Saddle-Node Bifurcation
Definition
Simple Definition
Saddle-node bifurcation is a bifurcation in a dynamical system where fixed points are created or annihilated as a parameter changes1.
Complex Definition
$$ \dot{x} = f \left( x , r \right) \qquad , x \in \mathbb{R}^{n} , r \in \mathbb{R}^{1} $$ Let us assume that in a given dynamical system, $f$ is smooth with respect to $x$ and $\alpha$. If $\bar{x}$ is a hyperbolic fixed point of this system, and one of the eigenvalues of its Jacobian matrix, denoted as $D f \left( \bar{x} \right)$, is $\lambda_{k}$. The bifurcation associated with the appearance or disappearance of $\lambda_{k} = 0$ is called a saddle-node bifurcation2.
Normal Form
The saddle-node bifurcation has the following normal form: $$ \dot{x} = r + x^{2} $$
Diagram
The bifurcation diagram of a saddle-node bifurcation is as follows:
Explanation
The saddle-node bifurcation is also known as a fold bifurcation, tangent bifurcation, or blue sky bifurcation, especially the bifurcation point is referred to as a turning point or limit point.
This type of bifurcation is frequently mentioned when explaining bifurcations,
Fold?
As can be seen in the bifurcation diagram, it’s nicknamed for the way the curve folds. This is especially more effective terminology in contexts related to hysteresis phenomena.
Blue Sky?
This term can be used in the sense that a fixed point suddenly appears as if lightning strikes out of a clear blue sky. Considering the bifurcation diagram, if we imagine gradually decreasing $r$ from $r > 0$, initially there are no fixed points, but suddenly a saddle node appears at $r = 0$.