Fractional-Memory Deformation of an Apsidal Precession Resonance: Finite-Time Trapping and Trapped-Interval Diagnostics

Abstract

A fractional-memory deformation of a reduced apsidal precession resonance model is developed as a controlled phenomenological extension of an established Hamiltonian resonant backbone. The conservative reference system is taken to be a one-degree-of-freedom Hamiltonian apsidal resonance with a positive-action resonance crossing, a libration center, a saddle point, a separatrix, and a finite classical resonance width. Fractional memory is therefore not introduced to generate the resonance, but to deform an existing resonant structure. The ordinary secular equations are replaced by Caputo fractional equations, yielding a history-dependent extension of the conservative reduced model rather than a first-principles derivation of astrophysical memory. The integer-order limit is used as a validation case, while fractional orders below unity are used to examine memory-modified dynamics. Numerical integrations are performed using a fractional Adams-Bashforth-Moulton predictor-corrector scheme. Finite-time diagnostics are introduced to measure libration-envelope deformation, projected phase-space changes, finite-time trapped-interval response, separatrix-region ambiguity, clean versus marginal behavior, and trapping-time transitions. The conservative Hamiltonian resonance is accurately recovered in the integer-order limit. For fractional orders below unity, the resonant backbone is found to remain visible in the clean regime, while the observable libration envelope is deformed and the projected phase-space curves no longer behave as closed Hamiltonian level sets. Near the classical separatrix, delayed escape and ambiguous finite-time trapping are observed, indicating increased separatrix-region ambiguity under the adopted diagnostic protocol. In the marginal near-boundary regime, the measured finite-time trapped set can approach the singular action boundary, making the measured trapped interval boundary-sensitive rather than an invariant Hamiltonian resonance width. These results provide a reduced fractional-dynamical framework for studying memory-deformed resonant trapping, while mechanism-specific astrophysical memory kernels remain a subject for future work.