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Bifurcation analysis for Chen’s system with delayed feedback and its application to control of chaos. (English) Zbl 1112.37303

Summary: Time-delayed feedback has been introduced as a powerful tool for control of unstable periodic orbits or control of unstable steady states. In the present paper, regarding the delay as parameter, we investigate the effect of delay on the dynamics of Chen’s system with delayed feedback. We first consider the effect of delay on the steady states, and then investigate the existence of local Hopf bifurcations. By using the normal form theory and center manifold argument, we derive the explicit formulas determining the stability, direction and other properties of bifurcating periodic solutions. Finally, we give several numerical simulations, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a stable periodic orbit.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34K18 Bifurcation theory of functional-differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
93C23 Control/observation systems governed by functional-differential equations
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