Phase portrait

A phase portrait is a geometric depiction of trajectories and equilibrium points in a dynamical system’s state space. It shows how states evolve over time without requiring a single initial condition.

Dynamical system: A system whose state changes over time according to a specified rule. A phase portrait summarizes the possible time evolutions of such a system.
Trajectory (dynamical systems): The path traced by a system’s state as time evolves from an initial condition. The collection of trajectories forms the central visual content of a phase portrait.
Phase plane: A two-dimensional state space, often formed from a variable and its derivative. Planar systems can be drawn directly, making their orbit geometry especially clear.
Time series: A sequence of observations recorded at successive times. Unlike a portrait, a time series shows values against time for selected trajectories.
Phase space: The space of all possible states of a dynamical system. Each point in a phase portrait represents one system state.
Stability theory: The study of how solutions respond to small changes in initial conditions or parameters. Nearby trajectories reveal whether an equilibrium attracts or repels states.
Pendulum: A suspended mass that swings under gravity, often modeled as a dynamical system. Its phase portrait distinguishes oscillation from complete rotation.
Bifurcation diagram: A plot showing how long-term states change as a system parameter varies. It tracks parameter-dependent changes that a single phase portrait holds fixed.
Vector field: A function assigning a vector to each point in a space. For continuous systems, its vectors indicate the local direction of trajectories.
Nullcline: A curve or surface on which one component of a system’s velocity vanishes. Nullclines help locate equilibria and infer flow direction in planar portraits.

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