How can I get the aliens to start doing abductions again?

Solution 1:

If you let a UFO go by without intercepting it, it's likely a battleship will come by later and shoot down a satellite.

If you have at least two countries uncovered by satellites, you'll get abduction missions.

So, you can wait for the aliens to shoot down two of your satellites and then abduction missions should start appearing again.

Solution 2:

I can confirm this works.

I left one country (Egypt) in my home continent (Africa) without a satellite hoping to continue getting abduction missions. This didn't work however.

I then let a UFO escape and a Battleship turned up later and I left this alone too. It shot down the satellite over Russia.

Russia went up to a panic level of 5 because of this. Shortly afterwards however I got a Bomb disposal mission in Russia which reduced panic levels and then an UFO landing.

The next mission after that was an Abduction mission as you can see:

This leads me to think when playing Iron-man it may be worth leaving the remaining countries in your home continent without satellites. The extra cash, meld and rewards would be a decent boost. That is assuming they don't replace normal UFO missions which they may well do.