Fractal Calculus Facilitates Rethinking ‘Hard Problems’: A New Research Paradigm

Abstract

This paper introduces a non-standard research technique to clarify how complex phenomena, such as those that are abundantly present in human physiology, can be faithfully described using fractal dynamical models with and without stochastic forces. This method for conducting research involves tracing the historical evolution of understanding an empirical medical process facilitated by the fractal-order calculus perspective. Herein, we trace the analysis of the time series for heart rate variability (HRV) developed for diagnosing the cardiovascular health of a patient. This is performed herein by introducing four (one empirical, which entails three theoretical fractal models) distinct but related fractal models, each one introduced to solve a particular problem arising from a fundamental defect in the previous model, but in generalizing a model at one stage to resolve the problem associated with the defect, another is invariably introduced by the replacement model. It is through the utilization of the fractal-order calculus that the necessity for rethinking how to systematically incorporate additional layers of complexity is revealed, ultimately resulting in a ‘complete’ description of its empirical dynamics in fractal terms.