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At base level the study of Dynamical Systems means the study of time-dependent phenomena. These studies originated from the study of differential equations but are nowadays a very broad and active research area within Mathematics. It is usually divided in subfields like

• Smooth dynamics : the study of systems like differential equation systems or more generally differentiable flows on smooth manifolds etc.
• Complex dynamics : the study of map iterations on the complex plane. This is very different from Complexity or Complex Systems which typically study Kolmogorov or other complexity in sequences, emergent properties of systems etc. (these two fields may perhaps be best viewed as stradling on the border between Dynamics and Computation).
• Measurable dynamics, also known as Ergodic Theory: the study of systems whose evolution preserves a measure and hence probabilies of events can be assessed. The roots of this particular brand of Dynamics are in Statistical Mechanics.
• Topological or symbolic dynamics: the study of shift-properties of symbol sequences often obtained in first coding a smooth or measurable system. The dynamics is either a time- or space-like action on the sequences.