Flash is a proprietary format, and the folks at Macromedia guard it zealously.
So there is NO open-source Flash renderer. You have to use a Linux plug in, provided by Macromedia, that happens to work with Open source browsers like Mozilla and Firefox.
There *are* open-source libraries to help you *create* Flash files, but none to help you *display* them.
(If that's in fact what you're asking. You might instead be asking this:
http://www.w3schools.com/flash/flash_inhtml.asp)
Sorry .. PSM
PS:
What happened with that "Midnight slowdown"?
PPS:
"Moho" is an excellent animation package for Linux. It also happens to generate Flash (among many other formats). It's not free, but it's VERY good and (IMHO) a bargain at $99.00. Check it out:
http://www.lostmarble.com