The biggest risk, critics say, is that Mr. Trump will talk himself into a war. Only slightly less dangerously, he could weaken the nation’s standing by backing off from a threat to use force.
“In Beijing, Moscow, Tehran, they are recalibrating their strategies — you can’t deny it — because they don’t have any idea of how Trump will respond,” said Senator Mark Warner of Virginia, the highest-ranking Democrat on the intelligence committee.
“That might be great in the short term,” he added, “but it’s not really a long-term strategy for asserting leadership in a world desperate for American leadership.” Mr. Warner, who criticized Mr. Obama for his failure to act more strongly in Syria, said: “China, Russia and Iran have real, long-term strategies. Why don’t we have one, too?”
Mr. Trump did not time the strike against Syria to impress Mr. Xi, according to White House officials. But he clearly recognized that disclosing the news during their dinner in Palm Beach, Fla., had a dramatic flair that would establish his toughness and unpredictability, while also pressuring Beijing to tame North Korea, its misbehaving client state.
The president’s defenders say those qualities will help restore America’s place in the world. “He’s far more in keeping with 70 years of postwar American leadership than Obama was,” said Senator Tom Cotton, an Arkansas Republican and staunch Trump ally.
But Mr. Trump’s show of strength in the Middle East was undercut in his response to North Korea by one of his administration’s all-too-common errors. After Mr. Trump warned that “we’re sending an armada” to the waters off the Korean Peninsula, the Carl Vinson, the aircraft carrier that leads the strike group, was photographed sailing through Indonesia, thousands of miles away.
“Your words have to match your actions,” said Senator Jack Reed, a Rhode Island Democrat and former Army Ranger who is the ranking member of the Armed Services Committee. “If it’s just bluffing, well, that’s dangerous. If it’s because the president was not informed and a mistake because he had bad information, that’s problematic, too.”
In South Korea, feelings were raw, with newspaper headlines branding the episode “Trump’s lie over the Carl Vinson” and politicians warning that they might never again be able to trust the president’s word.
Mr. Trump has pivoted to foreign affairs after a succession of humbling domestic policy defeats — discovering, as his predecessors did, that presidents can operate with more latitude in matters of war and peace than on tax policy or health care legislation.
In a series of taunts, Twitter messages and hawkish pronouncements by surrogates like Vice President Mike Pence, Mr. Trump has overturned Theodore Roosevelt’s dictum to “speak softly and carry a big stick.” But his bombastic statements have often been paired with policy reversals, on matters like NATO, which he once wanted to mothball and now supports, or Russia, which he once saw as a potential ally and now views with suspicion.
Though Mr. Trump’s words can be harsh and intemperate, his actions have proved less so. As a result, diplomats say, leaders are not yet able to draw firm conclusions about his foreign policy.
“There is the impression that President Trump is moving away from his campaign statements and pivoting back to the Republican mainstream on major foreign and security issues,” said Peter Wittig, the German ambassador to the United States. “But people in Europe aren’t connecting the dots and saying, ‘This is the new Trump doctrine.’”
Foreign-policy theorists sometimes compare Mr. Trump’s erratic approach to that of President Richard M. Nixon, who pursued what he called the “madman theory” of statecraft. By behaving vaguely unhinged — obsessed with Communism, his finger poised unsteadily on the nuclear button — Nixon hoped to force North Vietnam into negotiations to end the Vietnam War.
“It was aimed at both our allies and adversaries, and it appears to have worked, to some degree,” said Eric S. Edelman, a former under secretary of defense for policy during George W. Bush’s administration who now teaches at Johns Hopkins University.
But Mr. Edelman drew some critical distinctions between the two presidents. Nixon’s “madman” act generally masked a calculated strategy, which is not yet evident in Mr. Trump’s approach. Nixon’s national-security team was better coordinated than Mr. Trump’s, at least so far. And even in Nixon’s case, the madman strategy worked better later in his presidency, when he and his aides were more seasoned.
Mr. Trump won praise for his missile strike on Syria, even from those who have criticized his approach to other crises. Though the president moved swiftly — and by all accounts, emotionally — after a deadly chemical weapons attack by Syria’s president, Bashar al-Assad, the attack was measured, well planned and followed by an aggressive White House effort to establish Russia’s complicity with the Assad government.
“That missile strike certainly had to get Putin’s attention, and it did show we were determined to enforce international norms on chemical weapons,” said Antony J. Blinken, who was deputy secretary of state and deputy national security adviser in the Obama administration. “Equally important was the effort to tie Russia to the use of chemical weapons.”
Mr. Blinken has more reservations about how Mr. Trump has approached North Korea. While in the White House, Mr. Blinken helped coordinate a two-pronged pressure campaign against the North Korean government. The first part involved leaning on China to use its vast leverage over Pyongyang. The second involved persuading other countries that do business with North Korea to refuse entry to its guest workers; expel its diplomats, who are engaged in illicit activities; and deny landing rights to its state airline.
Mr. Trump has opted for a noisier, more direct approach, threatening North Korea with military action if it does not curb its provocations. But behind the hard-line rhetoric, the president is actually pursuing a strategy not unlike that of his predecessor: tightening the economic vise on Pyongyang in the hopes of forcing it to make concessions.
The trouble with Mr. Trump’s approach, Mr. Blinken said, is the gap between his words and his actions. “You risk others miscalculating on the basis of bravado,” he said. “We always thought it was better to talk softly but clearly, and to carry a big stick.”