Abstract
We discuss the dynamic behavior of a new Lorenz-like chaotic system with distributed delayed feedback by the qualitative analysis and numerical simulations. It is verified that the equilibria are locally asymptotically stable whenα∈(0,α0)and unstable whenα∈(α0,∞); Hopf bifurcation occurs whenαcrosses a critical valueα0by choosingαas a bifurcation parameter. Meanwhile, the explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Furthermore, regardingαas a bifurcation parameter, we explore variation tendency of the dynamics behavior of a chaotic system with the increase of the parameter valueα.
Highlights
As one of the important discoveries in 21st century, chaos has been extensively investigated in many fields over the last several decades, which has been widely applied in secure communication, signal processing, radar, image processing, power system protection, flow dynamics, and so on
Many schemes have been presented to carry out chaos control [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] of which using time-delayed controlling forces proves to be a simple and viable method for a continuous dynamical system
Theorem 4. μ2 determines the direction of the Hopf bifurcation, if μ2 > 0 (μ2 < 0); β2 determines the stability of the bifurcating period solutions: the bifurcation period solutions are orbitally stable if β2 < 0 (β2 > 0), and T2 determines the period of the bifurcating periodic solutions: the period increases if T2 > 0 (T2 < 0)
Summary
As one of the important discoveries in 21st century, chaos has been extensively investigated in many fields over the last several decades, which has been widely applied in secure communication, signal processing, radar, image processing, power system protection, flow dynamics, and so on. The conditions of Hopf bifurcation occurring and the stability analysis of the equilibrium points have been studied in detail in [19]. The signal error of Mathematical Problems in Engineering current state and past state of the continuous time system will be given distributed delay feedback to the system itself. We present the Hopf bifurcation of a Lorenz-like system with the distributed delay. We display numerical simulation of Hopf bifurcation in delayed feedback Lorenz-like system, and give the theoretical proof. Regarding the delay variable τ as a branch of parameters, when τ passes through a critical value, the stability of the equilibrium point will change from instability to stability, and the chaos phenomenon of the system disappears.
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