Shadowing for differential equations with grow-up

TL;DR

This paper develops new shadowing properties for differential equations with grow-up, including nonuniform and weighted shadowing, and applies compactification techniques to analyze such systems.

Contribution

It introduces nonuniform and weighted shadowing concepts for grow-up differential equations and establishes shadowing lemmas for these properties.

Findings

Proves analogs of shadowing lemma for nonuniform shadowing.

Establishes weighted shadowing for flows.

Uses compactification to transfer results to original systems.

Abstract

We consider the problem of shadowing for differential equations with grow-up. We introduce so-called nonuniform shadowing properties (in which size of the error depends on the point of the phase space) and prove for them analogs of shadowing lemma. Besides, we prove a theorem about weighted shadowing for flows. We compactify the system (using Poincare compactification, for example), apply the results about nonuniform or weighted shadowing to the compactified system, and then transfer the results back to the initial system using the decompactification procedure.