# Energy transfer between modes in a nonlinear beam equation

## TL;DR

This paper investigates energy transfer between modes in a nonlinear beam equation, analyzing the existence of nonlinear modes and how energy transfer depends on system geometry, supported by numerical experiments.

## Contribution

It provides new insights into mode interactions and energy transfer mechanisms in nonlinear beam equations, extending understanding of stability and instabilities in such systems.

## Key findings

- Energy transfer depends on the geometry of the energy function.
- Numerical experiments confirm theoretical predictions.
- Results suggest similar behavior in more complex systems.

## Abstract

We consider the nonlinear nonlocal beam evolution equation introduced by Woinowsky- Krieger. We study the existence and behavior of periodic solutions: these are called nonlinear modes. Some solutions only have two active modes and we investigate whether there is an energy transfer between them. The answer depends on the geometry of the energy function which, in turn, depends on the amount of compression compared to the spatial frequencies of the involved modes. Our results are complemented with numerical experiments, overall, they give a complete picture of the instabilities that may occur in the beam. We expect these results to hold also in more complicated dynamical system