The Poincaré map

A Poincaré map is a tool used in dynamical systems to analyze complex, repeating behavior by reducing it to simpler snapshots. Imagine tracking a point moving through a space; instead of following every movement, you record where it intersects a particular surface or section each time it loops back. This creates a sequence of points that captures the system's recurring patterns, making it easier to study stability, chaos, or periodic cycles. Essentially, it translates continuous, complicated motion into a manageable set of data, revealing the core structure of the system’s behavior over time.