# Averaging method for systems with separatrix crossing

## TL;DR

This paper extends the averaging method to systems with a single rotating phase that experience separatrix crossings, providing new estimates to justify the method's application in these complex scenarios.

## Contribution

It introduces estimates enabling the use of averaging in systems with separatrix crossings, a situation previously problematic for standard averaging techniques.

## Key findings

- Established estimates for averaging with separatrix crossings
- Extended applicability of averaging to new classes of systems
- Improved understanding of system evolution near separatrices

## Abstract

The averaging method provides a powerful tool for studying evolution in near-integrable systems. Existence of separatrices in the phase space of the underlying integrable system is an obstacle for application of standard results that justify using of averaging. We establish estimates that allow to use averaging method when the underlying integrable system is a system with one rotating phase, and the evolution leads to separatrix crossings.