A dynamical system combines a state space with a rule for evolution in time. A discrete rule sends a complete state to , while a continuous model can specify . The state must include enough information for the chosen rule: a harmonic oscillator needs both position and velocity, because equal positions with opposite velocities have different immediate futures.
The same model can be studied through several questions. An equilibrium point stays fixed, a periodic orbit repeats, and a basin of attraction describes which initial states approach a selected long-term behavior. Changing a parameter can cause a bifurcation. These qualitative properties can be useful even when no explicit formula for every trajectory is available. Deterministic evolution also requires well-defined forward solutions; merely writing a differential equation does not establish existence or uniqueness of its initial value problem.
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