tThe sSaudi ideology hasquietly changed smoothly over the last years of king abdullahin King Abdullah's final years The Saudi ideology quietly changed in King Abdullah's final years

If this change has relevance for the present, then you would write "has changed" (present perfect). But if you're chiefly commenting on that period of history without linking it to the present, then you would just say "changed" (simple past).

the saudi ideology has changed smoothly over the last years of king abdullah the saudi ideology has changed smoothly over the last years of king abdullah

Saudi's political ideology has swiftly changed during the last years of King Abdullah's reign.