Poincaré Map (Poincaré Return Map)

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

A Poincaré map is the discrete first-return map $P:\Sigma\to\Sigma$ induced by a Poincaré section $\Sigma$ : starting from a crossing $\mathbf{x}_k\in\Sigma$ , integrate the flow until the next crossing in the prescribed direction; that next crossing is $\mathbf{x}_{k+1}=P(\mathbf{x}_k)$ . Iterating $P$ converts the analysis of periodic, quasi-periodic and chaotic motion of the continuous flow into a problem in discrete dynamical systems (Poincaré 1892; Parker & Chua 1989).

Closely related but distinct concepts:

Concept	Emphasis	Object
Poincaré section	Geometry	The hypersurface $\Sigma$ on which crossings are recorded
Poincaré map	Dynamics	The discrete map $P$ and the patterns formed by its iterates

Section versus map is analogous to the cutting plane versus the pattern seen after projecting crossings onto it.

Dimensionality and visualization

Given a constraint on the Jacobi constant $C$ , the section reduces the flow dimension by one, and the map operates on a $(N-2)$ -dimensional state (Haapala & Howell 2014):

Planar CR3BP: the map is 2D; a planar projection fully represents the state, and contour intersections directly identify connections.
Spatial CR3BP: the map is 4D and cannot be fully represented by a planar projection. Glyph representations attach a vector (or a chain of vectors) to each base point: the base point encodes position $(y,z)$ , the vector encodes the in-plane velocity $(\dot y,\dot z)$ , and additional links encode out-of-plane components. Glyph maps make heteroclinic connections between halo orbits visually identifiable (Haapala & Howell 2014; Whittington 2022).

Fixed points and stability

Periodic orbits of the continuous flow correspond to fixed points (or $k$ -cycles) of $P$ . The stability type is read off the eigenvalues of the monodromy matrix $DP$ linearized about the fixed point:

Center-type fixed point: stable periodic orbit; surrounding iterates form closed curves (quasi-periodic tori).
Saddle-type fixed point: unstable periodic orbit (e.g. a Lyapunov orbit); iterates align with the stable/unstable manifolds.

Iterates form closed curves on invariant tori (quasi-periodic motion) or fill regions densely (chaotic orbits). Locating periodic orbits via the map is often the first step of a multiple-shooting or continuation scheme.

Specialized maps

Periapse map

Defined on the periapse section $\Sigma=\{\dot\rho=0,\ddot\rho>0\}$ . In the planar problem its projection into configuration space fully represents the state and reveals escape/capture structure near the smaller primary (Villac & Scheeres 2004; Paskowitz & Scheeres 2006). Variants named by central body (perigee map, perilune map, apse map) are the same construction with a different reference primary; the perilune map is widely used to screen lunar-gravity-assist + WSB capture transfers from the Earth and to analyze the perilune distribution of DRO family members (Scott & Spencer 2010).

Tisserand–Poincaré (T-P) graph

An extension of the Tisserand graph (a patched-conic gravity-assist sequencing tool) to the CR3BP, introduced by Campagnola & Russell (2010). Axes are osculating periapsis and apoapsis distances (or period) relative to the primary; contours of the Tisserand parameter $T=3-V_\infty^2$ are sampled once per revolution at a fixed Poincaré crossing (typically the negative- $x$ axis). The T-P graph covers the regime $T<3$ where $V_\infty$ becomes imaginary and the patched-conic Tisserand graph fails, enabling systematic design of high-altitude flyby sequences in planetary-moon tours (Lantoine & Russell 2010; Yang et al. 2023; Shen et al. 2026).

Applications

Heteroclinic and homoclinic connections: on the $x=1-\mu$ map, intersections of the unstable manifold of one periodic orbit with the stable manifold of another identify maneuver-free transfers; planar cases reduce to contour intersections, spatial cases use glyph inspection followed by differential correction (Gómez et al. 2001; Haapala & Howell 2014).
Transfer initial-guess generation: the map compresses a high-dimensional solution space into a 2D image, allowing interactive selection of transfer candidates that are then refined by differential correction or multiple shooting.
Long-term-capture orbit search: periapse maps classify non-transit (long-term-capture) trajectories; periodic orbits are seeded from nearby mirror configurations and refined by continuation (Haapala & Howell 2014).
DRO family analysis: a perilune map of DRO members shows the distribution of perilune states versus orbit parameter, identifying windows suitable for lunar-gravity-assist insertion.

Numerical notes

A typical Earth–Moon map for ~1000 manifold trajectories integrated over ~1.2 years takes 2–3 seconds in MATLAB with C-integration subroutines; Sun–Earth maps over ~100 years take a comparable time (Haapala & Howell 2014). Symplectic integrators are preferred for very long integrations to suppress energy drift.

References

Poincaré H. Les méthodes nouvelles de la mécanique céleste. Gauthier-Villars, 1892.
Parker T S, Chua L O. Practical Numerical Algorithms for Chaotic Systems. Springer, 1989.
Gómez G, Llibre J, Martínez R, Simó C. Dynamics and Mission Design near Libration Points. Vol. II. World Scientific, 2001.
Villac B F, Scheeres D J. On the concept of periapsis in Hill's problem. Dynamics & Control of Systems, 2004.
Paskowitz M E, Scheeres D J. Geometry of quasiperiodic orbits in the Hill problem. Celestial Mechanics and Dynamical Astronomy, 2006.
Campagnola S, Russell R P. The Tisserand-Poincaré graph for multi-body gravity assists. AAS/AIAA Astrodynamics Specialist Conference, 2010.
Haapala A F, Howell K C. Representations of higher-dimensional Poincaré maps with applications to spacecraft trajectory design. Acta Astronautica, 2014, 96: 23–46.
Scott C J, Spencer D B. Transfer and capture into distant retrograde orbits via Poincaré and Periapsis maps. JGCD, 2010. doi:10.2514/1.47791.
Whittington T R. Multi-body trajectory design in the Earth-moon region utilizing Poincaré maps. M.S. thesis, Purdue University, 2022.
Yang J, et al. Review of trajectory design and optimization for Jovian system exploration. Acta Astronautica, 2023.