Saddle-node bifurcation

In the mathematical area of bifurcation theory a saddle-node bifurcation, tangential bifurcation or fold bifurcation is a local bifurcation in which two fixed points (or equilibria) of a dynamical system collide and annihilate each other. The term 'saddle-node bifurcation' is most often used in reference to continuous dynamical systems. In discrete dynamical systems, the same bifurcation is often instead called a fold bifurcation. Another name is blue skies bifurcation in reference to the sudden creation of two fixed points.

If the phase space is one-dimensional, one of the equilibrium points is unstable (the saddle), while the other is stable (the node).

Saddle-node bifurcations may be associated with hysteresis loops and catastrophes.

A typical example of a differential equation with a saddle-node bifurcation is:

Here ${\displaystyle x}$ is the state variable and ${\displaystyle r}$ is the bifurcation parameter.

In fact, this is a normal form of a saddle-node bifurcation. A scalar differential equation ${\displaystyle {\tfrac {dx}{dt}}=f(r,x)}$ which has a fixed point at ${\displaystyle x=0}$ for ${\displaystyle r=0}$ with ${\displaystyle {\tfrac {\partial f}{\partial x}}(0,0)=0}$ is locally topologically equivalent to ${\displaystyle {\frac {dx}{dt}}=r\pm x^{2}}$, provided it satisfies ${\displaystyle {\tfrac {\partial ^{2}\!f}{\partial x^{2}}}(0,0)\neq 0}$ and ${\displaystyle {\tfrac {\partial f}{\partial r}}(0,0)\neq 0}$. The first condition is the nondegeneracy condition and the second condition is the transversality condition.

...
Wikipedia