A new class of generalized ordinary differential equations with applications

TL;DR

This paper introduces a novel class of generalized ordinary differential equations based on parametric b-measures, providing new tools for analyzing complex dynamical systems with applications to slow-fast systems.

Contribution

It develops a new framework for generalized ODEs using parametric b-measures, enabling advanced analysis of nonautonomous dynamical systems.

Findings

New class of generalized ODEs defined via parametric b-measures

Application to fast variables in slow-fast dynamical systems

Enhanced analysis of Carathéodory ODEs with equicontinuous bounds

Abstract

The space of parametric b-measures endowed with appropriate topologies is introduced to define a new class of generalized ODEs given by parametric b-measures. This framework offers a new approach for dealing with precompact families of Carath\'eodory ODEs using nonautonomous dynamical systems techniques. An application to the study of the dynamics of the fast variables of a slow-fast system of ODEs, where the fast motion is determined by a Carath\'eodory vector field with equicontinuous $m$-bounds and bounded $l$-bounds, is given.