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.
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A saddle-node bifurcation creates or destroys a pair of equilibria when a parameter passes through a critical value. The scalar normal formhas equilibria for . The derivative makes locally attracting and repelling. At they meet at a nonhyperbolic equilibrium; for 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.