I found these caterpillars chowing down on my Jerusalem Artichokes this morning. Its early fall, September, and they managed to eat about an 1/8th of one plant so far.
I think I have correctly identified them as Gorgone Checkerspot caterpillars. I took some of the caterpillars off the plant and placed them in a jar near a wasp nest. I haven't seen the wasps eating them yet.
If anyone knows a natural predator please let me know.
If they are concentrated in one place on the plant, I would just cut that piece off and crush it. JA are wicked strong plants and will easily regrow the missing bits.
If they are all over the plants spread out then you may just have to go on a seek and destroy mission. This depends on how many plants you're dealing with and how bad the damage is, as well as how much you depend on that harvest.
If you leave them there and keep an eye on them, you'll probably find that something likes to eat them. But that requires standing by while the caterpillars eat your food.
If the stand is small, your fingers are the predator. If the stand is big, chop it. The size of the stalk near the end of it's life (september) is not really important for the size of the tuber. The tuber has more or less reached maturity. And they just re-grow stalk and leaves as needed. Since you're not eating the leaves, you could probably just let the nibblers eat. It's much more of a problem on brassicas when you are eating the leaves so they need protection.
Birds are probably a predator, but I'm guessing you haven't researched which birds eat those things and planted the right plants to attract those birds. That would be a long-term solution.
Do you have something you can feed them to? That might be worth the effort of picking them off.
Another option is finding out what that animal REALLY likes to eat and planting a catch crop. You're currently feeding it apples but you could be growing ice-cream for it, which would divert them away from your sunchokes.
I was thinking that the sunchokes may be the catch crop, to keep them off of other things. Doing a little googling I found that they only eat sunflowers, and the like, so not to much to worry about in my mind. As already said your food crop is under ground.
I would let them do their thing and see what mother nature does with them in the years ahead.
I did some research for beetles that were sucking my tomatoes dry and I found that they prefer Sorghum. Maybe your caterpillers do to.
You might try to diversify your Asteraceae. I'm looking into a second Sunchoke-type plant, Heliopsis helianthoides (edible roots). And regular sunflower too. Maybe Solidago if you can get a use out of it (kind of creeping, and caterpillars ignore it in my garden).