A General View on Double Limits in Differential Equations

Abstract

In this paper, we review several results from singularly perturbed differential equations with multiple small parameters. In addition, we develop a general conceptual framework to compare and contrast the different results by proposing a three-step process. First, one specifies the setting and restrictions of the differential equation problem to be studied and identifies the relevant small parameters. Second, one defines a notion of equivalence via a property/observable for partitioning the parameter space into suitable regions near the singular limit. Third, one studies the possible asymptotic singular limit problems as well as perturbation results to complete the diagrammatic subdivision process. We illustrate this approach for two simple problems from algebra and analysis. Then we proceed to the review of several modern double-limit problems including multiple time scales, stochastic dynamics, spatial patterns, and network coupling. For each example, we illustrate the previously mentioned three-step process and show that already double-limit parametric diagrams provide an excellent unifying theme. After this review, we compare and contrast the common features among the different examples. We conclude with a brief outlook, how our methodology can help to systematize the field better, and how it can be transferred to a wide variety of other classes of differential equations.

Figures (28)

• Figure 1: Partitioning of the positive quadrant $\mathcal{K}$ near the doubly-singular limit $\varepsilon\rightarrow 0$ and $\delta\rightarrow 0$ into three different regions (I)--(III), which are non-equivalent under a property $\mathcal{P}$. The thick lines (in blue) indicate hard boundaries between the different regions, e.g. between (II) and (III) there is a precise curve separating these regions. The thin line (in red) indicates that the boundary is only asymptotic up to a constant between two regions. Dashed lines (in black) indicate an unclassified axis (such as the vertical axis in this figure). The circle at the origin also means that at this point a classification with respect to $\mathcal{P}$ is not known and/or may not even be possible.
• Figure 2: Classification diagram with respect to the property $\mathcal{P}_{[-1,1]}$. In region II we have no zeros while in region I we have two zeros (counting multiplicity).
• Figure 3: Classification diagram with respect to the property $\mathcal{P}_{\textnormal{cvx}}$. We just have a single region as $f(x;\varepsilon,\delta)=\varepsilon x^2-\delta$ is always convex on $\mathcal{K}$.
• Figure 4: Classification diagram with respect to the property $\mathcal{P}_{\partial\partial}$. The two regions correspond to the two possible partial derivative values at the origin of the function \ref{['eq:Clairaut']} given for our elementary analysis problem. The thin line (red) could have been chosen at any fixed slope as it is an asymptotic subdividing line of the form $\{\delta= \kappa \varepsilon,\varepsilon>0\}$ for some fixed constant $\kappa>0$.
• Figure 5: Sketch of the possible dynamics of \ref{['eq:tc']} in $(x,y)$-coordinates. The critical manifold $\mathcal{C}_0$ is shown in gray (repelling parts with dashed lines and attracting parts with solid lines). Three trajectories (green, cyan, magenta) for $0<\varepsilon\ll1$ are indicated for three different choices of $\delta$ (corresponding to the exchange-of-stability, canard, and critical transition cases respectively). Double arrows show the direction of the fast subsystem flow for orientation; the slow subsystem dynamics on $\mathcal{C}_0$ is always directed upwards at unit speed.