# Bifurcating solutions in a non-homogeneous boundary value problem for a nonlinear pendulum equation

## TL;DR

This paper investigates bifurcations in a nonlinear pendulum equation with non-homogeneous boundary conditions, analyzing how solutions change as the interval size varies, using phase space, time maps, asymptotics, and numerical methods.

## Contribution

It introduces a detailed bifurcation analysis for a nonlinear pendulum with mixed boundary conditions, combining analytical and numerical techniques.

## Key findings

- Identification of bifurcation points as interval size varies
- Characterization of solution branches and their stability
- Numerical confirmation of analytical results

## Abstract

Motivated by recent studies of bifurcations in liquid crystals cells [1,2] we consider a nonlinear pendulum ordinary differential equation in the bounded interval $(-L, L)$ with non-homogeneous mixed boundary conditions (Dirichlet an one end of the interval, Neumann at the other) and study the bifurcation diagram of its solutions having as bifurcation parameter the size of the interval, $2L$, and using techniques from phase space analysis, time maps, and asymptotic estimation of integrals, complemented by appropriate numerical evidence.