Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 2 / 2A / a

An equilibrium of an autonomous differential equation is a constant $y_{\ast }$ with $f(y_{\ast })=0$. Put $y=y_{\ast }+\eta$. For differentiable $f$, the linear stability analysis givesThus the linear perturbation is proportional to $e^{f^{\prime }(y_{\ast })x}$. More directly, continuity of $f^{\prime }$ makes $f(y)$ point toward $y_{\ast }$ on both sides when $f^{\prime }(y_{\ast })<0$, and away from it when $f^{\prime }(y_{\ast })>0$. HenceThe sign refers to evolution toward increasing $x$. If $f^{\prime }(y_{\ast })=0$, linear stability analysis is inconclusive: the leading nonzero nonlinear term or the signs of $f$ on either side are needed. For example ${\eta }^{\prime }=-{\eta }^3$ is attracting whereas ${\eta }^{\prime }={\eta }^3$ is repelling, despite the same zero linear coefficient.