Mathematical Model of Fractional Duffing Oscillator with Variable Memory

The article investigates a mathematical model of the Duffing oscillator with a variable fractional order derivative of the Riemann-Liouville type. The study of the model is carried out using a numerical scheme based on the approximation of the fractional derivative of the Riemann-Liouville type by a discrete analog-the fractional derivative of Grunwald-Letnikov. The adequacy of the numerical scheme is verified using specific examples. Using a numerical algorithm, oscillograms and phase trajectories are constructed depending on the values of the model parameters. Chaotic regimes of the Duffing fractional oscillator are investigated using the Wolf-Bennetin algorithm. The forced oscillations of the Duffing fractional oscillator are investigated using the harmonic balance method. Analytical formulas for the amplitude-frequency, phase-frequency characteristics, and also the quality factor are obtained. It is shown that the fractional Duffing oscillator possesses different modes: regular, chaotic, multi-periodic. The relationship between the order of the fractional derivative and the quality factor of the oscillatory system is established.