Mathematical Model of Fractional Duffing Oscillator with Variable Memory

Abstract

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.

A heuristic review on the homotopy perturbation method for non-conservative oscillators

“…One of the most effective analytical methods for fractional oscillators is the homotopy perturbation method. Ex12: The fractional damping Duffing equation is described as 68,69…”

“…Ex12: The fractional damping Duffing equation is described as 68,69 where μ, ω02, and Q are constants. The fractional derivative obeys the definition of the Riemann–Liouville time-fractional derivative.…”

Implicit Finite-Difference Scheme for a Duffing Oscillator with a Derivative of Variable Fractional Order of the Riemann-Liouville Type

“…For an explicit scheme, the issues of stability and convergence of [39] are theoretically justified. In the works [40,41], the properties of forced oscillations of a Duffing oscillator with a fractional derivative of variable order of the Riemann-Liouville type are investigated using amplitude-frequency (AFC), phase-frequency characteristics (PFC) and Q-factor. It turned out that the order of the fractional derivative affects the rate of attenuation of oscillations.…”

“…Moreover, unlike the explicit scheme, the stability and convergence of the implicit one does not depend on the constraints on the step of the dishonest grid. In this article, by analogy with the works [35,37,[39][40][41], an implicit finitedifference scheme for solving the Duffing equation with a fractional derivative of variable order of the Riemann-Liouville type is investigated, the issues of stability and convergence of the numerical scheme are substantiated, and chaotic regimes and bistability of oscillations are investigated.…”

Interesting and complex behaviour of Duffing equations within the frame of Caputo fractional operator

“…The considered model is analyzed with different numerical approaches by scholars to present their viewpoint and also derived some essential results which play a key role in generalizing the physical phenomena with Duffing equations. For instance, the block-pulse functions are employed by authors in [44] to derive the numerical solution for cubic-quintic-hepatic nonlinearities, the spectra of the maximum Lyapunov exponents are unified with the numerical algorithm by researchers in [45], to capture the complex nature for the model related different order, in [46] authors derived some interesting results with bifurcation for system associated to fractional-order damping, by considering the time-delayed position feedback researcher in [47] investigated the interesting consequences associated with duffing oscillator. The authors in [48] derived results for chaos nature with threshold condition and presented some essential results.…”