Poincaré Maps with the Theory of Functional Connections

TL;DR

This work shows that Poincaré maps can be constructed with high precision using the Theory of Functional Connections (TFC), by embedding constraints directly into constrained functionals that represent system trajectories. The approach formulates surface-crossing problems as both IVPs and BVPs, achieving linear convergence to Taylor expansions via carefully chosen support functions and enabling exact intersections with surfaces of section without external interpolation or adaptive-step integration. The IVP formulation yields efficient, high-accuracy surface crossings, while the BVP formulation permits exact periodic and recurrence analyses, demonstrated in the CRTBP and in non-autonomous four-body settings, including time-discretized (stroboscopic) maps. Collectively, the method provides a powerful, computationally efficient alternative for constructing Poincaré maps, computing periodic orbits, and analyzing complex astrodynamical dynamical structures with machine-level accuracy at modest cost.

Abstract

Poincaré maps play a fundamental role in nonlinear dynamics and chaos theory, offering a means to reduce the dimensionality of continuous dynamical systems by tracking the intersections of trajectories with lower-dimensional section surfaces. Traditional approaches typically rely on numerical integration and interpolation to detect these crossings, which can lead to inaccuracies and computational inefficiencies. This work presents a novel methodology for constructing Poincaré maps based on the Theory of Functional Connections (TFC). The constrained functionals produced by TFC yield continuous and differentiable representations of system trajectories that exactly satisfy prescribed constraints. The computation of Poincaré maps is formulated as either an initial value problem (IVP) or a boundary value problem (BVP). For IVPs, initial conditions are embedded into the functional, and the intersection time with a specified section surface is determined. We demonstrate linear convergence to the Taylor series, thereby enabling accurate interpolation without resorting to numerical integration or external optimization. For BVPs, periodicity conditions are encoded to identify periodic orbits in a Three-Body Problem context. Furthermore, by enforcing periodic constraints, we show how to construct first recurrence maps. The methodology is also extended to non-autonomous systems, demonstrated through applications to a Four-Body Problem. The proposed approach achieves machine-level accuracy with modest computational effort, eliminating the need for variable transformations or iterative integration schemes with adaptive step-sizing. The results illustrate that TFC offers a powerful and efficient alternative framework for constructing Poincaré maps, computing periodic orbits, and analyzing complex dynamical systems, particularly in astrodynamical contexts.

Figures (4)

• Figure 1: Planar periodic solutions in the CRTBP: Lyapunov orbits are shown in red for periods ranging from 12 days (innermost) to 23 days (outermost); Distant Retrograde Orbits (DROs) around the Moon are shown in green for periods between 2 and 21 days; and Earth-centered periodic orbits are depicted in blue for periods from 2 to 13 days. The period interval between successive orbits is 1 day. These solutions were obtained via numerical convergence using the constrained functional in Eq. \ref{['eq:ce2c']}.
• Figure 2: Orbits for $T=13.75$ days obtained using the constrained functional shown in Eq. (\ref{['eq:cf3']}) is shown in the left side. The corresponding pairs $x$ and $\dot{x}$ evaluated at $t=0$ obtained from the technique are shown on the right side. Besides satisfying the constraints shown in Eq. (\ref{['eq:constr2']}), they also satisfy the conditions $x(0)=x(T)$, $\dot{x} (0) =-\dot{x} (T)$, and $\dot{y} (0) = \dot{y} (T)$. The solution $L_1$H (in red) corresponds to the Lyapunov orbit, representing a particular case where $\dot{x} (0) =-\dot{x} (T)=0$. The lines are drawn to guide the eyes.
• Figure 3: Several solutions for $T=13.75$ days obtained using the constrained functional shown in Eq. (\ref{['eq:cf3']}). The solution in green corresponds to a Distant Retrograde Orbit, which is a particular solution where $\dot{x} (0)=-\dot{x} (T)=0$.
• Figure 4: Planar perturbed periodic solutions in the biplanar bicircular 4BP: Lyapunov orbits are shown in red; Distant Retrograde Orbits (DROs) around the Moon are shown in green; and Earth-centered periodic orbits are depicted in blue. The period of these orbits is 29.52887871613042 days, which corresponds to the period of the Sun's motion (and its gravitational influence on the spacecraft) in the Earth-Moon rotating frame. These solutions were obtained via numerical convergence using the constrained functionals shown in Eqs. \ref{['eq:cf3']} and \ref{['eq:cf3y']}.