In complex systems, processes operate across multiple time scales, such as rapid fluctuations in environmental conditions, intermediate responses like population dynamics, and slower shifts such as ecosystem succession or climate change. These dynamics are often nested, with fast processes embedded within slower ones. Fine-scale, rapid changes can accumulate over time to influence large-scale trends, while slower processes provide the conditions for fast dynamics to unfold. This interplay between processes at different time scales can lead to transient behaviors, where a system remains in one dynamic state for an extended period before abruptly shifting to another [4]. In ecological systems, these dynamics often manifest as long transients--periods of apparent stability followed by sudden regime shifts. These shifts can occur without any obvious external trigger, driven instead by internal processes or responses to environmental variability [6]. During these phases, a system may exhibit consistent behavior over time before transitioning to a different dynamic regime, which could involve altered oscillatory patterns or a completely new structure. Such transitions are difficult to predict, as they are nonlinear, involve systems operating at multiple interacting scales, and are influenced by stochasticity [5, 2]. Understanding these multiscale and nonlinear interactions is essential for anticipating regime shifts, which are often most consequential at the coarsest time scales, where changes in slow-moving processes like ecosystem succession or long-term climate changes lead to impactful, irreversible transitions [6].

The full-dimensional (metric, Euclidean, least squares) multidimensional scaling stress loss function is combined with a quadratic external penalty function term. The trajectory of minimizers of stress for increasing values of the penalty parameter is then used to find (tentative) global minima for low-dimensional multidimensional scaling. This is illustrated with several one-dimensional and two-dimensional examples.