Which companies develop software for airplane systems?
This entire branch is known as  avionics (aviation electronics). Limited work is generally done by the aircraft producers themselves, and the majority is subcontracted to specialist companies. A name which does come to mind is Thales group, who are behind much of the Airbus A380 avionics. Other ones are Rockwell Collins, Honeywell and Garmin. These are however though often at the top of an iceberg of various subsystems and controls.



A huge increase in this field (particularly in small aircraft but now also coming to larger ones) has been in tablet computers and the like in the past few years, and while I'm not a pilot I think it's hardly at the limit of what is possible. An iPad has more processing power than most if not virtually all aircraft flying today. Many customers today are disappointed if they don't get WiFi and inflight entertainment onboard.

With the increase in drones and unmanned aircraft, it's certainly not a shrinking field. A large area of this includes complex calculations and plenty of sensors. And more drones and unmanned aircraft require advanced communication and control systems. I've even heard about dynamic models of aircraft who adapt the plane's control to maintain control in the event of structural failure or serious damage. Only having scratched the field personally it's a very broad field with plenty.