Abstract
In this article, for delayed Nicholson’s blowflies equation, we propose a hybrid control nonstandard finite-difference (NSFD) scheme in which state feedback and parameter perturbation are used to control the Neimark-Sacker bifurcation. Firstly, the local stability of the positive equilibria for hybrid control delay differential equation is discussed according to Hopf bifurcation theory. Then, for any step-size, a hybrid control numerical algorithm is introduced to generate the Neimark-Sacker bifurcation at a desired point. Finally, numerical simulation results confirm that the control strategy is efficient in controlling the Neimark-Sacker bifurcation. At the same time, the results show that the NSFD control scheme is better than the Euler control method.
Highlights
The delay differential equation (DDE)x(t) = ax(t – τ )e–bx(t–τ) – cx(t), ( . )which is one of the important ecological systems, describes the dynamics of Nicholson’s blowflies equation
The results show that the dynamic behavior of a controlled system can be changed by choosing appropriate control parameters
The results show that the nonstandard finite-difference (NSFD) control scheme is better than the Euler control method
Summary
We construct a hybrid control nonstandard finite-difference scheme in which state feedback and parameter perturbation are used to control the Neimark-Sacker bifurcations. In Section , we analyze the distribution of the characteristic equation associated with a hybrid control delay differential equation with Nicholson’s blowflies equation, and we obtain local stability of the equilibria and existence of the Hopf bifurcation. In Section , the direction and stability of bifurcating periodic solutions from the Neimark-Sacker bifurcation of a controlled delay equation are determined by using the theories of discrete systems. Proof A Neimark-Sacker bifurcation occurs when two roots of the characteristic equation
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