Fractional Discrete-Time Analysis of an Emotional Model Built on a Chaotic Map through the Set of Equilibrium and Fixed Points

Fractional discrete systems can enable the modeling and control of the complicated processes more adaptable through the concept of versatility by providing system dynamics’ descriptions with more degrees of freedom. Numerical approaches have become necessary and sufficient to be addressed and employed for benefiting from the adaptability of such systems for varied applications. A variety of fractional Layla and Majnun model (LMM) system kinds has been proposed in the current work where some of these systems’ key behaviors are addressed. In addition, the necessary and sufficient conditions for the stability and asymptotic stability of the fractional dynamic systems are investigated, as a result of which, the necessary requirements of the LMM to achieve constant and asymptotically steady zero resolutions are provided. As a special case, when Layla and Majnun have equal feelings, we propose an analysis of the system in view of its equilibrium and fixed point sets. Considering that the system has marginal stability if its eigenvalues have both negative and zero real portions, it is demonstrated that the system neither converges nor diverges to a steady trajectory or equilibrium point. It, rather, continues to hover along the line separating stability and instability based on the fractional LMM system.