 Saddle-node bifurcation

A saddle-node bifurcation creates or destroys a pair of equilibria when a parameter passes through a critical value. The scalar normal formhas equilibria $x_{\pm }=\pm \sqrt{\mu }$ for $\mu >0$. The derivative $-2x$ makes $x_{+}$ locally attracting and $x_{-}$ repelling. At $\mu =0$ they meet at a nonhyperbolic equilibrium; for $\mu <0$ there is no equilibrium nearby.

The diagram is a fold in the equilibrium branches. A system slowly following the attracting branch can lose its resting state when the fold is crossed, producing a sudden departure even though the parameter changes smoothly. In higher dimensions the normal form describes the critical direction; stability in the remaining directions must be checked separately. Hysteresis requires additional branch structure and does not follow from one isolated fold alone.

A saddle-node bifurcation is a concept from dynamical systems theory and is a type of bifurcation that occurs in a system when two steady states (or equilibrium points) collide and annihilate each other as a parameter is varied. This typically leads to significant changes in the behavior of the system.