We study stability of so-called synchronous slowly oscillating periodic solutions (SOPSs) for a system of identical delay differential equations (DDEs) with linear decay and nonlinear delayed negative feedback that are coupled through their nonlinear term. Under a row sum condition on the coupling matrix, existence of a unique SOPS for the corresponding scalar DDE implies existence of a unique synchronous SOPS for the coupled DDEs. However, stability of the SOPS for the scalar DDE does not generally imply stability of the synchronous SOPS for the coupled DDEs. We obtain an explicit formula, depending only on the spectrum of the coupling matrix, the strength of the linear decay and the values of the nonlinear negative feedback function near plus/minus infinity, that determines the stability of the synchronous SOPS in the asymptotic regime where the nonlinear term is heavily weighted. We also treat the special cases of so-called weakly coupled systems, near uniformly coupled systems, and doubly nonnegative coupled systems, in the aforementioned asymptotic regime. Our approach is to estimate the characteristic (Floquet) multipliers for the synchronous SOPS. We first reduce the analysis of the multidimensional variational equation to the analysis of a family of scalar variational-type equations, and then establish limits for an associated family of monodromy-type operators. We illustrate our results with examples of systems of DDEs with mean-field coupling and systems of DDEs arranged in a ring.

对于具有线性衰减和非线性延迟负反馈且通过其非线性项耦合的相同延迟微分方程（DDE）系统，我们研究了所谓的同步缓慢振荡周期解（SOPS）的稳定性。在耦合矩阵的行总和条件下，对应标量DDE的唯一SOPS的存在意味着耦合DDE的唯一同步SOPS的存在。但是，标量DDE的SOPS的稳定性通常并不意味着耦合DDE的同步SOPS的稳定性。我们获得一个明确的公式，仅取决于耦合矩阵的频谱，线性衰减的强度以及正负无穷大附近的非线性负反馈函数的值，它决定了在非线性项被严重加权的渐近状态下同步SOPS的稳定性。在上述渐近状态下，我们还处理所谓的弱耦合系统，近均匀耦合系统和双非负耦合系统的特殊情况。我们的方法是估计同步SOPS的特征（Floquet）乘数。我们首先将多维变分方程的分析简化为一类标量变分类型方程的分析，然后为一个相关的单峰型算子族建立极限。我们以具有均场耦合的DDE系统和成环布置的DDE系统为例来说明我们的结果。在上述渐近状态下，近似均匀耦合系统和双非负耦合系统。我们的方法是估计同步SOPS的特征（Floquet）乘数。我们首先将多维变分方程的分析简化为一类标量变分类型方程的分析，然后为一个相关的单峰型算子族建立极限。我们以具有均场耦合的DDE系统和成环布置的DDE系统为例来说明我们的结果。在上述渐近状态下，近似均匀耦合系统和双非负耦合系统。我们的方法是估计同步SOPS的特征（Floquet）乘数。我们首先将多维变分方程的分析简化为一类标量变分类型方程的分析，然后为一个相关的单峰型算子族建立极限。我们以具有均场耦合的DDE系统和成环布置的DDE系统为例来说明我们的结果。我们首先将多维变分方程的分析简化为一类标量变分类型方程的分析，然后为一个相关的单峰型算子族建立极限。我们以具有均场耦合的DDE系统和成环布置的DDE系统为例来说明我们的结果。我们首先将多维变分方程的分析简化为一类标量变分类型方程的分析，然后为一个相关的单峰型算子族建立极限。我们以具有均场耦合的DDE系统和成环布置的DDE系统为例来说明我们的结果。