Shortly after Mr. Kim’s statement was released, his foreign minister, Ri Yong-ho, delivered prepared remarks to reporters outside his hotel in New York, saying it was up to Mr. Kim to decide what to do, but that North Korea might conduct the “biggest ever hydrogen bomb test in the Pacific.”
Mr. Ri could not have made such an alarming comment without approval from Mr. Kim, although some analysts question whether North Korea has the technology or political daring to conduct an atmospheric nuclear test, something the world has not seen for decades.
Mr. Trump responded on Friday by further personalizing the dispute. On Twitter, the president pronounced Mr. Kim to be “obviously a madman.”
North Korea has often issued statements in the names of its government and its People’s Army, and since taking power in late 2011, Mr. Kim has delivered an annual New Year’s Day speech. But Friday’s statement was the first by Mr. Kim directed openly at a foreign head of state. Mr. Kim’s father and grandfather, who ruled North Korea before him, never made such a statement, South Korean officials said.
In effect, Mr. Kim, whose cultlike leadership rests upon his perceived daring toward North Korea’s external enemies, has turned the nation’s standoff with the United States into a personal duel with Mr. Trump, analysts said.
The North Korean news media carried photographs of Mr. Kim sitting in his office and reading his statement, but his voice was not broadcast. On the country’s state-run Central TV, a female announcer read his statement.
“This is totally unprecedented,” said Paik Hak-soon, a longtime North Korea analyst at the Sejong Institute, a think tank outside Seoul, referring to Mr. Kim’s statement. “The way North Korea’s supreme leadership works, Kim Jong-un has to respond more assertively as its enemy gets more confrontational, like Trump has.
“There is no backing down in the North Korean rule book,” Mr. Paik said. “It’s the very core of their leadership identity and motive.”
Until now, Mr. Kim himself has appeared to refrain from personal attacks on the American president, even as Mr. Trump has called him a “maniac,” a “total nut job,” and, most recently, “Rocket Man.”
On Friday, Mr. Kim said he took Mr. Trump’s latest assault personally and accused him of making “the most ferocious declaration of a war in history.”
Mr. Kim also suggested Mr. Trump’s belligerent rhetoric signaled American weakness rather than resolve. “A frightened dog barks louder,” he said. And the Korean word that translated to “dotard” was “neukdari” — a lazy, useless and demented person.
Koh Yu-hwan, a professor of North Korean studies at Dongguk University in Seoul, said that Mr. Kim, faced with Mr. Trump’s threat of annihilation, could respond only with equal force.
“When Trump stood before the United Nations General Assembly and threatened to totally destroy his country, Kim Jong-un had to take that as the United States telling the world of its intention for possible military action,” Mr. Koh said. “He had to respond in kind, launching the same kind of verbal bombs.”
Analysts said that by putting his reputation on the line with his statement, Mr. Kim was now far more unlikely to stand down. Instead, his government was likely to conduct more nuclear and missile tests, they said.
“Trump shot himself in the foot with his unabashedly undiplomatic United Nations General Assembly speech,” said Lee Sung-yoon, a Korea expert at the Fletcher School of Law and Diplomacy at Tufts University. “By threatening to totally destroy North Korea, he created the impression around the world that it is actually the United States — instead of North Korea — that’s motivated by aggression. In effect, Trump gave Kim Jong-un a freebie for another major provocation. Kim will oblige, and claim that it was in ‘self-defense’ against Trump’s unnerving threats.”
Daryl G. Kimball, executive director of the Arms Control Association, compared the Korean standoff to the October 1962 crisis over Soviet missiles in Cuba, urging the United Nations secretary general, António Guterres, to convene the six parties that were previously involved in talks on denuclearizing the Korean Peninsula — China, Japan, North Korea, Russia, South Korea and the United States — to discuss reducing fever-pitch tensions.
“We are in a cycle of escalation that leads to a very bad end,” Mr. Kimball said.
North Korea has conducted all of its six nuclear tests within deep underground tunnels to diminish the spread of radioactive materials, and has stepped up the pace of its missile tests. Some analysts fear that the next step might be for North Korea to try to prove that it can deliver a nuclear warhead on a long-range missile, no matter how dangerous and provocative that might be.
It has been 37 years since any nation tested a nuclear weapon in the planet’s atmosphere, reflecting the nearly universal opposition to such tests over fears of the effects of radioactive fallout on human health and the environment. The last atmospheric test took place in 1980, when China fired what experts believed to be a nuclear-tipped ballistic missile into a desert salt flat more than 1,300 miles west of Beijing.
Shin Beom-chul, a security expert at the government-run Korea National Diplomatic Academy in Seoul, said that even if North Korea wanted to conduct an atmospheric nuclear test in the Pacific, it did not have the ability to dispatch test-monitoring ships to the open ocean while the United States military was on the prowl.
Mr. Shin said North Korea probably would not risk the radioactive fallout and other grave dangers involved in a nuclear missile test. The country has yet to master the technologies needed to prevent the warhead at the tip of its long-range ballistic missile from burning up while re-entering Earth’s atmosphere, South Korean officials said.
“What if the nuclear missile goes wrong midflight and detonates over Japan? It would mean a nuclear war,” Mr. Shin said. “More likely, North Korea will graduate its provocations, as if moving on steppingstones.”
Analysts said North Korea had been escalating tensions in stages in what they called a “salami tactic,” as in slice by slice.
Kim Dong-yub, a defense analyst at the Seoul-based Institute for Far Eastern Studies of Kyungnam University, said that North Korea would probably try to disprove skeptics in the West over its ability to strike long-range targets by firing its Hwasong-14 intercontinental ballistic missile over Japan and farther into the Pacific — but without a nuclear payload.
Some analysts said the North Korean leader was acting more defensively than offensively, with his threats aimed at forcing the Trump administration to ease sanctions. On Thursday, Mr. Trump issued an executive order empowering his government to punish international banks and other entities that trade with North Korea.
But other analysts warned that North Korea’s determination to improve its nuclear capabilities — and act offensively — had long been underestimated.
“If we follow what North Korea has been doing, it will be almost certain that it will fire its missile sooner or later to demonstrate an ICBM range,” Mr. Kim, the Kyungnam University analyst, said. “I don’t think the missile will carry a nuclear warhead, but I can’t shake off the fear that it might, because North Korea has time and again carried things beyond my expectation.”