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

Wuce Xing,Enli Chen,Yujian Chang,Meiqi Wang

doi:10.1155/2019/1230194

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.

Highlights

Ere has been some research on chaotic motions and synchronization control of chaos to the fractional-order Duffing system
The necessary condition for generating chaos in sense of Smale horseshoes in a typical Duffing oscillator with the fractional-order derivative under harmonic excitation is investigated based on the Melnikov method. e paper is organized as follows
(1) e necessary condition for generating chaos in the sense of Smale horseshoes is obtained based on the Melnikov method. e relationship between each parameter is established, and the effect of each parameter on the chaotic threshold is studied, respectively

Summary

Necessary Condition for Chaos of Fractional-Order Duffing Oscillators

Duffing oscillators are one of the most typical and important objects in nonlinear dynamics. The ordinary Duffing oscillator with a fractional-order derivative under harmonic excitation is described with mx€ − k1x + k3x3 + c1x_ + hDpt [x(t)] F cos(ωt), (1). When the Hamilton function H(x, y) 0, there are two homoclinic orbits connecting saddle points. E homoclinic orbits of the system described in equation (1) can be obtained by solving the following expression:. An analysis of equation (10) leads to the conclusion that a plus sign of x(t) represents the positive axis of a homoclinic orbit and a minus sign represents the negative part. An analysis of equation (10) leads to the conclusion that a plus sign of x(t) represents the positive axis of a homoclinic orbit and a minus sign represents the negative part. e plus sign of y(t) represents the upper half of a homoclinic orbit, and minus sign represents the lower half

Melnikov Function Determines the Chaotic System’s

G Circle 1

F Circle 3

Conclusions