We study a discrete non-autonomous system whose autonomous counterpart (with the frozen bifurcation parameter) admits a saddle-node bifurcation, and in which the bifurcation parameter slowly changes in time and is characterized by a sweep rate constant $\epsilon$. The discrete system is more appropriate for modeling realistic systems since only time series data is available. We show that in contrast to its autonomous counterpart, when the time mesh size $\Delta t$ is less than the order $O(\epsilon)$, there is a bifurcation delay as the bifurcation time-varying parameter is varied through the bifurcation point, and the delay is proportional to the two-thirds power of the sweep rate constant $\epsilon$. This bifurcation delay is significant in various realistic systems since it allows one to take necessary action promptly before a sudden collapse or shift to different states. On the other hand, when the time mesh size $\Delta t$ is larger than the order $o(\epsilon)$, the dynamical behavior of the solution is dramatically changed before the bifurcation point. This behavior is not observed in the autonomous counterpart. Therefore, the dynamical behavior of the system strongly depends on the time mesh size. Finally. due to the very discrete feature of the system, there are no efficient tools for the analytical study of the system. Our approach is elementary and analytical.

翻译：本文研究了一类离散非自治系统，其自治对应系统（在冻结分岔参数下）存在鞍结分岔，且分岔参数随时间缓慢变化，变化速率由常数$\epsilon$表征。鉴于实际系统中仅能获取时间序列数据，离散系统更适合建模现实场景。研究表明：当时间网格尺寸$\Delta t$小于$O(\epsilon)$量级时，与自治系统不同，当分岔时变参数穿越分岔点时会出现分岔延迟现象，且该延迟与扫频速率$\epsilon$的三分之二次方成正比。这种分岔延迟在现实系统中具有重要价值，因其为系统在发生突发性崩溃或状态跃迁前预留了采取必要措施的响应时间。当时间网格尺寸$\Delta t$大于$o(\epsilon)$量级时，分岔点前解的动力学行为会发生剧烈改变，此类现象在自治系统中未被观测到。因此，系统动力学行为强烈依赖于时间网格尺寸。最后，鉴于系统的高度离散特性，目前缺乏有效的分析工具进行解析研究。本文采用基础性解析方法展开研究。