PoincaréSchnitte

PoincaréSchnitte, or Poincaré sections, are a tool in the qualitative theory of dynamical systems named after Henri Poincaré. They provide a way to study the long-term behavior of continuous-time systems by reducing their flow to a discrete map.

Consider a flow φ_t on a manifold M generated by a differential equation x' = f(x). A PoincaréSchnitte is a transversal submanifold Σ of codimension 1 such that almost every trajectory intersects Σ in forward time.

The Poincaré map P: Σ → Σ is defined by P(x) = φ_{τ(x)}(x), where τ(x) > 0 is the first return time with φ_{τ(x)}(x) ∈ Σ. The map encodes the recurrence of the flow and turns a continuous problem into a discrete one.

Fixed points of P correspond to periodic orbits of the original system; the eigenvalues of the derivative DP at a fixed point indicate stability. In Hamiltonian dynamics, Poincaré maps preserve a measure and often reveal invariant structures such as tori and chaotic regions.

Historical note: Poincaré introduced the method in the late 19th century to analyze celestial mechanics and other mechanical systems; it has since become a standard tool in the study of chaos, bifurcations, and numerical experiments.

Poincaré sections are widely used in physics and mathematics, including celestial mechanics, billiards, and the study of geodesic flows.