Abstract
We investigate the local Hopf bifurcation in Genesio system with delayed feedback control. We choose the delay as the parameter, and the occurrence of local Hopf bifurcations are verified. By using the normal form theory and the center manifold theorem, we obtain the explicit formulae for determining the stability and direction of bifurcated periodic solutions. Numerical simulations indicate that delayed feedback control plays an effective role in control of chaos.
Highlights
Since the pioneering work of Lorenz 1, much attention has been paid to the study of chaos
In order to gain further insights on the control of chaos via time-delayed feedback, in this paper, we aim to investigate the dynamical behaviors of Genesio system with timedelayed controlling forces
Genesio system, which was proposed by Genesio and Tesi 2, is described by the following simple three-dimensional autonomous system with only one quadratic nonlinear term: xy, yz, 1.1 zax by cz x2, ISRN Mathematical Physics z(t)
Summary
Since the pioneering work of Lorenz 1 , much attention has been paid to the study of chaos. There are many methods in controlling chaos, among which using time-delayed controlling forces serves as a good and simple one. In order to gain further insights on the control of chaos via time-delayed feedback, in this paper, we aim to investigate the dynamical behaviors of Genesio system with timedelayed controlling forces. Many researchers have studied this system from many different points of view; Park et al 3–5 investigated synchronization of the Genesio chaotic system via backstepping approach, LMI optimization approach, and adaptive controller design. Chen and Han 7 investigated controlling and synchronization of Genesio chaotic system via nonlinear feedback control. Inspired by the control of chaos via time-delayed feedback force 8 and following the idea of Pyragas 9 , we consider the following Genesio system with delayed feedback control: xt y t , ytztMyt − y t − τ , 1.2 zt ax t by t cz t x2 t , where τ > 0 and M ∈ R
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