returnmap

A return map, also known as a Poincaré map in dynamical systems, is a method for converting a continuous-time flow into a discrete-time map by recording where trajectories intersect a chosen cross-section of the phase space. It reduces the study of continuous dynamics to iterations of a map on a lower-dimensional space.

Construction: Consider a continuous dynamical system with flow φt(x). Choose a transversal cross-section Σ such that trajectories intersect Σ transversely. For a point x in Σ, define the first return time T(x) as the smallest t > 0 for which φt(x) also lies in Σ. The return map R: Σ → Σ is then R(x) = φT(x)(x), provided the return exists. In discrete-time systems, a return map can be formed by taking the state at consecutive visits to a chosen section of the state space.

Properties: The return map captures recurrence properties of the original flow and can reveal periodic, quasi-periodic, or chaotic behavior. Fixed points of the return map correspond to periodic orbits of the original system, and the local stability of a fixed point is related to the eigenvalues of the derivative of R. The dimensionality of the map is typically one less than that of the ambient flow.

Variants: The first return map records the first intersection after leaving the section, while higher-order return maps track subsequent intersections. The concept is closely related to the Poincaré map and to stroboscopic maps used for periodically forced systems.

Applications: Return maps are used to analyze periodic orbits, bifurcations, and chaotic dynamics in mechanical, electrical, and biological systems. They enable visualization and reduction of complex continuous dynamics to discrete, often lower-dimensional representations, aiding both qualitative and quantitative study.

Examples: In the Lorenz attractor, a suitable Poincaré section yields a two-dimensional return map that reflects the underlying chaotic dynamics. Fixed points of the map indicate periodic orbits of the original flow. See also Poincaré map, cross-section, and discrete dynamical system.