Nonlinear Dynamics of the Menstrual Cycle: Fractional Modeling, Cycle Energy, and Contraceptive Suppression

The human menstrual cycle is a nonlinear endocrine oscillator that can be modeled using classical mathematical methods.Most mathematical models describe menstrual dynamics using integer-order differential equations.However, there are many clinical observations that suggest that the endocrine system has memory properties, and the effects of hormonal contraceptives can last beyond the time frame suggested by the classical models.In this paper, we propose a fractional dynamical model of the human menstrual cycle that takes into account the effects of hormonal contraceptives.The fractional derivative is used to model the memory and delayed response characteristics of the endocrine system.In addition, we propose a new geometric invariant called the Hormonal Cycle Energy (HCE).The HCE is defined as a phase integral that represents the strength of endocrine oscillations.The stability and bifurcation analysis indicate that increasing the exogenous hormone dosage leads to deformation and eventual collapse of the limit cycle via a Hopf-type bifurcation.The fractional order analysis indicates that the fractional memory affects the level of suppression, the time to recover from the suppression after the withdrawal of exogenous hormones, and the hysteresis in the recovery of the hormonal cycle.The persistent homology analysis indicates that the physiological cycles exhibit nontrivial topological features, while the suppressed cycles have trivial topology and zero HCE.The proposed model combines fractional calculus, dynamical systems theory, persistent homology, and topological data analysis to investigate endocrine suppression.The results also suggest that the HCE can serve as a biomarker of endocrine vitality and recovery.