Blowup for flat slow manifolds

TL;DR

This paper extends the blowup method to analyze flat slow manifolds with infinite co-dimension, which appear in various complex systems, providing new tools for understanding their dynamics.

Contribution

It introduces a novel extension of the blowup method to flat slow manifolds that lose hyperbolicity beyond any algebraic order, applicable in multiple real-world models.

Findings

Successfully applied to a simple model system

Demonstrated on regularized piecewise smooth systems

Validated on aircraft landing dynamics model

Abstract

In this paper we present a method for extending the blowup method, in the formulation of Krupa and Szmolyan, to flat slow manifolds that lose hyperbolicity beyond any algebraic order. Although these manifolds have infinite co-dimension, they do appear naturally in certain settings. For example in (a) the regularization of piecewise smooth systems by $\tanh$, (b) a model of aircraft landing dynamics, and finally (c) in a model of earthquake faulting. We demonstrate the approach on a simple model system and the examples (a) and (b).