According to the official specs, the application works on Windows and Mac OS X. more precisely on the following versions of these two operating systems:
Windows OS: Windows 7, 8, Vista, XP (64bit & 32bit)
Mac OS: Mac OS 10.7 / 10.8 / 10.9
and it supports:
iPhone Series: iPhone, iPhone 3G, iPhone 3GS, iPhone 4, iPhone 4S, iPhone 5, iPhone 5s, iPhone 5c.
No, the application will not work on Mac OS X 10.5.8 as it's not compatible.