We suggest a mechanism which leads to 3+1 space-time dimensions. The Universe assumed to have nine spatial dimensions is regarded as a special nonlinear oscillatory system—a kind of Einstein solid. There are p-brane solutions which manifest as phase oscillations separating different phase states. The presence of interactions allows for bifurcations of higher-dimensional spaces to lower-dimensional ones in the form of brane junctions. We argue this is a natural way to select lower dimensions.