延迟Gompertz模型的数值分支和混合控制

为了研究物种的稳定性问题,要求缩小或者扩大生物系统的稳定区域,通过混合控制欧拉法研究了一个时滞Gompertz模型,运用状态反馈和参数扰动控制得到了Neimark-Sacker分支的理想结果。根据Hopf分支理论得到了连续系统平衡点的稳定性,通过混合控制欧拉算法得到了离散系统在要求的分支点所产生的Neimark-Sacker分支,利用中心流形定理和正规形方法,给出了确定分支周期解的分支方向与稳定性的计算公式。采用数值模拟验证了所得结果的正确性。研究结果表明,对于延迟Gompertz模型系统,如果选择合适的控制参数,就能够使分支点提前或者延迟。研究方法在理论和数值模拟方面都得到了良好的预期结果,为解决相关的控制问题提供了新的方法,对其他领域的控制问题研究具有一定的借鉴意义。

Numerical bifurcation and hybrid control of a Gompertz model with time delay

In order to study the stability of species, the biological systems are required to reduce or expand the stable region. For a Gompertz model with time delay, a hybrid control Euler method is proposed in which state feedback and parameter perturbation are used to control the Neimark-Sacker bifurcation. The local stability of the equilibria is discussed according to Hopf bifurcation theory. For controlling Neimark-Sacker bifurcation, the hybrid control numerical algorithm is introduced to generate the Neimark-Sacker bifurcation at a desired bifurcation point. The explicit algorithms for determining the direction of the bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form method and center manifold theorem. Numerical examples are provided to illustrate the theoretical results. The research results show that the branch point can be in advance or delay for the delay Gompertz model system through choosing appropriate control parameters. The algorithm has obtained good results both in theory and numerical performance, which provides a new method and has certain theoretical significance for its application in many control problems.