Relating dynamics and structure of discrete and continuous time systems

Abstract: In numerous subjects there are instances of dynamical variables that change in discrete time. One such example is a population's size, which changes in discrete generations which we label by $n$ ($= 0, 1, 2, . ..$). There are some exactly soluble cases where the population's size at discrete time $n$, written $x_{n}$, can be related to a different problem, where change occurs in continuous time, $t$, and the population's size is then written as $x(t)$. The relation is that $x(t = n)$ precisely coincides with $x_{n}$. In such a case, we say the two models are 'equivalent', even though $x_{n}$ obeys a difference equation and $x(t)$ obeys a differential equation. A comparison of the difference equation and the differential equation allows us to make a correspondence between coefficients in these equations. This exposes structural similarities and sometimes appreciable structural differences of discrete and continuous time dynamics. We present cases of such equivalences for both time homogeneous and time inhomogeneous problems. Beyond this, when the equation that $x_{n}$ obeys is modified so there is no exact solution, we show how to construct an approximate continuous time solution that can capture rapid change, such as growth, of $x_{n}$. Additionally, in some discrete time problems, the solution has an oscillatory component that changes sign as $n \rightarrow n + 1$. We show that the way an equivalent continuous time solution replicates this oscillatory behaviour is to become complex. We give concrete examples of this 'complexification' of the continuous time solution, which is real at $t = 0,1,2,...$ but is complex away from the integers.