Reasoning allows artificial intelligence models to revisit and correct their mistakes, enabling recent frontier advances in mathematical theorem solving, software engineering, and autonomous task planning. Reasoning models are widely observed to reason for longer on harder tasks, but the general mechanism responsible for these slowdowns is unknown. Here, we show that reasoning models exhibit transient chaos, a physical consequence of the computational complexity of difficult tasks. As a consequence, we show that diverse leading reasoning models are dynamical systems with fractal basins, with fractality increasing with task difficulty across diverse tasks like Sudoku and maze solving, visual puzzles, and mathematical logic. We show that transient chaos emerges due to reasoning becoming trapped for extended durations near saddle points, which we show correspond to nearly-correct attempted solutions of the underlying problem. Our results show that reasoning slowdowns are an inevitable consequence of problem hardness in modern artificial intelligence models, and establish reasoning traces as a rich new class of dynamical system.