= Dynamical system
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A <dynamical system> combines a <state space> with a rule for evolution in <time>. A discrete rule sends a complete state $x_n$ to $x_{n+1}=F(x_n)$, while a continuous model can specify $\dot x=f(x)$. 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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