# Loss of energy concentration in nonlinear evolution beam equations

## TL;DR

This paper introduces a new concept of instability based on energy loss in prevailing modes for nonlinear beam equations, linking infinite-dimensional stability to finite-dimensional approximations and analyzing different nonlinearities.

## Contribution

It defines a novel instability criterion for nonlinear beam solutions and provides a reduction method to finite-dimensional systems for stability analysis.

## Key findings

- Instability can be characterized by energy concentration loss in prevailing modes.
- Theoretical reduction from infinite to finite-dimensional stability analysis.
- Numerical simulations illustrate the impact of different nonlinearities.

## Abstract

Motivated by the oscillations that were seen at the Tacoma Narrows Bridge, we introduce the notion of solutions with a prevailing mode for the nonlinear evolution beam equation $$ u_{tt} + u_{xxxx} + f(u)= g(x, t) $$ in bounded space-time intervals. We give a new definition of instability for these particular solutions, based on the loss of energy concentration on their prevailing mode. We distinguish between two different forms of energy transfer, one physiological (unavoidable and depending on the nonlinearity) and one due to the insurgence of instability. We then prove a theoretical result allowing to reduce the study of this kind of infinite-dimensional stability to that of a finite-dimensional approximation. With this background, we study the occurrence of instability for three different kinds of nonlinearities $f$ and for some forcing terms $g$, highlighting some of their structural properties and performing some numerical simulations.