We describe the phase space structures related to the semi-major axis of Molniya-like satellites subject to tesseral and lunisolar resonances. In particular, the questions answered in this contribution are: (1) we study the indirect interplay of the critical inclination resonance on the semi-geosynchronous resonance using a hierarchy of more realistic dynamical systems, thus discussing the dynamics beyond the integrable approximation. By introducing ad hoc tractable models averaged over fast angles, (2) we numerically demarcate the hyperbolic structures organising the long-term dynamics via fast Lyapunov indicators cartography. Based on the publicly available two-line elements space orbital data, (3) we identify two satellites, namely Molniya 1-69 and Molniya 1-87, displaying fingerprints consistent with the dynamics associated to the hyperbolic set. Finally, (4) the computations of their associated dynamical maps highlight that the spacecraft are trapped within the hyperbolic tangle. This research therefore reports evidence of actual artificial satellites in the near-Earth environment whose dynamics are ruled by hyperbolic manifolds and resonant mechanisms. The tools, formalism and methodologies we present are exportable to other region of space subject to similar commensurabilities as the geosynchronous region.

Phase space of the resonant integrable approximation. The width of the separatrix (red curve) allows excursion of the semi-major axis up to 2Δa = 54 km within the libration domain. The oscillations near the elliptic fixed-point (blue point) have a period of about 1.76 years.

(Left) Poincare' section associated to (35) computed for g(0) = 0. The unstable fixed point is labeled with the red cross, the blue circle surrounds the stable periodic orbit. The phase space is similar to the integrable approximation but contains a thin chaotic layer (scattered erratic points) surrounding the unperturbed separatrix. Each considered initial condition has been iterated 100 times under P. (Right) Details of finite pieces of the stable manifold W^s(x_u).

Composite plot illustrating the mechanisms of the intermittency phenomena. The red line represents the separatrix of the integrable model T. Realisations of the stable and unstable manifolds, for ε ≠ 0; ε << 1, are not shown for the sake of readability. One hyperbolic orbit trapped in the hyperbolic tangle is highlighted in the phase space, with a color code depending on the regime of the resonant angle. When the resonant angle circulates (grey color), the action takes negative Λ’s. When the angle librates (black color), the action variable performs the full homoclinic loop and exhibit larger variations.

Composite plot highlighting the main features of Molniya semi-major axis dynamics. The global FLI map and a magnified portion near the saddle-like structure detail the hyperbolic structure. Initial conditions within the hyperbolic layer display intermittency phenomena, whilst stable orbits display regular oscillations. This is exemplified for two orbits whose initial conditions are labeled with the white stars. The width of the layer, for a fixed u1 but varying ω, might exhibit a complex geometry. For Molniya’s prototypical range of values of ω, materialised by the white shaded-line region around ω = 270^o, the width is limited to a few kilometer in the semi-major axis only. See text for details.

Time history of the semi-major axis and resonant angle u₁ extracted from the TLE data for the satellite Molniya 1-69.

Time history of the semi-major axis and resonant angle u₁ extracted from the TLE data for the satellite Molniya 1-87.

Dynamical maps for Molniya 1-69 and Molniya 1-87. The locations of Molniya 1-69 and Molniya 1-87 are marked through the black circle. Both satellites reside within the hyperbolic tangle.

Intersections of the forward in time FLIs with the plane (a; u₁) for model S computed on a 500 × 500 grid of initial conditions for i₀ ∈ {62.5; 63.4; 64.3; 65.2} deg.

Intersections of the forward in time FLIs with the plane (a; u₁) for model J computed on a 500 × 500 grid of initial conditions for i₀ ∈ {62.5; 65.2} deg.