I have an 05 gixxer 600 and when i start it it sounds like it wants to turn off, but then it idols down?

when it idols down it kind of sputters like it wants to cut off. i was riding the other night and i was in traffic and the bike temperature was going up(around 220) and all of a sudden the bike cut off. It started back up but seem to still want to do the same when i came to a light and sat there for a minute. Lately when i start it I sometimes have to try twice to get it started( this is when it is cold). I am from florida so it is not like it gets cold here. Any idea of what the problem might be? I purchased this bike used from a private seller about four months ago. ANY AND ALL SUGGESTIONS WELCOME.