-doubling bifurcations and chaos is proposed. The control method is then applied to a discrete small-world network model. Qualitative analyses and simulations show that under a generic condition, the bifurcations and the chaos can be delayed or eliminated completely. In addition, the periodic orbits embedded in the chaotic attractor can be stabilized.