Limitations of the asymptotic approach to dynamics

Standard dynamical systems theory is centred around the coordinate-invariant asymptotic-time properties of autonomous systems. We identify three limitations of this approach. Firstly, we discuss how the traditional approach cannot take into account the time-varying nature of dynamics of open systems. Secondly, we show that models with explicit dependence on time exhibit stark dynamic phenomena, even when they cannot be defined for infinite time. We see a bifurcation occurring in nonautonomous finite-time systems that cannot be identified by classical methods for infinite-time autonomous systems. Thirdly, even when a time-varying model can be extended to infinite time, the classical infinite-time approach is likely to miss dynamical phenomena that are more readily understood within the framework of finite-time dynamics. We conclude the potentially crucial importance of a nonautonomous finite-time approach to real-world, open systems.