Simplicity rider is possessed!?

I have a 2 year old Simplicity ZT 1644 with a 16 hp Kohler engine. I used the mower Sunday with no issues at all. Went back out to mow tonight (Wed.) and when I hit the key it fired right up but died almost immediately. I hit the key again and nothing! I lifted the cover and checked the cables at the battery which were tight when I heard a click from the solenoid. I realized that I has left the key in the run position. I closed the cover and heard the click again. Climbed on and turned key to the start position and again nothing. I left the key in the run position and just sat there. Now here is the weird part, about ever 25-40 seconds the solenoid clicks. I cant tell you this is normal or not cuz I am not in the habit of leaving the key in the run position without the engine running. If I hold the key securely in the start position at about the same 25-40 seconds it will engage the starter and even try to fire momently but dies right away. I checked and charged the battery, pulled the wires from the ignition switch and cleaned the connections with electric contact cleaner. Still no luck. I am fairly certain it is electrical, like something is overheating, maybe a relay? I am a auto repair shop owner but I have NO small engine experience at all. Any help would be appreciated.