Computing the location and the direction of bifurcation

We consider positive solutions of the Dirichlet problem \[ u^{\prime\prime}(x)+\lambda f(u(x))=0 on (-1,1) u(-1)=u(1)=0. \] depending on a positive parameter $\lambda$. Each solution $u(x)$ is an even function, and hence it is uniquely identified by $\alpha=u(0)$. We present a formula, which allows to compute all $\alpha$’s where a turn may occur, and then we give another formula, which allows to compute the direction of the turn. As an application, we present a computer assisted proof of the exact bifurcation diagram in case $f(u)$ is any cubic with real and distinct roots. Another application is a computer assisted proof of a conjecture by S.-H. Wang \cite{W1}, related to gas combustion.

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Published 1 January 2005