Yes, CURRENT_TIMESTAMP (without parens) is what the SQL standard (SQL2008) prescribes (and it looks like mysql does actually implement it).

(In law-abiding systems like postgres, now() and current_timestamp are synonymous)

However, I don't see any CURRENT_TIMESTAMP in sqlite.

update:

I now actually tried it: turns out that SQLite does implement current_timestamp after all (without parens).

DBD::SQLite::VERSION = 1.27
[download]

So, the best solution for the OP: use the standard current_timestamp (in both mysql and sqlite) instead of non-standard now().