A DEP CARJ FLT CREW GETS SOME ASSISTANCE FROM DEP CTLR WHEN THEIR FMS AND DIRECTIONAL GYROS CREATE A NAV PROB FOR THEM AFTER LIFTOFF FROM JFK; NY.

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IMMEDIATELY AFTER LIFTOFF; WE GOT HEADING DISAGREEMENT WARNINGS ON BOTH PLT EFIS DISPLAYS. THIS IMPLIES A LOSS OF HEADING ACCURACY; WHICH WAS NOT CRITICAL AT THAT MOMENT. OUR DEP PROC SIMPLY REQUIRED A TURN TO TRACK DIRECTLY TO A VOR; THEN OUTBOUND ON A RADIAL. THE FO DID A GOOD JOB OF STAYING FOCUSED ON FLYING THE ACFT; WHILE I EXECUTED THE NORMAL CHKLISTS; TALKED TO ATC; AND ATTEMPTED TO CORRECT OUR COMPASS PROB. WE WERE WELL INTO THE DEP PROC; WHEN OUR FMS SYS BEGAN WARNING US TO 'CHK POS.' ON TOP OF THIS; THE COMPASS SYS STUBBORNLY RESISTED MY ATTEMPT TO FIX IT. USUALLY THIS IS A SIMPLE TOGGLE AND SWITCH THROW. THE CTLR THEN TURNED US TO THE FIRST ENRTE FIX. WHEN THE FO LEVELED THE TURN; I XCHKED THE MAGNETIC COMPASS AGAINST THE EFIS AND COMPLETED THE HEADING CORRECTION; WHICH CLRED THE HEADING WARNING. THE FO WAS HEADING DIRECTLY TO THE FIX; BUT ATC CALLED TO WARN US THAT WE WERE NOT QUITE HEADED PROPERLY. SHE (THE CTLR) SAID 'YOUR FMS IS WRONG; TURN 10 DEGS R!' I COULD NOT TELL HOW GOOD OR BAD OUR FMS INFO WAS; BUT WE DID CONTINUE TO HAVE OUR 'CHK POS' WARNING EVEN THOUGH I HAD REPEATEDLY TRIED TO RESET THE SYS RELATIVE TO LCL NAVAIDS. I TOLD THE CTLR THAT WE HAD BEEN HAVING A HEADING PROB. ANOTHER CTLR CAME ON FREQ TO SIMPLY DIRECT US TO THE NEXT ON COURSE VOR. WE DID THIS ON 'RAW DATA' EASILY. AFTER NUMEROUS TRIES; I WAS FINALLY ABLE TO HELP THE FMS FIND ITSELF; AND WE NAVED SUCCESSFULLY FOR THE REMAINDER OF THE FLT USING IT. IT IS SAD THAT OUR FMS AND HEADING BECOMES OCCASIONALLY LOST IN NEW YORK (ESPECIALLY AT LGA) AND FLORIDA. OBVIOUSLY; WE ARE CAPABLE OF FLYING WITHOUT FMS; BUT THE SYS SEEMS TO BE ALMOST PREDICTABLY UNRELIABLE AT TIMES. THESE FAILURES USUALLY CORRECT THEMSELVES; BUT I WAS CAUGHT OFF-GUARD BY THE PERSISTENCE OF THIS MULTIPLE FAILURE. I SHOULD HAVE NOTIFIED ATC IMMEDIATELY THAT OUR FMS/HEADING INFO WAS SUSPECT. I AM NOT LOOKING FORWARD TO USING THIS SYS IN IMC IN HIGHLY MOUNTAINOUS TERRAIN. OUR COMPANY IS AWARE OF THE WEAKNESS OF THIS SYS.