Synchronization of the Circadian Rhythms with Memory: A Simple Fractional-Order Dynamical Model Based on Tow Coupled Oscillators

Abstract

Disruptions of the circadian rhythm are associated with internal desynchronization. It affects some internal functions of our body and behavior that are important to our health. Our modern lifestyle has contributed to millions of people developing some circadian rhythm disruptions, making the subject very important clinically as well as economically. Motivated by studying a simple mathematical model that can reveal some features of internal synchronization desynchronization, in this contribution, we extend the coupling oscillator phase model proposed by Strogatz [14] in the sense that memory is considered in the modeling. Such memory is a result of the introduction of Caputo-type fractional derivatives in the coupling oscillators’ phase model dynamics, resulting in a fractional phase model. We show that the proposed fractional coupling oscillator phase model is well-posed. Furthermore, we analyze
the synchronization phenomena. We obtain the synchronized solutions explicitly when the memory is equally distributed between the oscillators. In contrast, when distinct levels of memory are imposed in the modeling, we obtain lower and upper bounds because any existing synchronized solution must be confined in between. We present numerical realizations that support the theoretical findings in great detail.