Poincar\'e Maps with the Theory of Functional Connections

TL;DR

This paper introduces a new method for constructing Poincaré maps using the Theory of Functional Connections, enabling highly accurate, efficient analysis of nonlinear dynamical systems without traditional numerical integration.

Contribution

The work develops a TFC-based approach for Poincaré maps that improves accuracy and efficiency, especially in complex systems like the Three-Body and Four-Body Problems.

Findings

Achieves machine-level accuracy in Poincaré map construction.

Eliminates the need for numerical integration and adaptive step-sizing.

Effectively identifies periodic orbits and recurrence maps.

Abstract

Poincar\'e 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 Poincar\'e 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 Poincar\'e 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 Poincar\'e maps, computing periodic orbits, and analyzing complex dynamical systems, particularly in astrodynamical contexts.