Considering a quantized chaotic system, we analyze the evolution of its eigenstates as a result of varying a control parameter. As the induced perturbation becomes larger, there is a crossover from a perturbative to a nonperturbative regime, which is reflected in the structural changes of the local density of states. The full scenario is explored for a physical system: an Aharonov-Bohm cylindrical billiard. As we vary the magnetic flux, we discover an intermediate twilight regime where perturbative and semiclassical features coexist. This is in contrast with the simple crossover from a Lorentzian to a semicircle line shape which is found in random-matrix models.