BEIRUT, Lebanon — Many Arab governments grumbled quietly in 2011 as the United States left Iraq, fearful it might fall deeper into chaos or Iranian influence. Now, the United States is back and getting a less than enthusiastic welcome, with leading allies like Egypt, Jordan and Turkey all finding ways on Thursday to avoid specific commitments to President Obama’s expanded military campaign against Sunni extremists.

As the prospect of the first American strikes inside Syria crackled through the region, the mixed reactions underscored the challenges of a new military intervention in the Middle East, where 13 years of chaos, from Sept. 11 through the Arab Spring revolts, have deepened political and sectarian divisions and increased mistrust of the United States on all sides.

“As a student of terrorism for the last 30 years, I am afraid of that formula of ‘supporting the American effort,’ ” said Diaa Rashwan, a scholar at the Al-Ahram Center for Political and Strategic Studies, a government-funded policy organization in Cairo. “It is very dangerous.”