Attractor

An attractor is a set of states toward which nearby trajectories of a dynamical system evolve over time. It may be a fixed point, a repeating orbit, or a more complex invariant set.

Stable fixed point: An equilibrium state that nearby trajectories approach as time passes. This is the simplest attractor: motion settles to one state.
Dynamical system: A rule that specifies how a system's state changes over time. Attractors describe the long-term behavior generated by these evolution rules.
Lorenz system: A three-variable system of differential equations that exhibits chaotic dynamics for suitable parameters. Its trajectories approach the famous butterfly-shaped strange attractor.
Repeller: An invariant set from which nearby trajectories move away over time. It reverses the defining local behavior of an attractor.
Limit cycle: An isolated, closed trajectory that nearby trajectories approach in a continuous-time dynamical system. It attracts trajectories into sustained periodic motion rather than rest.
Phase space: A space whose points represent all possible states of a dynamical system. An attractor is a region or set of points in this state space.
Van der Pol oscillator: A nonlinear oscillator with damping that depends on amplitude and produces self-sustained oscillations. Its stable limit cycle explains how oscillations can persist without periodic external forcing.
Saddle point: An equilibrium stable in some directions and unstable in others. Nearby trajectories approach along some directions but escape along others, so it is not an attractor.
Strange attractor: An invariant set with complex geometry that supports chaotic dynamics and attracts nearby trajectories. It captures long-term motion that is bounded, aperiodic, and sensitive to initial conditions.
Trajectory: The path traced by a system's state as it evolves from an initial condition. Attraction is defined by how trajectories move toward a set.

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