Poincarékaart

A Poincaré map is a technique used in the study of dynamical systems. It is named after the French mathematician Henri Poincaré. Instead of observing the entire continuous trajectory of a system over time, a Poincaré map samples the trajectory at discrete points, typically when it crosses a specific surface within the system's phase space. This surface is called the Poincaré section.

The resulting map is a set of points, each representing the state of the system at the moment of crossing the section. The evolution of these points from one iteration of the map to the next provides insight into the long-term behavior of the system. For example, if a system is periodic, its Poincaré map might consist of a single point that is revisited repeatedly. If the system exhibits chaotic behavior, the Poincaré map can reveal complex fractal structures.

Poincaré maps are particularly useful for analyzing systems with a high number of degrees of freedom or those that are difficult to study directly due to their continuous nature. They simplify the analysis by reducing the dimensionality of the problem, transforming a continuous-time problem into a discrete-time one. This technique is widely applied in fields such as celestial mechanics, fluid dynamics, and the study of oscillations.