Understanding how complex systems respond to change is of fundamental importance in the natural sciences. There is particular interest in systems whose classical newtonian motion becomes chaotic^{,,,,,,,,,,,,,,,,,,,,,} as an applied perturbation grows. The transition to chaos usually occurs by the gradual destruction of stable orbits in parameter space, in accordance with the Kolmogorov–Arnold–Moser (KAM) theorem^,,,,,,—a cornerstone of nonlinear dynamics that explains, for example, gaps in the asteroid belt. By contrast, ‘non-KAM’ chaos switches on and off abruptly at critical values of the perturbation frequency^,,,. This type of dynamics has wide-ranging implications in the theory of plasma physics, tokamak fusion, turbulence^,,, ion traps, and quasicrystals^,. Here we realize non-KAM chaos experimentally by exploiting the quantum properties of electrons in the periodic potential of a semiconductor superlattice^,,,,, with an applied voltage and magnetic field. The onset of chaos at discrete voltages is observed as a large increase in the current flow due to the creation of unbound electron orbits, which propagate through intricate web patterns^,,,,,,,,, in phase space. Non-KAM chaos therefore provides a mechanism for controlling the electrical conductivity of a condensed matter device: its extreme sensitivity could find applications in quantum electronics and photonics.