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.

中文翻译：

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具有多个延迟的惯性两神经元系统中的多个分叉和共存。

在本文中，我们构造了一个具有多个时滞的惯性两神经元系统，该系统由三个一阶时滞微分方程描述。通过采用多种类型的分叉，神经系统呈现出与平衡，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简和正态形式方法分析了微分平衡的干草叉分叉。该系统呈现超临界和亚临界干草叉分叉的不同序列。此外，从平凡平衡分叉的非平凡平衡出现了次级干草叉分叉。该系统表现出多种平衡的稳定共存。使用干草叉分叉曲线，我们将参数平面划分为不同的区域，对应于不同数量的平衡。为了获得时延对系统动力学行为的影响，我们使用系统的特征方程分析了平衡稳定性。通过霍普夫分叉，该系统示出了平凡平衡附近的周期轨道。我们在延迟平面中给出稳定性区域，以说明稳定性切换。说明该神经系统具有Hopf–Hopf分叉点。在这些分叉点附近，存在与两个周期轨道的共存。最后，我们提出了一些混合动力共存。系统在干草叉-霍夫夫分叉点附近具有周期轨道和平衡的稳定共存。此外，系统的多个频率会引起准周期性行为的出现。