Subharmonic Resonance of Duffing Oscillator With Fractional-Order Derivative

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
Nguyen Van Khang ◽  
Truong Quoc Chien
Keyword(s):  
Analytical Solution ◽  
Fractional Order ◽  
Duffing Oscillator ◽  
Existence Condition ◽  
Lyapunov Theory ◽  
Subharmonic Resonance ◽  
System Stability ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Approximately Analytical Solution

In this paper, the subharmonic resonance of Duffing oscillator with fractional-order derivative is investigated using the averaging method. First, the approximately analytical solution and the amplitude–frequency equation are obtained. The existence condition for subharmonic resonance based on the approximately analytical solution is then presented, and the corresponding stability condition based on Lyapunov theory is also obtained. Finally, a comparison between the fractional-order and the traditional integer-order of Duffing oscillators is made using numerical simulation. The influences of the parameters in fractional-order derivative on the steady-state amplitude, the amplitude–frequency curves, and the system stability are also investigated.

Resonance Oscillation of Third-Order Forced van der Pol System With Fractional-Order Derivative

2016 ◽  
Vol 11 (4) ◽  
Author(s):  
Nguyen Van Khang ◽  
Bui Thi Thuy ◽  
Truong Quoc Chien
Keyword(s):  
Fractional Order ◽  
Frequency Equation ◽  
Lyapunov Theory ◽  
Van Der Pol Oscillator ◽  
System Stability ◽  
Order Derivative ◽  
Resonance Oscillation ◽  
Third Order ◽  
Van Der Pol ◽  
Fractional Order Derivative

This study aims to investigate the harmonic resonance of third-order forced van der Pol oscillator with fractional-order derivative using the asymptotic method. The approximately analytical solution for the system is first determined, and the amplitude–frequency equation of the oscillator is established. The stability condition of the harmonic solution is then obtained by means of Lyapunov theory. A comparison between the traditional integer-order of forced van der Pol oscillator and the considered fractional-order one follows the numerical simulation. Finally, the numerical results are analyzed to show the influences of the parameters in the fractional-order derivative on the steady-state amplitude, the amplitude–frequency curves, and the system stability.

scholarly journals Subharmonic Resonance of Van Der Pol Oscillator with Fractional-Order Derivative

2014 ◽  
Vol 2014 ◽  
pp. 1-17 ◽  
Author(s):  
Yongjun Shen ◽  
Peng Wei ◽  
Chuanyi Sui ◽  
Shaopu Yang
Keyword(s):  
Steady State ◽  
Fractional Order ◽  
Averaging Method ◽  
Frequency Equation ◽  
Lyapunov Theory ◽  
Subharmonic Resonance ◽  
Order Derivative ◽  
Van Der Pol ◽  
Fractional Order Derivative ◽  
Equivalent Linear

The subharmonic resonance of van der Pol (VDP) oscillator with fractional-order derivative is studied by the averaging method. At first, the first-order approximate solutions are obtained by the averaging method. Then the definitions of equivalent linear damping coefficient (ELDC) and equivalent linear stiffness coefficient (ELSC) for subharmonic resonance are established, and the effects of the fractional-order parameters on the ELDC, the ELSC, and the dynamical characteristics of system are also analysed. Moreover, the amplitude-frequency equation and phase-frequency equation of steady-state solution for subharmonic resonance are established. The corresponding stability condition is presented based on Lyapunov theory, and the existence condition for subharmonic resonance (ECSR) is also obtained. At last, the comparisons of the fractional-order and the traditional integer-order VDP oscillator are fulfilled by the numerical simulation. The effects of the parameters in fractional-order derivative on the steady-state amplitude, the amplitude-frequency curves, and the system stability are also studied.

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

2019 ◽  
Vol 2019 ◽  
pp. 1-16 ◽  
Author(s):  
Wuce Xing ◽  
Enli Chen ◽  
Yujian Chang ◽  
Meiqi Wang
Keyword(s):  
Numerical Simulation ◽  
Fractional Order ◽  
Chaotic Motion ◽  
Duffing Oscillator ◽  
Threshold Value ◽  
Order Derivative ◽  
Necessary Condition ◽  
Threshold Curve ◽  
Fractional Order Derivative ◽  
Simulation Results

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.

scholarly journals Numerical Solution of the Bagley-Torvik Equation Using the Integer-Order Derivatives Expansion

2018 ◽  
Vol 6 (2) ◽  
Author(s):  
Afrah Sadiq Hasan
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Numerical Solution ◽  
Fractional Order ◽  
Infinite Series ◽  
Exact Analytical Solution ◽  
Second Order ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Integer Order

Numerical solution of the well-known Bagley-Torvik equation is considered. The fractional-order derivative in the equation is converted, approximately, to ordinary-order derivatives up to second order. Approximated Bagley-Torvik equation is obtained using finite number of terms from the infinite series of integer-order derivatives expansion for the Riemann–Liouville fractional derivative. The Bagley-Torvik equation is a second-order differential equation with constant coefficients. The derived equation, by considering only the first three terms from the infinite series to become a second-order ordinary differential equation with variable coefficients, is numerically solved after it is transformed into a system of first-order ordinary differential equations. The approximation of fractional-order derivative and the order of the truncated error are illustrated through some examples. Comparison between our result and exact analytical solution are made by considering an example with known analytical solution to show the preciseness of our proposed approach.

scholarly journals Fractional Model for a Class of Diffusion-Reaction Equation Represented by the Fractional-Order Derivative

2020 ◽  
Vol 4 (2) ◽  
pp. 15 ◽  
Author(s):  
Ndolane Sene
Keyword(s):  
Analytical Solution ◽  
Diffusion Equation ◽  
Fractional Order ◽  
Integration Method ◽  
Diffusion Reaction ◽  
Order Derivative ◽  
Reaction Equation ◽  
Fractional Model ◽  
Fractional Order Derivative ◽  
The Laplace Transform

This paper proposes the analytical solution for a class of the fractional diffusion equation represented by the fractional-order derivative. We mainly use the Grunwald–Letnikov derivative in this paper. We are particularly interested in the application of the Laplace transform proposed for this fractional operator. We offer the analytical solution of the fractional model as the diffusion equation with a reaction term expressed by the Grunwald–Letnikov derivative by using a double integration method. To illustrate our findings in this paper, we represent the analytical solutions for different values of the used fractional-order derivative.

scholarly journals A new general fractional-order wave model involving Miller-Ross kernel

2019 ◽  
Vol 23 (Suppl. 3) ◽  
pp. 953-957
Author(s):  
Linming Dou ◽  
Xiao-Jun Yang ◽  
Jiangen Liu
Keyword(s):  
Analytical Solution ◽  
Fractional Order ◽  
Wave Model ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Complex Processes ◽  
First Time

In the paper we consider a general fractional-order wave model with the general fractional-order derivative involving the Miller-Ross kernel for the first time. The analytical solution for the general fractional-order wave model is investigated in detail. The obtained result is given to explore the complex processes in the mining rock.

scholarly journals Novel Approaches for Getting the Solution of the Fractional Black–Scholes Equation Described by Mittag-Leffler Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Ndolane Sene ◽  
Babacar Sène ◽  
Seydou Nourou Ndiaye ◽  
Awa Traoré
Keyword(s):  
Analytical Solution ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Fractional Integration ◽  
Order Derivative ◽  
Graphical Representations ◽  
Fractional Order Derivative ◽  
Black Scholes

The value of an option plays an important role in finance. In this paper, we use the Black–Scholes equation, which is described by the nonsingular fractional-order derivative, to determine the value of an option. We propose both a numerical scheme and an analytical solution. Recent studies in fractional calculus have included new fractional derivatives with exponential kernels and Mittag-Leffler kernels. These derivatives have been found to be applicable in many real-world problems. As fractional derivatives without nonsingular kernels, we use a Caputo–Fabrizio fractional derivative and a Mittag-Leffler fractional derivative. Furthermore, we use the Adams–Bashforth numerical scheme and fractional integration to obtain the numerical scheme and the analytical solution, and we provide graphical representations to illustrate these methods. The graphical representations prove that the Adams–Bashforth approach is helpful in getting the approximate solution for the fractional Black–Scholes equation. Finally, we investigate the volatility of the proposed model and discuss the use of the model in finance. We mainly notice in our results that the fractional-order derivative plays a regulator role in the diffusion process of the Black–Scholes equation.

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