Alanna Haslam-Hyde | Searcharxiv

arXiv2026

Exponential Dichotomies for Elliptic Equations on Multidimensional Domains

The existence of exponential dichotomies has been well-established as a powerful tool to study existence, stability, and bifurcations of coherent structures. Currently, the application of exponential dichotomies to elliptic problems posed on multi-dimensional domains is predominately limited to the context of cylindrical spatial domains. Recent work by Beck et. al. (2021) has shown how to extend the method of spatial dynamics, in which one views a spatial variable as a time-like evolutionary variable, to general multi-dimensional spatial domains. In this paper, we show that exponential dichotomies exist for a class of spatial dynamical systems arising in this more general setting, thus allowing for their use in future analyses of coherent structures.

math.AP↗

arXiv2026

A Radial and Tangential Framework for Studying Transient Reactivity in Two-Dimensional Systems

Even if a linear system of ordinary differential equations has a globally attracting equilibrium at the origin, small disturbances from the equilibrium may lead to large transient excursions before the system stabilizes. This counter-intuitive phenomenon of transient amplification is called reactivity and is often associated with systems that are non-normal. Here, we establish a new framework for analyzing reactivity and transient dynamics in two-dimensional linear ODEs. Our work is facilitated by decomposing the corresponding vector field into sinusoidal radial and tangential components. Using this decomposition, we introduce a structure of orthovectors and orthovalues as dual to the eigenstructure. Since diagonalization masks transient reactivity, we combine the eigenstructure and the orthostructure to propose alternative matrix forms which capture both transient and asymptotic behavior and which highlight reactivity features more directly. Leveraging these matrix forms, we analytically quantify the maximal amplification in globally attracting systems, and we provide new insight into how a nonautonomous linear system can be unstable, even when all the frozen-time systems are stable.

math.DS↗