Computational Singular Perturbation Method for Nonstandard Slow-Fast Systems

Abstract

The computational singular perturbation (CSP) method is an algorithm which iteratively approximates slow manifolds and fast fibers in multiple-timescale dynamical systems. Since its inception due to Lam and Goussis [17], the convergence of the CSP method has been explored in depth; however, rigorous applications have been confined to the standard framework, where the separation between 'slow' and 'fast' variables is made explicit in the dynamical system. This paper adapts the CSP method to nonstandard slow-fast systems having a normally hyperbolic attracting critical manifold. We give new formulas for the CSP method in this more general context, and provide the first concrete demonstrations of the method on genuinely nonstandard examples.

Beyond Slow-Fast: Relaxation Oscillations in Singularly Perturbed Nonsmooth Systems

“…Thus in this case, we were able to address (Q1) via the application of GSPT for slow-fast systems in nonstandard form; an approach which has been shown to be successful in a wide variety of applications already in e.g. [43,58,59,60,50,55,80,108].…”

“…1.1. Our review is based on that in [53,108]; see also [80,19]. We also refer to [43,108] throughout, including remarks where they are relevant for generalisations to R n .…”

“…There has been a significant amount of work on slow-fast systems in nonstandard form in the context of applications, see e.g. [43,58,50,57,59,75,80,108] and in particular [60], where oscillations which are not slow-fast are studied via an auxiliary (nonstandard) slowfast system, similarly to the two-stroke relaxation oscillators considered in this chapter. For more on a coordinate-free framework for GSPT in the context of planar nonstandard form problems (2.9), see [19].…”

“…Indeed, a combination of GSPT and blow-up has been applied to study a wide variety of slow-fast systems in nonstandard form 4 which do not satisfy (SF2); see e.g. [58,55,59,60,75,80,50] for applications and [43,108] for theoretical background. For problems in nonstandard form, the slow-fast structure cannot be reflected in terms of a separation of slow/fast variables.…”

The origin of CEMA and its relation to CSP

“…[16][17][18][19][20][21][22]. The usefulness of the method in analyzing multi-scale systems was examined and it was concluded that it can generate, order by order, the results of asymptotic analysis [23][24][25][26]. Concepts introduced by CSP were later employed for the development of other multi-scale algorithmic methodologies for the analysis of combustion processes, which all introduced new concepts and algorithmic tools; i.e., ILDM [27,28], G-Scheme [29,30] and TSR [31,32].…”