Question: Help for a mathematical question related bifurcation analysis

How I can obtain system (21) in the following pdf file?
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 questuin:
1-The change of variables performed in the neighborhood (w1 *, w2 *)
for system (20) or for system (7) ???
2-What does it mean in the neighborhood (w1 *, w2 *)?
3- How did obtaun Eq (21)?

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