Bicycle Mechanics - Fixed/single bike chain problem

Hello, I'm riding a single gear/fixed gear bike.
When I'm on a fixed gear pedaling, my chain seems to be twitching? in some parts of the pedal,
also when I flip for a single, and pedal, the chain does not seem to be a doing perfect motion, but a bit of going up and down by not much.
I don't know if I explained it clear enough, but any help would be very appreciated.
Thank you so much.

Chainrings and cranks are seldom absolutely round or concentric with the crank axis. What you are seeing is a result of this eccentricity. Sometimes you can minimize this by rotating the ring one or two bolts away from where it is presently mounted; sometimes not. If it works for you, it's the best and cheapest fix. If not, you need to choose between living with it or replacing the chainring with a different one, which may or may not solve the problem. Usually this is only a cosmetic problem, but sometimes it can be bad enough to allow the chain to derail.

Another thing to keep in mind is that road cranks and chainrings are sometimes not machined to the same standards as track cranks and chainrings, since road bikes usually have a derailleur to maintain sufficient chain tension. This is one of the reasons why track equipment tends to cost more than road equipment.

As John explained the slack/tight effect is because of non-concentricity in the chainring, and to a lesser extent the rear sprocket. It's important that there's at least a tiny bit of slack at the tightest point, so take a moment to check that by watching the loop as you turn the cranks and feeling for some vertical play when it's tightest.

The twitching, if I'm reading you right, might be that the chain isn't spooling smoothly onto the chainring. It might be a chain-line issue, too tight a chain, or lack of lubrication that makes the chain run roughly. Check all three and all should be well.