Scaling behavior of the terminal transient phase

Transient chaos can emerge in a variety of diverse systems, e.g., in chemical reactions, population dynamics, neuronal activity, or cardiac dynamics. The end of the chaotic episode can either be desired or not, depending on the specific system and application. In both cases, however, a prediction of the end of the chaotic dynamics is required. Despite the general challenges of reliably predicting chaotic dynamics for a long time period, the recent observation of a "terminal transient phase" of chaotic transients provides new insights into the transition from chaos to the subsequent (nonchaotic) regime. In spatially extended systems and also low-dimensional maps it was shown that the structure of the state space changes already a significant amount of time before the actual end of the chaotic dynamics. In this way, the terminal transient phase provides the conceptual foundation for a possible prediction of the upcoming end of the chaotic episode a significant amount of time in advance. In this study, we strengthen the general validity of the terminal transient phase by verifying its existence in another spatially extended model (Gray-Scott model) and the Henon map, where in the latter case the underlying mechanisms can be understood in an intuitive way. Furthermore, we show that the temporal length of the terminal transient phase remains approximately constant, when changing the system size (Gray-Scott) or parameters (Henon map) of the investigated models, although the average lifetime of the observed chaotic transients sensitively depends on these variations. Since the timescale of the terminal transient phase is in this sense relatively robust, this insight might be essential for possible applications, where the ratio between the length of the terminal transient phase and the relevant timescale of the dynamics may probably be crucial when a reasonable prediction (thus a sufficient time before) the end of the chaotic episode is required