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Abstract

The fractional calculus is useful to model non-local phenomena. In this paper a new modified version of the van der Pol type oscillator is proposed, introducing fractional-order time derivatives into the state-space model and the trirhythmicity in an enzymatic-substrate reaction, described by a fractional-order extended van der Pol equation under periodic excitation is investigated. The fractional derivatives are introduced in the trirhythmic system in order to model the memory property of the biological system. The presence of fractional derivatives requires the use of suitable criteria, which usually makes the analytical work much hard. The residue harmonic balance method is used to obtain the periodic solutions. Highly accurate limited cycle frequency and amplitude are captured. Numerically, we used the predictor-corrector algorithm to solve the fractional trirhythmic system and the results agree with the analytical solutions for a wide range of parameters. Based on the obtained solutions, the effects of the damping, the initial conditions and the periodic force on the oscillators are investigated. When the system parameters are identical, the steady state responses and their stability are qualitatively different. The initial approximations are obtained by solving a few harmonic balance equations. They are improved iteratively by solving nonlinear differential equations of increasing dimension. The second-order solutions accurately exhibit the dynamical phenomena when taking the fractional derivative and periodic force as bifurcation parameters. When the damping is described with the periodic excitation, the stable steady state response is more complex because this force influences past history into account implicitly. Numerical examples taking periodic excitation and fractional derivative are respectively given for feature extraction and convergence study. Based on the obtained response, we show that the stability and variation of response for this trirhythmic self-sustained system is significantly dependent to the initial conditions.