Song Zigen, Zhen Bin, Hu Dongpo
1College of Information Technology, Shanghai Ocean University, Shanghai, 201306 China.
2School of Environment and Architecture, University of Shanghai for Science and Technology, Shanghai, 200093 China.
Cogn Neurodyn. 2020 Jun;14(3):359-374. doi: 10.1007/s11571-020-09575-9. Epub 2020 Mar 6.
In this paper, we construct an inertial two-neuron system with multiple delays, which is described by three first-order delayed differential equations. The neural system presents dynamical coexistence with equilibria, periodic orbits, and even quasi-periodic behavior by employing multiple types of bifurcations. To this end, the pitchfork bifurcation of trivial equilibrium is analyzed firstly by using center manifold reduction and normal form method. The system presents different sequences of supercritical and subcritical pitchfork bifurcations. Further, the nontrivial equilibrium bifurcated from trivial equilibrium presents a secondary pitchfork bifurcation. The system exhibits stable coexistence of multiple equilibria. Using the pitchfork bifurcation curves, we divide the parameter plane into different regions, corresponding to different number of equilibria. To obtain the effect of time delays on system dynamical behaviors, we analyze equilibrium stability employing characteristic equation of the system. By the Hopf bifurcation, the system illustrates a periodic orbit near the trivial equilibrium. We give the stability regions in the delayed plane to illustrate stability switching. The neural system is illustrated to have Hopf-Hopf bifurcation points. The coexistence with two periodic orbits is presented near these bifurcation points. Finally, we present some mixed dynamical coexistence. The system has a stable coexistence with periodic orbit and equilibrium near the pitchfork-Hopf bifurcation point. Moreover, multiple frequencies of the system induce the presentation of quasi-periodic behavior. The system presents stable coexistence with two periodic orbits and one quasi-periodic behavior.
在本文中，我们构建了一个具有多个延迟的惯性双神经元系统，该系统由三个一阶延迟微分方程描述。通过采用多种类型的分岔，该神经系统呈现出平衡点、周期轨道甚至准周期行为的动态共存。为此，首先利用中心流形约化和范式方法分析了平凡平衡点的叉形分岔。该系统呈现出超临界和亚临界叉形分岔的不同序列。此外，从平凡平衡点分岔出的非平凡平衡点呈现出二次叉形分岔。该系统展现出多个平衡点的稳定共存。利用叉形分岔曲线，我们将参数平面划分为不同区域，对应于不同数量的平衡点。为了获得时间延迟对系统动态行为的影响，我们利用系统的特征方程分析平衡点的稳定性。通过霍普夫分岔，该系统在平凡平衡点附近展示出一个周期轨道。我们给出延迟平面中的稳定区域以说明稳定性切换。该神经系统被证明具有霍普夫 - 霍普夫分岔点。在这些分岔点附近呈现出两个周期轨道的共存。最后，我们展示了一些混合动态共存情况。该系统在叉形 - 霍普夫分岔点附近具有周期轨道和平衡点的稳定共存。此外，系统的多个频率导致了准周期行为的出现。该系统呈现出两个周期轨道和一种准周期行为的稳定共存。
