Bifurcation Analysis of the Fractional Duffing System Based on the Improved Short Memory Principle Method

2021 ◽  
Author(s):  
Ruiqun Ma ◽  
Bo Zhang ◽  
Haiwei Yun ◽  
Jinglong Han
Keyword(s):  
System Dynamics ◽  
Bifurcation Diagram ◽  
Fractional Order ◽  
Chaotic Motion ◽  
Excitation Frequency ◽  
Nonlinear Dynamic Analysis ◽  
Frequency Change ◽  
Duffing System ◽  
Short Memory ◽  
Short Memory Principle

Abstract In this study, the improved short memory principle method is introduced to the analysis of the dynamic characteristics of the fractional Duffing system, and the basis for the improvement of the short memory principle method is provided. The influence of frequency change on the dynamic performance of the fractional Duffing system is studied using nonlinear dynamic analysis methods, such as phase portrait, Poincare map and bifurcation diagram. Moreover, the dynamic behaviour of the fractional Duffing system when the fractional order and excitation amplitude change is investigated. The analysis shows that when the excitation frequency changes from 0.43 to 1.22, the bifurcation diagram contains four periodic and three chaotic motion regions. Periodic motion windows are found in the three chaotic motion regions. Results confirm that the frequency and amplitude of the external excitation and the fractional order of damping have a greater impact on system dynamics. Thus, attention should be paid to the design and analysis of system dynamics.

Nonlinear Dynamics of Duffing System With Fractional Order Damping

2010 ◽  
Vol 5 (4) ◽  
Author(s):  
Junyi Cao ◽  
Chengbin Ma ◽  
Hang Xie ◽  
Zhuangde Jiang
Keyword(s):  
Nonlinear Dynamics ◽  
Bifurcation Diagram ◽  
Fractional Order ◽  
Periodic Motion ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
Euler Method ◽  
Period Doubling Bifurcation ◽  
Damping Coefficient ◽  
Duffing System

In this paper, nonlinear dynamics of Duffing system with fractional order damping is investigated. The fourth-order Runge–Kutta method and tenth-order CFE-Euler method are introduced to simulate the fractional order Duffing equations. The effect of taking fractional order on system dynamics is investigated using phase diagram, bifurcation diagram and Poincaré map. The bifurcation diagram is introduced to exam the effect of excitation amplitude, frequency, and damping coefficient on the Duffing system with fractional order damping. The analysis results show that the fractional order damped Duffing system exhibits periodic motion, chaos, periodic motion, chaos, and periodic motion in turn when the fractional order varies from 0.1 to 2.0. The period doubling bifurcation route to chaos and inverse period doubling bifurcation out of chaos are clearly observed in the bifurcation diagrams with various excitation amplitude, frequency, and damping coefficient.

scholarly journals Complexity and Chimera States in a Network of Fractional-Order Laser Systems

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 341
Author(s):  
Shaobo He ◽  
Hayder Natiq ◽  
Santo Banerjee ◽  
Kehui Sun
Keyword(s):  
Numerical Simulation ◽  
Numerical Solution ◽  
Bifurcation Diagram ◽  
Fractional Order ◽  
Dynamical Model ◽  
Chimera States ◽  
Laser Systems ◽  
Complexity Algorithm ◽  
Derivative Order

By applying the Adams-Bashforth-Moulton method (ABM), this paper explores the complexity and synchronization of a fractional-order laser dynamical model. The dynamics under the variance of derivative order q and parameters of the system have examined using the multiscale complexity algorithm and the bifurcation diagram. Numerical simulation outcomes demonstrate that the system generates chaos with the decreasing of q. Moreover, this paper designs the coupled fractional-order network of laser systems and subsequently obtains its numerical solution using ABM. These solutions have demonstrated chimera states of the proposed fractional-order laser network.

scholarly journals Discretization of forced Duffing system with fractional-order damping

2014 ◽  
Vol 2014 (1) ◽  
Author(s):  
Ahmed MA El-Sayed ◽  
Zaki FE El-Raheem ◽  
Sanaa M Salman
Keyword(s):  
Fractional Order ◽  
Duffing System

Superharmonic Resonance of Fractional-Order Mathieu–Duffing Oscillator

2019 ◽  
Vol 14 (7) ◽  
Author(s):  
Jiangchuan Niu ◽  
Xiaofeng Li ◽  
Haijun Xing
Keyword(s):  
Fractional Order ◽  
Order Term ◽  
Duffing Oscillator ◽  
Steady State Solution ◽  
Harmonic Excitation ◽  
Frequency Equation ◽  
Superharmonic Resonance ◽  
Duffing System ◽  
Resonance Response ◽  
Kbm Method

The superharmonic resonance of fractional-order Mathieu–Duffing oscillator subjected to external harmonic excitation is investigated. Based on the Krylov–Bogolubov–Mitropolsky (KBM) asymptotic method, the approximate analytical solution for the third superharmonic resonance under parametric-forced joint resonance is obtained, where the unified expressions of the fractional-order term with fractional order from 0 to 2 are gained. The amplitude–frequency equation for steady-state solution and corresponding stability condition are also presented. The correctness of the approximate analytical results is verified by numerical results. The effects of the fractional-order term, excitation amplitudes, and nonlinear stiffness coefficient on the superharmonic resonance response of the system are analyzed in detail. The results show that the KBM method is effective to analyze dynamic response in a fractional-order Mathieu–Duffing system.

scholarly journals Nonlinear Dynamic Analysis of Rotor Rub-Impact System

2019 ◽  
Vol 2019 ◽  
pp. 1-20
Author(s):  
Youfeng Zhu ◽  
Zibo Wang ◽  
Qiang Wang ◽  
Xinhua Liu ◽  
Hongyu Zang ◽  
...  
Keyword(s):  
Periodic Motion ◽  
Chaotic Motion ◽  
Nonlinear Dynamic Analysis ◽  
Period Doubling ◽  
Time History ◽  
Rotor System ◽  
Quasiperiodic Motion ◽  
Rotor Bearing ◽  
Motion Period ◽  
Bearing System

A dynamic model of a double-disk rub-impact rotor-bearing system with rubbing fault is established. The dynamic differential equation of the system is solved by combining the numerical integration method with MATLAB. And the influence of rotor speed, disc eccentricity, and stator stiffness on the response of the rotor-bearing system is analyzed. In the rotor system, the time history diagram, the axis locus diagram, the phase diagram, and the Poincaré section diagram in different rotational speeds are drawn. The characteristics of the periodic motion, quasiperiodic motion, and chaotic motion of the system in a given speed range are described in detail. The ways of the system entering and leaving chaos are revealed. The transformation and evolution process of the periodic motion, quasiperiodic motion, and chaotic motion are also analyzed. It shows that the rotor system enters chaos by the way of the period-doubling bifurcation. With the increase of the eccentricity, the quasi-periodicity evolution is chaotic. The quasiperiodic motion evolves into the periodic three motion phenomenon. And the increase of the stator stiffness will reduce the chaotic motion period.

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