We show that electrons in a semiconductor superlattice can be used to realize and exploit the unique dynamics of the driven harmonic oscillator that were discovered and explored by George Zaslavsky and colleagues. Under the action of an electric and tilted magnetic field, the semiclassical dynamics of electrons in an energy band of the superlattice exhibit non-KAM chaos, which strongly affects the electrical conductivity. At certain critical field parameters, the electron trajectories change abruptly from fully localized to completely unbounded, and map out intricate stochastic webs in phase space, which act as conduction channels for the electrons. Delocalization of the electron paths produces a series of strong resonant peaks in the electron drift velocity versus electric field curves. We use these drift velocity characteristics to make self-consistent drift-diffusion calculations of the current-voltage and differential conductance-voltage curves of the superlattices, which agree well with our experimental data and reveal strong resonant features originating from the sudden delocalization of the stochastic single-electron paths. We show that this delocalization has a pronounced effect on the distribution of space charge and electric field domains within the superlattices. Inter-miniband tunneling greatly reduces the amount of space-charge buildup, thus enhancing the domain structure and both the strength and number of the current resonances.