Air carrier Captain reported loss of flight director and autopilot system during climb. Flight returned to departure airport with no other issues.

Narrative

Noticed Flight Director (FD) was giving incorrect information on initial climb out. During the climb just after 1000'; the FD was in a nose down pitch attitude. Attempted to sync multiple times; and it would continue to descend immediately after syncing. Around 8;500' I requested autopilot. However; when I tried to capture with autopilot immediately after syncing the FD; the aircraft would immediately start to descend to a nose down pitch attitude while simultaneously rolling left. I leveled off at around 10;200; (I may have slightly deviated above by 100' ft before going back to 10;000'. ATC gave us a climb to 23;000'; during the climb I attempted to set the autopilot at least 2-3 times; no joy every time. Ultimately; we were still just shy of 700nm to ZZZ and climbing through 21;000'; I decided to excersice my Captain's Authority and request a return to field". I hand flown the aircraft for the remainder of the flight. Additionally; I noticed the FD was in a climbing attitude during the descend back to ZZZ1 and it was giving incorrect information during the approach as well. All checklists; briefs and other tasks were completed promptly without error. On final I called for flaps 45 just prior to the final approach fix and we exceeded the flaps 45 airspeed limitation by 1 or 2 kias (171-172) for like a sec or 2. Completed the flight with no other issues noted."