Threshold for Chaos of a Duffing Oscillator with Fractional‐Order Derivative

Abstract

In this paper, the necessary condition for the chaotic motion of a Duffing oscillator with the fractional-order derivative under harmonic excitation is investigated. The necessary condition for the chaos in the sense of Smale horseshoes is established based on the Melnikov method, and then the chaotic threshold curve is obtained. The largest Lyapunov exponents are provided, and some other typical numerical simulation results, including the time histories, frequency spectrograms, phase portraits, and Poincare maps, are presented and compared. From the analysis of the numerical simulation results, it could be found that, near the chaotic threshold curve, the system generates chaos via the period-doubling bifurcation, from single periodic motion to period-2 motion and period-4 motion to chaotic motion. The effects of fractional-order parameters, the stiffness coefficient, and the damping coefficient on the threshold value of the chaotic motion are analytically discussed. The results show that the coefficient of the fractional-order derivative has great effect on the threshold value of the chaotic motion, while the order of the fractional-order derivative has less. The analysis results reveal some new phenomena, and it could be useful for designing or controlling dynamic systems with the fractional-order derivative.

Immediate solution for fractional nonlinear oscillators using the equivalent linearized method

“…To highlight the main results of this work and to show the effectiveness of solving nonlinear fractional oscillators, two illustrative examples are studied to illustrate the aims: the fractional-Duffing oscillator 33 and the fractional Van-der-Pol-Duffing oscillator. 34 Example 1: Duffing oscillator with the fractional-order This example is concerned with how to solve the following class of second-order fractional-Duffing equations 37 using the non-perturbative approach…”

“…The comparison between the solutions (20) and their frequency (22) with the solution (36) and its frequency (37) shows that the damping factor in both solutions is identical, while the frequency ( 22) is more accurate than the frequency (37).…”

“…This example is concerned with how to solve the following class of second-order fractional-Duffing equations 37 using the non-perturbative approachwhere μ,italicω02 and Q are constants. Assuming that the above oscillator, subject to the initial conditions y(0)=A, truey˙(0)=0.…”

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

“…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. The motivation of the present study is to employ the most familiar scheme to examine complex nature and illustrate chaotic and strange behaviours.…”

Mathematical Model of Fractional Duffing Oscillator with Variable Memory

“…Similar studies were carried out in article [9], but with the Caputo derivative of fractional constant order. In article [10], on the basis of Melnikov's method, the necessary condition for a chaotic regime of a Duffing oscillator with a fractional order derivative under harmonic excitation is investigated.…”