Fractal Calculus Facilitates Rethinking 'Hard Problems'; A New Research Paradigm

This paper intoduces 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 his- torical evolution of understaning an empirical medical process faciltated by the fractal calculus perspective. Herein we trace the analysis of the time series for heart rate varability (HRV) developed for diagnosing the cardiovascular health of a patient. This is done herein by introducing four (one empirical which en- tails 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 re- placement model. It is through the utilization of the fractional calculus that the necessity for rethinking how to systematically incorporate additonal layers of complexity are revealed ultimately resulting in a 'complete' description of its empirical dynamics in fractal terms.