Memory effect in a self-sustained birhythmic biological system

In this paper, birhythmicity in an enzymatic-substrate reaction described by a fractional-order extended van der Pol equation is investigated. The fractional derivatives are introduced in the system equations in order to model the memory property of the biological system. The residue harmonic balance scheme is used to study the periodic motions of the considered fractional-order van der Pol equations. It is shown that depending on system parameters and the fractional derivative order, the bistability area strongly increased. This fractional oscillator is analytically mapped, onto an ordinary bistable systems with a two stable amplitude. The obtained results clearly show an interesting collapse and revival of birhythmicity with the variation of the fractional derivative order. The amplitude and frequency of the fractional order van der Pol oscillator are derived. The analysis of amplitude equation corroborates with the results obtained by numerical simulations of the fractional-order differential equations describing the system.