Gulfstream IV / G350 / G450
Flight Crew Multiengine; Flight Crew Instrument; Flight Crew Flight Instructor; Flight Crew Air Transport Pilot (ATP)
Flight Crew Last 90 Days 30; Flight Crew Total 5876; Flight Crew Type 1230
Time Pressure
Deviation - Altitude Undershoot; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance
Air Traffic Control Provided Assistance; Flight Crew FLC Overrode Automation; Flight Crew Returned To Clearance
Airspace Structure; Human Factors
Departing on the ZZZZZ departure out of ZZZ; per the SID; we took off on a heading of 343. At 9;100 feet I directed the PNF to dial in a heading 273 also per the SID. As we tracked towards the NW outbound course the winds were stronger than anticipated and by the time we would have intercepted the NW course outbound the 'smart turn' function started toward ZZZ1.  I was hand flying and decided to make the turn early so as to not fly over ZZZZZ as it would have overflown the point. The turn from ZZZZZ to ZZZ1 is fairly tight and I was concerned we would get spit out on the outside of the turn; so I followed the HUD for the 'smart turn' software and turned slightly inside ZZZZZ towards ZZZ1. As I started the left-hand turn towards ZZZ1; the Departure Controller mentioned something about a 'possible deviation' and told us to contact ZZZ Tower and gave us a land-line number. I was climbing at normal climb EPR and following HUD cues for the climb. We were also VFR; well clear of terrain (NO GPWS warnings or cautions activated) and there was no other traffic in visual range or on the TCAS. After calling Tower; we were essentially told there may or may not be more information about this possible deviation. No other information was given to us which increased stress and worry about a 'possible deviation' and its repercussions. Operations at ZZZ are difficult on a good day with the high altitude and mountainous terrain. This SID makes it even more difficult with three heading changes (one to 273; one to intercept the NW course outbound; and finally a sharp turn to ZZZ1 all while climbing to 16;000 feet).  ZZZ needs to put more emphasis on the FMS based RNAV departure procedure to reduce the workload on aircrew and increase safety overall.
Aircraft X departed ZZZ on the ZZZZZ departure. Aircraft checked on with me; I issued a climb to FL210. Initially; the pilot did not state the altitude they were climbing through; so I verified Mode C. Leaving about 11;000 feet the aircraft slowed its climb and increased speed. The aircraft did not intercept the I-PKN LDA outbound per the departure procedure. At the time the aircraft should have been intercepting the localizer; it was traveling at 310kts. 1 mile from the localizer is a MVA of 13;300 [feet]. The aircraft entered that MVA at 12;200 [feet] and 330kts. The time from crossing the localizer to entering the higher MVA was about 13 seconds. The aircraft continued its climb into another higher MVA of 14;200 [feet] and entered the MVA at 13;400 [feet] doing 335 kts. I noticed the aircraft about 1 miles from entering the 14;200 MVA and issued a turn away to lower terrain followed by a low altitude alert. I did not catch the deviating aircraft very quickly because STARS Auto Handoff was disabled and I was having difficulty re-enabling Auto Handoff. I was distracted by looking at the STARS Quick Reference Cards trying to find the command to re-enable the automation. In my opinion; regardless of how quickly the deviation was caught; the aircraft would have entered the 13;300 MVA; but if I wasn't distracted; entering the 14;200 MVA may have been avoidable. Always keep a scan vigilant even when there are only a few planes. Do not get distracted; operational distractions can happen. Use automation to the fullest.
G450 First Officer reported altitude/heading deviation during complicated departure while simultaneously ATC was experiencing departure radar automation issues.