Lear 60 First Officer reported their primary flight display would not sync with Captain's and the headings did not match during initial climb. The flight crew returned to the origin airport.

Narrative

The first issue occurred from our flight from ZZZ to ZZZ1 at approximately XA:18. Following the ZZZZZ Departure out of ZZZ; we activated the autopilot; which was paired with the navigation from the co-pilot's side. Upon reaching the first fix; we noticed the plane was not turning to the next fix. Noticing that FMS 1 was navigating correctly; we immediately switched the navigation source from the FMS 2 to FMS 1 (pilot's side). The plane then corrected course and flew the departure with no further issues. Our FMS did show an error message that FMS 2 on co-pilot's side was 'difference>12nms' from FMS 1. We continued to check the system during cruise flight down to ZZZ1 and the lateral difference between both FMS were eventually identical. We continued to the destination with out issue and landed safely. Our second flight plan was from ZZZ1 to ZZZ. We had conducted checks from after start through taxi phase to ensure the FMS and heading were reading correctly. No error was shown and the system was functioning as intended. During the take-off roll at approximately 100 kts; we had Amber Heading lights flash on both primary flight displays. Since our airspeed was about 90 kts; we elected to continue the take-off since it was a non critical item and should be fixed in the air. After take-off; the heading bugs on both primary flight displays were showing approximately 120 heading instead of 090 for a Runway XL departure. We advised ATC that we would need vectors and used the standby compass to follow ATC vectors. Due to the erroneous heading on PFD (Primary Flight Display) 1 and 2; we notified ATC that due to safety of flight; we would like to return to ZZZ1. We held for 45 minutes to burn required fuel to land below certified max landing weight. During both events; no deviations or limitations were conducted or exceeded.