Periodic orbit

A trajectory that returns to the same state after a fixed interval of time. Its repeated motion forms a closed path in the system’s state space.

State space: A space whose points represent all possible states of a system. A periodic orbit is a closed trajectory through states in this space.
Ordinary differential equation: An equation relating a function of one variable to its derivatives. Many continuous-time systems define trajectories by ordinary differential equations.
Simple harmonic motion: Oscillation in which a restoring force is proportional to displacement from equilibrium. Its sinusoidal trajectories are elementary periodic orbits.
Quasiperiodic motion: Motion combining incommensurate frequencies that never exactly repeats. It can look recurrent without returning to precisely the same state at a fixed period.
Poincaré–Bendixson theorem: A theorem constraining long-term trajectories of certain continuous systems in the plane. It gives conditions under which planar trajectories must approach a periodic orbit.
Period (dynamical systems): The smallest positive time interval after which a periodic trajectory repeats. It measures the time required for one complete traversal of the orbit.
Initial condition: A specified state from which a differential equation’s solution is determined. The initial state selects a trajectory whose later return can be tested.
Planetary orbit: The path followed by a planet under gravitational influence, often approximated as a repeating ellipse. An ideal bound two-body system traces a periodic orbit.
Chaotic attractor: An attracting invariant set with sensitive dependence and typically aperiodic trajectories. Its long-term trajectories generally do not close into periodic orbits.
Bifurcation theory: The study of qualitative changes in a system as parameters vary. Bifurcations can create, destroy, or change the stability of periodic orbits.

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