Suppose I have a diffeomorphism of a plane, $$\bar{x} = F(x,s,t)$$ where $x \in \mathbb{R}^{2}$ and $s \in [a,b] \subset \mathbb{R}$ and $t \in I_{2} \subset{ \mathbb{R}}$ are parameters.
Suppose that I have a fixed point $x_{0}$ for all $s$ and $t$.
I choose a value of $s$ as given and vary $t$. The range of $t$ is bounded (as a function) of $s$. For $s=a$ and $\mathbf{any}$ $t \in I_{2} \subset{ \mathbb{R}}$, $x_{0}$ is parabolic. For $a<s \leq b$ $\mathbf{and}$ some interval $t \in \tilde{I} \subset I$, where the end points of $\tilde{I}$ depend on $s$, the fixed point becomes elliptic (and hypebolic for $t$ outside the specified interval).
What is this bifurcation called?
Any links to relevant literature / articles would be helpful!