Memristors are nano-devices, which, in some of their physical realisations, feature multiple operating modes, allowing data to be alternatively sensed, stored, or processed. Their multi-functionality, which resembles the operating principles of neural structures in the brain, in conjunction with the ease of their fabrication, which is typically carried out on top of underlying CMOS circuitry, at the regularly-spaced spaced crossings of crossbar matrices, composed of two vertically-adjacent and perpendicular sets of parallel metal lines, makes their use in integrated circuit design ideally suitable for enhancing the performance of state-of-the-art computing machines without the need to shrink the sizes of CMOS transistors any further. Given the extraordinary opportunities, which memristors open up in the field of Artificial Intelligence, gaining a deep understanding of their inherently-nonlinear dynamics is of primary importance. Fading memory is the capability of a system to display a unique steady state under a stimulus from a given class. Fading memory effects in a memristive medium were first discovered in 2016 through a comprehensive experimental and theoretical investigation of the switching dynamics of a TaO_x ReRAM cell. Recently, a bifurcation study of a predictive model of the same nanostructure revealed how it may also exhibit bi-stability under suitable AC periodic excitation. Bi-stability is essentially a local form of fading memory. This paper gains a deeper insight into the local fading memory properties of the TaOx ReRAM cell, demonstrating how the existence domain for its memory state may be in fact partitioned in a tuneable number of basins of attraction, each of which includes all the initial conditions, from which the memristor would settle into one of the admissible locally-stable oscillatory steady states.