GLF4 FLT CREW EXCEEDS PROTECTED AIRSPACE WHEN TURNING FROM A699 TO AR16 AND LOSES SEPARATION WITH ACFT ON PARALLEL AR18.

Narrative

ON DEP FROM PBI; I WAS CLRED ON A699 TO PERMT INTXN AR16 TO ILM. ON CLBOUT ON A699 I WAS CLRED TO MAINTAIN FL280 AND TO INTERCEPT AR16 AT PERMT. THE ACFT WAS BEING FLOWN ON AUTOPLT AND FMS MODE (BLUE DATA) AND LEVEL AT FL280 APCHING PERMT. THE FMS SEQUENCED TO THE NEXT WAYPOINT; SNABS; WHICH IS ON AR16 AND ACFT BEGAN A SHALLOW L TURN TO INTERCEPT AR16 AT PERMT. APCHING PERMT I REALIZED THAT THE AUTOPLT WAS TURNING THE ACFT AT TOO SLOW OF A TURN RATE TO INTERCEPT AR16 PROPERLY. I SWITCHED THE AUTOPLT TO HDG MODE AND INITIATED A TURN TOWARD THE NW TO INTERCEPT AR16. ZMA CALLED AND ASKED IF WE WERE INTERCEPTING AR16 AND THE FO REPLIED THAT WE WERE IN A TURN TO INTERCEPT. ATC THEN RESPONDED AND INFORMED US THAT THEY SHOWED THE ACFT 3 MI E OF AR16 AND TO MAINTAIN A 340 DEG HDG TO INTERCEPT AR16 WHICH WAS COMPLIED WITH. I WAS THEN INSTRUCTED TO CONTACT THE ZMA WATCH DESK UPON MY ARR AT TEB. I GAVE MIAMI REQUESTED PIC INFO AND WAS THEN TOLD THAT MY ACFT CAME TO WITHIN 4.7 MI OF AN ACFT ON AR18 ALSO AT FL280. AT NO TIME DID I RECEIVE A TCAS ALERT; WARNING OR RESOLUTION. I WAS FLYING THE ACFT AND THE PIC. THERE WERE NO NEGATIVE FACTORS AFFECTING PERFORMANCE. ACTION TO OVERRIDE AUTOMATION SHOULD HAVE BEEN TAKEN SOONER.