BEIJING — As Kim Jong-un, the young leader of North Korea, consolidates his grip on power, China is showing signs of increasing frustration at the bellicose behavior of its longtime ally.


Since succeeding his father, Kim Jong-il, six months ago, Mr. Kim has quickly alienated the Obama administration and put North Korea on track to develop a nuclear warhead that could hit the United States within a few years, Chinese and Western analysts say.


Most surprising, though, is how Mr. Kim has thumbed his nose at China, whose economic largess keeps the government afloat. For example, shortly after Mr. Kim took over, a Chinese vice minister of foreign affairs, Fu Ying, visited Pyongyang, North Korea’s capital, and sternly warned him not to proceed with a ballistic missile test. The new leader went ahead anyway.


Now, the Obama administration and the Chinese government, who warily consult each other on North Korea, are waiting to see if Mr. Kim will follow in his father’s footsteps and carry out a nuclear test, which would be North Korea’s third. The previous tests were in 2006 and 2009.


This month, the North Korean news agency said there were no plans for a third test “at present,” a statement analysts said suggested Mr. Kim was just waiting for a moment that better suited him.


“We have made this absolutely clear to them; we are against any provocation,” Cui Tiankai, another Chinese vice minister of foreign affairs, said in a recent interview when asked about a possible third nuclear test by North Korea. “We have told them in a very direct way, time and again, we are against it.”


Asked why China did not punish North Korea for its actions, Mr. Cui replied: “It’s not a question of punishment. They are a sovereign state.”


China backed sanctions against North Korea at the United Nations Security Council after the first two nuclear tests, he said. “If they refuse to listen to us,” he added, “we can’t force them.”


Mr. Kim’s erratic behavior unfolded early on. In late February, his government signed an agreement with the United States to freeze its nuclear weapons and ballistic missile programs, giving hope that he would turn out to be more open to change than his father. But six weeks later, Mr. Kim ripped up the accord and, without informing China, ordered the missile test that Washington viewed as a test run for launching a nuclear weapon.


The missile test, in April, was a failure, but that did little to alleviate concerns within the Obama administration that Mr. Kim was intent on pushing ahead with its nuclear weapons program. “The North is on track to build a warhead that could in a few years hit any regional target and eventually the United States,” said Evans J. R. Revere, a former United States principal deputy assistant secretary of state for East Asian and Pacific affairs.


Since the failed missile test, Mr. Kim has formalized North Korea as a “nuclear armed state” in the Constitution, another signal that the government has no intention of giving up its nuclear program, Mr. Revere said. With virtually no contact between the United States and North Korea, Mr. Revere argued, it is time for Washington to toughen its approach.


In a series of quick maneuvers, Mr. Kim, whose exact age is not known (he is believed to be 28 or 29), assumed the mantle of power immediately after his father’s death and cast aside early assumptions that his tenure would be a regency largely run by his elderly relatives.


The China News Service, a state-run agency, headlined an article last week: “Smooth transfer of power six months after Kim Jong-il’s death. North Korea enters era of Kim Jong-un.” The top North Korean Army generals, some of them in their 80s, have joined ranks around Mr. Kim, presenting a unified command, said Daniel A. Pinkston of the International Crisis Group in Seoul, who has written a forthcoming report by the group on North Korea.


At a congress of the ruling Communist Party in April, members of the Kim family were appointed to senior positions in the Politburo. The new appointees included Kim Kyong-hui, a younger sister of Mr. Kim’s father. Her husband, Chang Song-taek, also won a spot on the Politburo.