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In this paper, we explore the chaotic signatures of the geodesic dynamics forparticles moving in the slowly rotating Hartle-Thorne spacetime; an approximatesolution of vacuum Einstein field equations describing the exterior of amassive, deformed, and slowly rotating compact object. We employ the numericalstudy to examine the geodesics of prolate and oblate deformations for genericorbits and find the plateaus of the rotation curve, which are associated withthe existence of Birkhoff islands in the Poincare surface of the section, wherethe ratio of the radial and polar frequency of geodesics remains constantthroughout the island. We investigate various phase space structures, includinghyperbolic points and chaotic regions in the neighborhood of resonant islands.Moreover, chaotic behavior is observed to be governed by the stickinessphenomenon, where chaotic orbits remain attached to stable ones for an extendedduration before eventually diverging and are attracted toward the surface ofthe neutron star. The precision of the numerical integration used to simulatethe particle's trajectories plays a crucial role in the structures of thePoincare surface of the section. We present a comparison of several efficientstructure-preserving numerical schemes of order four applied to the considerednon-integrable dynamical system and we investigate which schemes possess thecanonical property of the Hamiltonian flow. Among the class of non-symplecticintegrators, we employ the explicit Runge-Kutta method and explicit generallinear method with a standard projection technique to project the numericalsolution onto the desired manifold. The projection scheme admits theintegration without any drift from the desired manifold and is computationallycost-effective. We are concerned with two crucial aspects -- long-termbehaviour and CPU time consumption.

Structure-preserving numerical simulations of test particle dynamics around slowly rotating neutron stars within Hartle-Thorne approach