By applying a positive potential on an electrode immersed in a dc discharge plasma, a complex space charge structure (CSCS), consisting from a positive ion-enriched plasma region confined by an electrical double layer (DL), appears. Under certain experimental conditions, a more complex structure was observed, which was called multiple double layers (MDLs). Depending on the experimental parameters (especially the geometry of the experiment, the gas nature and pressure and the plasma density), this structure can appear in two geometrical forms: concentric (as several bright plasma shells attached to the electrode) and non-concentric (as a network of luminous plasma spots, each near others, almost equally distributed on the electrode surface). Here we present experimental results concerning the dynamic states of multiple double layers in dc discharge plasma. These dynamic states were recorded as strongly nonlinear oscillations of the current collected by the electrode. The oscillations appear because the plasma system performs fast transitions between states characterized by different electrical conductivities. This bistability was analyzed by the modern techniques provided by the nonlinear dynamics. In the case of non-concentric MDLs, a scenario of transition to chaos by torus breakdown was emphasized.