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.

Dead zones and phase reduction of coupled oscillators

“…We illustrate these mechanisms explicitly in Propositions 5 and 6 for an example of weakly coupled relaxation oscillators 14 , a dynamical system with two small parameters 18 . The geometry that shapes the limit cycle oscillators yields an explicit calculation of phase response curves and thus allows for a concrete analysis of emergent dead zones.…”

Discrete Geometric Singular Perturbation Theory

“…The latter part of assertion (iii) as well as assertion (iv) in Theorem 5.14 hold only for fixed h > 0 and asymptotically small ε ∼ 0. They do not account for asymptotic dependence in the case that both ε → 0 and h → 0, in which case one has an instance of the general class of double singular limit problems described in [43].…”