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Precompact families of Carathéodory differential equations revisited

Sylvia Novo, Rafael Obaya, Ana M. Sanz

TL;DR

This work develops a generalized ODE framework based on parametric $b$-measures to study precompact families of Carathéodory ODEs through nonautonomous dynamics. By embedding Carathéodory maps into a space of $b$-measures and using extended topologies, the authors establish a compactification (hull) and a continuous skew-product flow on it, enabling analysis via ergodic measures and attractors. They prove fundamental results on integrability along curves, topological compactness under equicontinuity/boundedness assumptions, and continuity of time translations, then apply the framework to reformulate Carathéodory dynamics as generalized ODEs and to slow-fast systems, with numerical illustrations. The approach relaxes classical regularity requirements while preserving key dynamical tools, broadening the applicability to more general time-dependent vector fields and providing a robust link between Carathéodory theory and nonautonomous dynamics.

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 Carathéodory 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 Carathéodory vector field with equicontinuous $m$-bounds and bounded $l$-bounds, is given.

Precompact families of Carathéodory differential equations revisited

TL;DR

This work develops a generalized ODE framework based on parametric -measures to study precompact families of Carathéodory ODEs through nonautonomous dynamics. By embedding Carathéodory maps into a space of -measures and using extended topologies, the authors establish a compactification (hull) and a continuous skew-product flow on it, enabling analysis via ergodic measures and attractors. They prove fundamental results on integrability along curves, topological compactness under equicontinuity/boundedness assumptions, and continuity of time translations, then apply the framework to reformulate Carathéodory dynamics as generalized ODEs and to slow-fast systems, with numerical illustrations. The approach relaxes classical regularity requirements while preserving key dynamical tools, broadening the applicability to more general time-dependent vector fields and providing a robust link between Carathéodory theory and nonautonomous dynamics.

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 Carathéodory 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 Carathéodory vector field with equicontinuous -bounds and bounded -bounds, is given.
Paper Structure (8 sections, 27 theorems, 86 equations, 1 figure)

This paper contains 8 sections, 27 theorems, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Integration of parametric b-measures along curves
  3. Topologies in the space of parametric b-measures
  4. Continuity of time translations and the hull of a parametric b-measure
  5. The skew-product flow induced by a class of generalized ODEs given by $N$-dim parametric b-measures on $\mathbb{R}^N$
  6. Application to precompact families of Carathéodory ODEs
  7. Numerical examples
  8. Application to slow-fast systems of ODEs

Key Result

Proposition 2.5

Given a b-measure $\nu_0\in \mathcal{M}_c(\mathcal{B})$ that has an m-bound $m_0\in\mathcal{M}^+_c$ over $\mathcal{B}$ and a map $\phi\colon \mathbb{R}^N\to[0,1]$ of class $C^1$ with compact support, the map $\,\phi\,\nu_{0}\colon \mathbb{R}^N\mapsto \mathcal{M}_c(\mathcal{B})$, $y\to \phi(y)\nu_{0

Figures (1)

  • Figure 1: The left-hand panel shows the solution of the initial value problem $y'= f_{\text{sum}}(t)$, $y(0)=0$, where $f_{\text{sum}}(t)$ is defined as in \ref{['eq:cantor-translations']}. The right-hand panel displays a selection of solutions of the differential equation $y' = -y^2+ 2 + f_{\text{sum}}(t)$. The red curve approximates the graph of the hyperbolic attracting solution, while the blue one approximates the graph of the hyperbolic repelling solution. The black curves are examples of eight more solutions with different initial conditions at time zero.

Theorems & Definitions (66)

  • Definition 2.1
  • Example 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Remark 2.9
  • Lemma 2.10
  • ...and 56 more