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Title: A bounded-time approach to dynamics analysis

Abstract: Traditional analysis of dynamics is in terms of coordinate-invariant long-time-asymptotic properties of time-independent (or more general infinite-time) dynamical systems. However, through a simple and instructive model of phase-stabilisation of an oscillator by slow-timescale external forcing, we identify three limitations of such traditional approaches, that can be overcome by analysis in terms of dynamical systems defined on a bounded time-interval. Firstly and most fundamentally, a prescribed indefinite-time deterministic or stochastic model will not take into account the true time-variable nature of dynamics of open systems. Secondly, bounded-time dynamical models that admit no natural extension to infinite time can exhibit important dynamical phenomena, which will not be identifiable by application of asymptotic-time methods. Thirdly, even when a time-dependent model is well-defined on infinite time, an asymptotic approach may obscure dynamical phenomena that are more readily understood within the framework of finite-time dynamical systems. We conclude that theoretical and experimental analysis of dynamics throughout the sciences may crucially need to rely on approaches that are rooted in concepts of finite-time nonautonomous dynamics.

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