# Stratified Lie systems: Theory and applications

## TL;DR

This paper introduces stratified Lie systems, extending classical Lie systems to include stratified structures, and explores their solutions, geometric properties, and applications to Lax pairs and Hamiltonian systems.

## Contribution

It develops the theory of stratified Lie systems, generalizing Lie systems to stratified manifolds and analyzing their solutions and geometric structures.

## Key findings

- Solutions can be characterized using stratified Lie algebra structures
- Applications to Lax pairs demonstrate the framework's utility
- Poisson structures induced by r-matrices are compatible with stratified Lie systems

## Abstract

A stratified Lie system is a nonautonomous system of first-order ordinary differential equations on a manifold $M$ described by a $t$-dependent vector field $X=\sum_{\alpha=1}^rg_\alpha X_\alpha$, where $X_1,\ldots,X_r$ are vector fields on $M$ spanning an $r$-dimensional Lie algebra that are tangent to the strata of a stratification $\mathcal{F}$ of $M$ while $g_1,\ldots,g_r:\mathbb{R}\times M\rightarrow \mathbb{R}$ are functions depending on $t$ that are constant along integral curves of $X_1,\ldots,X_r$ for each fixed $t$. We analyse the particular solutions of stratified Lie systems and how their properties can be obtained as generalisations of those of Lie systems. We illustrate our results by studying Lax pairs and a class of $t$-dependent Hamiltonian systems. We study stratified Lie systems with compatible geometric structures. In particular, a class of stratified Lie systems on Lie algebras are studied via Poisson structures induced by $r$-matrices.