On non-existence of bifurcations in one-dimensional Bratu equation

TL;DR

This paper proves analytically that the one-dimensional Bratu equation does not have bifurcations, contrasting with numerical discretizations that can produce spurious bifurcation points.

Contribution

It provides an analytical proof showing the absence of bifurcations in the continuous one-dimensional Bratu equation, clarifying discrepancies with numerical methods.

Findings

Finite difference discretization introduces spurious bifurcations.

Finite element approach also shows spurious bifurcations.

Analytical proof confirms no bifurcations in the continuous equation.

Abstract

In this paper, we revisit the classical problem of Bratu differential equation in one-dimension. While it is known that the finite difference discretized form of continuous Bratu equation gives rise to spurious bifurcations, we show that spurious bifurcation points exist even when the finite element approach is employed. We then present an analytical proof demonstrating that there are no bifurcations when the continuous Bratu equation is considered.