Car revs to 1000 RPM then drops when idling?

My car sputters, and noisily revs up to 1000 RPM then drops to about 500 RPM and then revs up to 1000 RPM again. It takes about 10 seconds each time to do this. The car shakes a little and worsens each cycle the longer I have to sit at a light. There's no Check Engine light on. None of the lights on the dashboard are on. It's a little automatic 2002 Ford Focus ZX3. I know there could be numerous problems, but could anyone happen to guess what it could be?

My first thing to check would be a vacuum leak, no question. You can get a short piece of tubing to use as a stethoscope. Put one end in you ear and with the other, move it around the different vacuum hoses and fittings, with the engine running. Careful of the fan! When you find a vacuum leak, it will be very pronounced and loud compared to all the other fittings and hoses.

This is Almost normal, because the O2 sensor is supposed to provide a feedback loop similar to this; but it should be fast enough and subtle enough so you don't notice.
So something is exaggerating it.
A vacuum leak can do it, such as stuck ERG, PCV, purge valve, etc.
So can a sticking or dirty idle solenoid control.
But a dirty fuel filter or weak pump can also do it.
Fuel pressure should be 32 psi I believe.
And seeing if the gas cap changes things, is a good idea, although caps do NOT have a vent any more and they vent through the purge valve after the charcoal canister.

Remove the gas cap just for a quick test. If the problem goes away, then it is the vent hole clogged in your gas cap. Just buy a new gas cap and problem is fixed. It is kind of rare for a gas cap to go bad, but it does happen and it is not something you would think to check. Another symptom of this is every time you remove the gas cap to fill up the tank… You hear a suction sound as you remove the cap. It makes that sound because as you drive, the gas level goes down, but no air comes into the tank because the gas cap has the clogged vent hole.

You have LOTS of money?(LOL)! This engine was co-designed by PORSCHE for Ford and will last all VERY LONG LONG TIME! Your issue NOW is only a dirty IDLE AIR CONTROL VALVE and THROTTLE BODY. Find a source for AMSOIL POWER FOAM ($8) and a can of GUMOPUT! The IACV a On AL FORDS is removebale for cleanign up! Clean teh passages UDNER it as well! Run the POWER FOAM through your PCV to clean the oil control rings on your pistons as well! Clean BOTH sides of the THROTTLE BODY butterfly plates! This will return NORMAL running to your fine ZX3/PORSCHE Use ONLY MOBIL ONE 0W-30 oil to prevent more sensor problems and change it ONCE a year only! Will keep your sensors happy and your converter lasting the longest! SEND ME THE MONEY you save on fuel for the next SIX MONTHS! This will RESTORE full power to engine and get you MORE smooth running! For MORE power, use BOSCH or DENSO iridium plugs WORTH EVERY RED CENT! These three things will make you fall in love with it all over again! TRULY!

My first guess would be a partially plugged fuel filter. Check when it was last replaced.