APPLIED MATHEMATICAL MODELLING OF SATELLITE ORBITAL STABILITY VIA DISCRETE TOPOLOGICAL SYNDETIC DIVERGENCE
Abstract. This paper introduces a newapplied mathematical modeling method for satellite orbital stability. We introduce the notion of "syndetic divergence" for topological transformation groups. The drawbacks of the traditional station-keeping techniques are cumulative orbital drift, which directly leads to highly inefficient fuel consumption. To address these issues, we analytically formulate the link between syndetic return intervals and velocity change (ΔV ). With the help of the strong transitivity property, we prove that the time intervals between maneuvers are strictly uniformly bounded. We then implement this theoretical framework in a computational model using two new numerical algorithms. The first algorithm gives a discrete numerical approximation scheme to ensure strict boundedness of the orbit. It is based on a topological contraction, using Banach’s fixed-point principle, for the numerical convergence. The second algorithm serves as a forward-looking computational model to anticipate divergence before breaches of the boundaries. A detailed step-by-step numerical example is provided to demonstrate the practical viability of the model. The proposed analytical approach provides long-term dynamical stability for low Earth orbit (LEO) and geostationary Earth orbit (GEO) satellites. This model significantly reduces the overall energy consumption and sets up a powerful predictive tracking model. 2020 Mathematics Subject Classification. Primary 54H20; Secondary 70M20, 54H25, 37M05. Key words and phrases. applied mathematical modelling; syndetic divergence; satellite orbital stability; fixed-point theorem; topological dynamics; predictive algorithms; station-keeping optimization.