  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.
