翻译成中文In dynamical systems theory, a stable manifold is a set of points in phase space that converges towards a stable equilibrium point or limit cycle. More specifically, it is a set of initial conditions for which the system will converge towards the stable equilibrium or limit cycle. The stable manifold can be thought of as a geometric structure that characterizes the behavior of the system near the equilibrium or limit cycle. In the context of the HH neuron model discussed in the referenced article, the stable manifold is the set of initial conditions for which the system will converge towards the stable resting state of the neuron. This is because the stable resting state is the only stable equilibrium point of the system. Trajectories that start to the left of the stable manifold will quickly converge towards the stable resting state, while trajectories that start to the right of the stable manifold will diverge from the resting state and eventually converge towards the unstable equilibrium point. Overall, the stable manifold is a fundamental concept in dynamical systems theory and plays an important role in understanding the behavior of nonlinear systems such as the HH neuron model.

稳定流形是动力系统理论中的一组相空间点，它们收敛于稳定的均衡点或极限周期。更具体地说，它是一组系统将收敛到稳定均衡或极限周期的初始条件。稳定流形可以被认为是一种几何结构，它描述了系统在均衡或极限周期附近的行为。在参考文章中讨论的HH神经元模型的上下文中，稳定流形是一组系统将收敛到神经元稳定静止状态的初始条件。这是因为稳定静止状态是系统唯一的稳定均衡点。在稳定流形的左侧开始的轨道将迅速收敛到稳定静止状态，而在稳定流形的右侧开始的轨道将从静止状态分离，最终收敛到不稳定的均衡点。总的来说，稳定流形是动力系统理论中的一个基本概念，在理解非线性系统（如HH神经元模型）的行为中起着重要作用。

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帮我地道的翻译：The differential variational inequalities ((DVIs), for short) are useful for the study of models involving both dynamics and constraints in the form of in￾equalities. They arise in many applications: electrical circuits with ideal diodes, Coulomb friction problems for contacting bodies, economical dynamics, dynamic traffic networks. Pang and Stewart [26], [27] established the existence, unique￾ness, and Lipschitz dependence of solutions subject to boundary conditions for (DVIs) in finite dimensional spaces. Han and Pang investigated a class of dif￾ferential quasi-variational inequalities in [11], and Li, Huang and O’Regan [18] studied a class of differential mixed variational inequalities in finite dimensional Well-Posedness of Differential Mixed Quasi-Variational-Inequalities 137 spaces. Gwinner [8] obtained an equivalence result between (DVIs) and projected dynamical systems. In [9] he also proved a stability property for (DVIs) by using the monotonicity method of Browder and Minty, and Mosco set convergence. Chen and Wang [4] studied dynamic Nash equilibrium problems which have the formulation of differential mixed quasi-variational inequalities. Elastoplastic contact problems can also be incorporated into (DMQVIs) formulation because general dynamic processes in the nonsmooth unilateral contact problems are governed by quasi-variational inequalities. A numerical study for nonsmooth contact problems with Tresca friction can be found in [10], Liu, Loi and Obukhovskii [19] studied the existence and global bifurcation for periodic solutions of a class of (DVIs) by using the topological degree theory for multivalued maps and the method of guiding functions. For more details about (DVIs) we refer to [3], [30], [12], [22]–[21].

差分变分不等式(DVIs)对于研究涉及动力学和不等式约束的模型非常有用。它们出现在许多应用中：带有理想二极管的电路、接触体的库仑摩擦问题、经济动力学、动态交通网络。Pang和Stewart(26,27)在有限维空间中建立了(DVIs)解的存在、唯一性和利普希茨依赖性的边界条件。Han和Pang在(11)中调查了一类差分拟变分不等式，Li、Huang和O'Regan在有限维空间中研究了一类差分混合变分不等式。Gwinner(8)得到了(DVIs)和投影动力系统之间的等价结果。在(9)中，他还通过使用Browder和Minty的单调性方法以及Mosco集收敛法证明了(DVIs)的稳定性性质。Chen和Wang(4)研究了动态Nash均衡问题，其公式为差分混合拟变分不等式。弹塑性接触问题也可以并入(DMQVIs)公式，因为非光滑单向接触问题中的一般动态过程受到拟变分不等式的控制。在(10)中可以找到Tresca摩擦下的非光滑接触问题的数值研究，Liu、Loi和Obukhovskii在(19)中使用多值映射的拓扑度理论和引导函数法研究了一类(DVIs)的周期解的存在和全局分支。关于(DVIs)的更多细节，我们可以参考(3)，(30)，(12)，(22)-(21)。