I would like to know which methods or tricks are there for investigating the stability of an equilibrium point of a 2-dimensional discrete dynamical system(f:$\mathbb{R}^2\to\mathbb{R}^2$)? The Internet provides a quite small amount of information, much less than about continuous systems.
It is clear that one should begin with considering a function $V(x,y)=|x|+|y|$, or $V(x,y)=x^2+y^2$, and maybe $V(x,y)=ax^2+bxy+cy^2$ and checking if $V(f(x,y))-V(y)$ is positively or negatively definite.
The first-order approximation method is also useful in simple cases. Sometimes we can also use the fact that an equilibrium point $(x,y)$ of $f$ is stable if and only if it is stable for $f^n$.
Are there any other ideas which can help one deal with hard tasks on equilibrium point stability in the discrete case? For example, such as investigating the stability of $(0,0)$ (constructing the right $V$), when
a) $f(x,y)=(-x+y^2,y/2+x^2)$
b) $f(x,y)=(-y+x^3-y^5,x-y^3+2x^5)$
(I know the solutions to these ones, these are just examples)