Slow passage through a saddle-node bifurcation in discrete dynamical systems

We study a discrete non-autonomous system whose autonomous counterpart (with the frozen bifurcation parameter) admits a saddle-node bifurcation in which the bifurcation parameter slowly changes in time and is characterized by a sweep rate constant ϵ. The discrete system is more appropriate for modeling realistic systems since only time series data are available. In contrast to its autonomous counterpart, we show that when the ratio ϵ/Δt is of the order O(1), there is a bifurcation delay as the bifurcation time-varying parameter varies through the bifurcation point. The delay is proportional to the two-thirds power of the sweep rate constant ϵ. 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 ratio ϵ/Δt is of the order o(1), the dynamical behavior of the system is dramatically changed before the bifurcation point. This behavior is not observed in the system's (continuous) autonomous counterpart. Therefore, the system's dynamical behavior strongly depends on the time mesh size, and the ratio ϵ/Δt can be viewed as a measurement of how much extent the system behaves like a discrete system. Finally, the inherently discrete nature of the system significantly increases the complexity of its analytical study. Our approach employs the construction of super- and sub-solutions, complemented by a careful phase portrait analysis, to investigate bifurcation delay phenomena.