Recently, I was on a flight from TNCM to MYBS. While flying along, enjoying the scenery, for no reason my landing gear deployed at 38,000 feet. By the time I was able to recover the resulting stall/spin, I was at 22,000 feet. (Okay, I was watching a movie at the time and the gear deployed while I wasn't watching the plane)Another time I was on a flight from KMIA to TXKF and the LNAV, RNAV, and A/P disconnected for no reason.Also, when I was flying from KJAX to TNCM, the airplane started rolling gently back and forth for no reason. The PFD showed no deviation.I have witnessed my landing lights turning on/off while the switches are in the off positiion.This only happens when I am flying from/to the Caribbean islands, Bermuda, and Florida.Anyone else noticing any strange behavior in this area?