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Poincaré maps with the theory of functional connections
Authors: de Almeida Jr, A.; Mortari, D.
Ref.: Nonlinear Dyn. 114(3), 222 (2026)
Abstract: Poincar & eacute; 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, especially in systems characterized by long-term evolution or sensitivity to initial conditions. This work presents a novel methodology for constructing Poincar & eacute; 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 & eacute; maps is formulated as either an initial value problem (IVP) or a boundary value problem (BVP). For IVPs, we embed initial position and velocity constraints into the functional and determine the exact intersection time with a prescribed section surface using root-finding methods. We demonstrate linear convergence to the Taylor series, enabling accurate interpolation without requiring external integration or optimization procedures over short time intervals. For BVPs, periodicity conditions are encoded to identify periodic orbits such as families of Lyapunov and Distant Retrograde Orbits in a Circular Restricted Three-Body Problem context. Furthermore, by enforcing partially periodic constraints, we show how to construct first recurrence maps with selective control over specific components of position and/or velocity. The methodology is also extended to non-autonomous systems, demonstrated through applications to the Bicircular Biplanar 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 & eacute; maps, computing periodic orbits, and analyzing complex dynamical systems, particularly in astrodynamical contexts.
