It is almost always caused by a too lean pilot jet circuit. That means the brass pilot jet metering orifice (at the center of the end of the jet) and the pilot adjusting screw is slightly gummed up. The pick below is for the EF1000iS, but it is similar to the EF2000iS. To remove the pilot screw for cleaning, you have to remove the plastic limiter tab on the screw cap itself. I would either cut it off with a small Dremel cut-off disc -or- melt it away with a soldering gun. DO NOT cut it off with a knife or a pair of Dykes (wire cutters). Why? Because I found that the brass screw shaft has a very small necked down diameter under the plastic cap to punish people that try to force things, even just a little bit! Immediately after removing the tab, turn the screw IN slowly, counting the turns, so you know where to reset is as a starting point.
IMPORTANT: Be sure the carb's throttle plate tab is pushed all the way to the full throttle stop after remounting the carb (if removed) so the throttle stepping motor is correctly set. After the engine runs again with ECO idle immediately ON, after 2 minutes running (for ideal adjustment engine temp) set the pilot screw by turning it in (CW) until it falters and RPM cycles, then turn it out (CCW) until the RPM is stable and "sounds smooth".
If it still cycles you may have corrosion in the pilot jet metering orifice, something only "mechanical cleaning" or replacement can cure.
Thanks for this. Saved the day. One thing I would add is that it is near impossible on a 1000 to adjust the pilot and/or the idle screw with the throttle control and its mounting bracket on carb. After many times off and on and a whole lot of swearing I ended up taking off the throttle control and bracket. I re-installed the carb and started it up. (Make sure you hold the throttle open when starting.) Once it warmed up I adjusted the idle and pilot screw so that it ran smooth at a fast idle. One thing that helped is to find a setting between idle screw and pilot where the engine doesn't die after quickly coming off of full throttle. It won't be perfect but you will find a sweet spot where you need to feather the throttle (with finger) a minimal amount after closing from full. I found the pilot adjustment to be extremely sensitive so patience is a virtue. the difference between too lean and too rich was within a turn on mine. So finding the right idle setting was key and then smoothing out with pilot.
Once I was happy I took off carb and reinstalled throttle control. It started up and took a minute or two to come up to proper rpm. Not sure if it was not getting gas or the throttle control needed to calibrate? It behaved like it had no fuel. But after a minute it came up to normal and ran honestly better than even new out of the showroom.
Thanks for the awesome advice CycleRob!