论文标题
稳定性和HOPF分叉分析对人呼吸系统建模两个状态延迟差分方程
Stability and Hopf bifurcation analysis of a two state delay differential equation modeling the human respiratory system
论文作者
论文摘要
我们研究了描述二氧化碳和氧的平衡方程的两个状态模型。这些是非线性参数依赖性的，并且由于呼吸控制系统的传输延迟，它们以延迟微分方程为模型。因此，研究了两个状态的一个延迟模型的动力学。通过选择延迟作为参数，可以获得稳定性和HOPF分叉条件。我们注意到，随着延迟通过其临界值，正平衡失去了其稳定性，而HOPF分叉发生。还绘制了针对其他参数和分叉图的系统的稳定区域。构建了两个状态模型的三维稳定性图。我们发现延迟参数对稳定性有效，但对平衡状态没有影响。 HOPF分叉方向的显式推导以及分叉周期溶液的稳定性是在正常形式理论和中心歧管定理延迟微分方程的帮助下确定的。最后，进行了一些数值示例和模拟以确认分析结果。数值模拟验证了理论结果。
We study the two state model which describes the balance equation for carbon dioxide and oxygen. These are nonlinear parameter dependent and because of the transport delay in the respiratory control system, they are modeled with delay differential equation. So, the dynamics of a two state one delay model are investigated. By choosing the delay as a parameter, the stability and Hopf bifurcation conditions are obtained. We notice that as the delay passes through its critical value, the positive equilibrium loses its stability and Hopf bifurcation occurs. The stable region of the system with delay against the other parameters and bifurcation diagrams are also plotted. The three dimensional stability chart of the two state model is constructed. We find that the delay parameter has effect on the stability but not on the equilibrium state. The explicit derivation of the direction of Hopf bifurcation and the stability of the bifurcation periodic solutions are determined with the help of normal form theory and center manifold theorem to delay differential equations. Finally, some numerical example and simulations are carried out to confirm the analytical findings. The numerical simulations verify the theoretical results.
