# Bifurcation of equilibrium forms of an elastic rod on a two-parameter Winkler foundation

## TL;DR

This paper analyzes the bifurcation behavior of an elastic rod on a deformable foundation with two parameters, establishing conditions for bifurcation and demonstrating the existence of solution continua from bifurcation points.

## Contribution

It provides a rigorous mathematical framework for understanding bifurcation in elastic rods on foundations, using Brouwer degree to prove the existence of solution branches.

## Key findings

- Bifurcation occurs iff the linearized problem has nontrivial solutions.
- From each bifurcation point, a continuum of solutions branches off.
- The proof employs Brouwer degree to establish solution existence.

## Abstract

We consider two-parameter bifurcation of equilibrium states of an elastic rod on a deformable foundation. Our main theorem shows that bifurcation occurs if and only if the linearization of our problem has nontrivial solutions. In fact our proof, based on the concept of the Brouwer degree, gives more, namely that from each bifurcation point there branches off a continuum of solutions.