WASHINGTON — President Obama ordered his advisers last August to produce a secret report on unrest in the Arab world, which concluded that without sweeping political changes, countries from Bahrain to Yemen were ripe for popular revolt, administration officials said Wednesday. Mr. Obama’s order, known as a Presidential Study Directive, identified likely flashpoints, most notably Egypt, and solicited proposals for how the administration could push for political change in countries with autocratic rulers who are also valuable allies of the United States, these officials said.