# Effective chaos for the Kirchhoff equation on tori

## TL;DR

This paper demonstrates the existence of effective chaos in solutions to the Kirchhoff equation on tori, showing that Sobolev norms can oscillate chaotically over long time scales due to symbolic dynamics in an effective coupled pendulum model.

## Contribution

It introduces the concept of effective chaos for the Kirchhoff equation on tori and constructs solutions with prescribed chaotic oscillations in Sobolev norms.

## Key findings

- Sobolev norms oscillate chaotically over long time scales.
- Chaotic dynamics are modeled by an effective system akin to two coupled pendulums.
- Existence of symbolic dynamics explains the chaotic behavior.

## Abstract

We consider the Kirchhoff equation on tori of any dimension and we construct solutions whose Sobolev norms oscillates in a chaotic way on certain long time scales. The chaoticity is encoded in the time between oscillations of the norm, which can be chosen in any prescribed way. This phenomenon, that we name as effective chaos (it occurs over a long, but finite, time scale), is consequence of the existence of symbolic dynamics for an effective system. Since the first order resonant dynamics has been proved to be essentially stable, we need to perform a second order analysis to find an effective model displaying chaotic dynamics. More precisely, after some reductions, this model behaves as two weakly coupled pendulums.