Dynamical system

A mathematical model in which a state changes over time according to a specified rule. Its evolution may be continuous, as in differential equations, or discrete, as in difference equations.

State space: The set of all possible states of a system, often represented as points in a mathematical space. Each point represents the system's complete state at a given time.
Initial condition: A specified value of a system's state at a starting time. Together with the evolution rule, it determines a trajectory.
Celestial mechanics: The study of the motions of celestial bodies under gravitational forces. Orbital motion is modeled by evolving positions and velocities under physical laws.
Static model: A mathematical representation of relationships among quantities without modeling their change over time. Unlike a dynamical system, it does not specify temporal evolution.
Ordinary differential equation: An equation relating a function of one variable to its derivatives. It specifies continuous-time evolution through the rate of change of state.
Trajectory (dynamical systems): The path traced by a system's state through its state space over time. Trajectories show the outcomes of applying the system's rule from particular states.
Population dynamics: The study of how population sizes and structures change over time. Population equations describe growth, competition, and predator–prey cycles.
Stochastic process: A collection of random variables indexed by time or another parameter. It introduces probabilistic evolution where a deterministic rule gives a fixed next state.
Difference equation: An equation defining a sequence by relating its terms at different indices. It specifies discrete-time evolution from one state to the next.
Fixed point (mathematics): A point left unchanged by a function or transformation. In a dynamical system, fixed points are states that do not evolve.

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