Help for analysis bifurcation process

How I can obtain system (21) in the following pdf file?

https://drive.google.com/file/d/10NofU9AzejTEdLr7094rtyzuSoxB7lVn/view?usp=sharing

In the first step, several changes of variables are done to obtain the system (20),
then changes the variables again repeated in the neighborhood (w1 *, w2 *) to gain Eq 21.
I have 3 questions:
1-The change of variables performed in the neighborhood (w1 *, w2 *)
for the system (20) or for the system (7) ???
2-What does it mean in the neighborhood (w1 *, w2 *)?
3- How did obtain Eq (21)?
Thanks