Poincaré map

A discrete map that sends each point on a section of state space to the trajectory’s next intersection with that section. It converts continuous-time dynamics into an iterated map.

Poincaré section: A lower-dimensional surface in state space chosen to intersect trajectories transversely. The map’s domain is this surface, where each return point is recorded.
Periodic orbit: A trajectory that repeats after a fixed positive time. Each periodic orbit produces a periodic point of the return map.
Dynamical system: A rule describing how a state changes over time or through successive steps. A Poincaré map is a discrete dynamical system derived from a continuous one.
Time-T map: The discrete-time map obtained by evolving a continuous system for a fixed duration T. Unlike a return map, it samples at fixed time intervals rather than section crossings.
First-return map: A map that sends a point to its first later intersection with a chosen section. This is the standard rule defining a Poincaré map.
Floquet theory: A framework for analyzing stability of solutions to linear differential equations with periodic coefficients. Its multipliers correspond to eigenvalues of the return map near a periodic orbit.
Phase space: A space whose points represent all possible states of a system. The chosen section lies inside the system’s phase space.
Stroboscopic map: A map that samples a periodically forced system at the same phase of each forcing cycle. It is a fixed-period sampling method, whereas a Poincaré map uses state-dependent return times.
Flow (mathematics): A continuous-time family of maps describing how states evolve under a differential equation. The Poincaré map samples this continuous evolution at successive crossings.
Bifurcation: A qualitative change in a system’s dynamics as a parameter varies. Changes in fixed points or cycles of the map reveal bifurcations in the flow.

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