(My first time posting.)
Consider a 2-d map:
$x_{n+1} = y_n$
$y_{n+1} = \mu_1 y_n + \mu_2 - x_n^2$
I was asked to find what parameter will give saddle-node, period-doubling and Naimark-Sacker bifurcations. Which is not too hard, I found the fixed point by letting:
$x_{n+1} = x_{n} =\bar{x}$ and $y_{n+1} = y_{n} =\bar{y}$
Then it was just a system of equation to solve. After which I perturbed the fixed points by $\delta x$ and $\delta y$, which I am able to find the Jacobian and the eigenvalues of it.
Next part is the problem, where the question asks for the normal form of each bifurcation.
I don't really have an idea on how to solve it. I know that I would like to reduce the equation into nonlinear terms which I cannot remove with linear terms, but that is about all I know. (Another thing I know is when the space is 1-d i.e. just $x$ and there is only one parameter i.e. $\mu_1$)
My question is: how exactly can I find the normal forms? Detailed steps will be greatly appreciated. It is also helpful to point me to some reference which has detailed examples on 2-d maps and 2 parameter family maps.
Thank you very much!