What is the minimum phase space dimension for chaos?
This is a slightly confusing topic, since the answer depends on the type of system considered. First consider a flow (or system of differential equations). In this case the Poincaré-Bendixson theorem tells us that there is no chaos in one or two-dimensional phase spaces. Chaos is possible in three-dimensional flows–standard examples such as the Lorenz equations are indeed three-dimensional, and there are mathematical 3D flows that are provably chaotic (e.g. the ‘solenoid’). Note: if the flow is non-autonomous then time is a phase space coordinate, so a system with two physical variables + time becomes three-dimensional, and chaos is possible (i.e. Forced second-order oscillators do exhibit chaos.) For maps, it is possible to have chaos in one dimension, but only if the map is not invertible.
