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Delayed feedback on the 3-D chaotic system only with two stable node-foci. (English) Zbl 1238.34122
Summary: We investigate the effect of delayed feedbacks on the 3-D chaotic system only with two stable node-foci by Q. Yang, Zh. Wei and G. Chen [Int. J. Bifurcation Chaos Appl. Sci. Eng. 20, No. 4, 1061–1083 (2010; Zbl 1193.34091)]. The stability of equilibria and the existence of Hopf bifurcations are considered. The explicit formulas determining the direction, stability and period of the bifurcating periodic solutions are obtained by employing the normal form theory and the center manifold theorem. Numerical simulations and experimental results are given to verify the theoretical analysis. Hopf bifurcation analysis can explain and predict the periodic orbit in the chaotic system with direct time delay feedback. We also find that the control law can be applied to the chaotic system only with two stable node-foci for the purpose of control and anti-control of chaos. Finally, some numerical simulations are given to illustrate the effectiveness of the results found.
MSC:
34H10Chaos control (ODE)
34K35Functional-differential equations connected with control problems
93B52Feedback control
37D45Strange attractors, chaotic dynamics
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