“Despite it all, China has been an increasingly responsible actor on Iran,” said James B. Steinberg, a former deputy secretary of state who made a number of trips to Beijing to air American concerns. “Despite some wobbles, they’ve played a positive role in constraining North Korea at times of crisis.”

The president’s Asia agenda, however, raises many questions. With deep cuts in the military budget looming, critics question whether the United States has the money to back up its words. A Pentagon preoccupied by Afghanistan and Iraq has done little planning to shift troops or ships — so little, in fact, that a Navy commander was called to the White House for his first meeting after Mr. Obama had already laid out the broader strategy.

America’s eastward shift has left the Chinese deeply suspicious of American motives, with some analysts in China arguing that the United States is trying to encircle the country. For all the talk of give-and-take, the Chinese rebuffed Mrs. Clinton during her recent visit to Beijing when she raised the disputes over the South China Sea.

“The Chinese feel a bit whiplashed,” said Michael J. Green, an Asia policy maker in the administration of George W. Bush who is now at the Center for Strategic and International Studies. “The hope and change of the first year, followed by the sharp-edged push-back of the second year, all of this, to the Chinese, looks like gross inconsistency and unpredictability.”

The President’s Asia

It is little surprise that Mr. Obama would look east. The president’s Asia, however, lies not on the wind-swept ramparts of the Great Wall of China but in the tropical swelter of Singapore and Indonesia. He identifies more with the languid rhythms of Jakarta, aides say, than with the crackling energy of Shanghai.

An adviser recalled a breakfast at a summit meeting in Toronto in 2010 that Mr. Obama shared with President Susilo Bambang Yudhoyono of Indonesia, which was so relaxed and serene that afterward the president’s hyperactive chief of staff, Rahm Emanuel, told him, “Now I see what your Asianness is about.”

Despite his preferences, Mr. Obama was determined not to antagonize China when he ran for president in 2008. Unlike Mr. Clinton, who referred to China’s leaders as the “butchers of Beijing” in 1992, Mr. Obama said little about China, and his thin record on foreign policy left few clues for the Chinese to size him up.