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.

Nonlinear Dynamics of Duffing System With Fractional Order Damping

2009 ◽  
Author(s):  
Junyi Cao ◽  
Chengbin Ma ◽  
Hang Xie ◽  
Zhuangde Jiang
Keyword(s):  
Nonlinear Dynamics ◽  
Fractional Order ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
Bifurcation Diagrams ◽  
Duffing System ◽  
Runge Kutta Method ◽  
Route To Chaos ◽  
Euler Methods ◽  
Period Motion

In this paper, nonlinear dynamics of Duffing system with fractional order damping is investigated. The four order Runge-Kutta method and ten order CFE-Euler methods are introduced to simulate the fractional order Duffing equations. The effect of taking fractional order on the system dynamics is investigated using phase diagrams, bifurcation diagrams and Poincare map. The bifurcation diagram is also used to exam the effects of excitation amplitude and frequency on Duffing system with fractional order damping. The analysis results show that the fractional order damped Duffing system exhibits period motion, chaos, period motion, chaos, period motion in turn when the fractional order changes from 0.1 to 2.0. A period doubling route to chaos is clearly observed.

THE EFFECT OF NONLINEAR DAMPING ON THE UNIVERSAL ESCAPE OSCILLATOR

1999 ◽  
Vol 09 (04) ◽  
pp. 735-744 ◽  
Author(s):  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Energy Dissipation ◽  
Period Doubling ◽  
Nonlinear Damping ◽  
Basins Of Attraction ◽  
Period Doubling Bifurcation ◽  
Damping Coefficient ◽  
Nonlinear Dissipation ◽  
Fractal Basin Boundaries ◽  
Basin Boundaries

This paper analyzes the role of nonlinear dissipation on the universal escape oscillator. Nonlinear damping terms proportional to the power of the velocity are assumed and an investigation on its effects on the dynamics of the oscillator, such as the threshold of period-doubling bifurcation, fractal basin boundaries and how the basins of attraction are destroyed, is carried out. The results suggest that increasing the power of the nonlinear damping, has similar effects as of decreasing the damping coefficient for a linearly damped case, showing the very importance of the level or amount of energy dissipation.

Dynamic Behaviors of a Fractional Order Viscoelastic Two-Member Truss

2011 ◽  
Vol 90-93 ◽  
pp. 951-957
Author(s):  
Yuan Ping Li ◽  
Wei Zhang
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Chaotic Motion ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
Fractional Dynamics ◽  
Dynamic Behaviors ◽  
Period Doubling Bifurcations

The fractional dynamics equation of a viscoelastic two-member truss system, in which fractional derivative model introduced to simulate the materials’ characteristics, is proposed. The simplified single DOF differential equation is developed combined with boundary conditions and symmetry. Dynamic behaviors of the fractional single DOF system with harmonic loads are discussed by numerical calculations. The results show that: the system may lead to chaotic motion via period-doubling bifurcations or intermittent routes; the dynamical character is greatly inflected by the varying of excitation amplitude or damping coefficient or fractional order.

An Unsymmetrical Motion in a Horizontal Impact Oscillator

2002 ◽  
Vol 124 (3) ◽  
pp. 420-426 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Numerical Investigation ◽  
Dynamic Systems ◽  
Periodic Motion ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Periodic Excitation ◽  
Periodic Motions ◽  
Discontinuous Systems ◽  
Impact Oscillator ◽  
Good Agreement

Stability and bifurcation for the unsymmetrical, periodic motion of a horizontal impact oscillator under a periodic excitation are investigated through four mappings based on two switch-planes relative to discontinuities. Period-doubling bifurcation for unsymmetrical period-1 motions instead of symmetrical period-1 motion is observed. A numerical investigation for symmetrical, period-1 motion to chaos is completed. The numerical and analytical results of periodic motions are in very good agreement. The methodology presented in this paper is applicable to other discontinuous dynamic systems. This investigation also provides a better understanding of such an unsymmetrical motion in symmetrical discontinuous systems.

scholarly journals Period-Doubling Bifurcation of Stochastic Fractional-Order Duffing System via Chebyshev Polynomial Approximation

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Youming Lei ◽  
Yanyan Wang
Keyword(s):  
Fractional Order ◽  
Chebyshev Polynomial ◽  
Polynomial Approximation ◽  
Period Doubling ◽  
Random Parameter ◽  
Fractional Order System ◽  
Period Doubling Bifurcation ◽  
Order System ◽  
Integer Order ◽  
Chebyshev Polynomial Approximation

Fractional-order calculus is more competent than integer-order one when modeling systems with properties of nonlocality and memory effect. And many real world problems related to uncertainties can be modeled with stochastic fractional-order systems with random parameters. Therefore, it is necessary to analyze the dynamical behaviors in those systems concerning both memory and uncertainties. The period-doubling bifurcation of stochastic fractional-order Duffing (SFOD for short) system with a bounded random parameter subject to harmonic excitation is studied in this paper. Firstly, Chebyshev polynomial approximation in conjunction with the predictor-corrector approach is used to numerically solve the SFOD system that can be reduced to the equivalent deterministic system. Then, the global and local analysis of period-doubling bifurcation are presented, respectively. It is shown that both the fractional-order and the intensity of the random parameter can be taken as bifurcation parameters, which are peculiar to the stochastic fractional-order system, comparing with the stochastic integer-order system or the deterministic fractional-order system. Moreover, the Chebyshev polynomial approximation is proved to be an effective approach for studying the period-doubling bifurcation of the SFOD system.

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.

Controlling Nonlinear Dynamics in a Two-Well Impact System.

1998 ◽  
Vol 08 (12) ◽  
pp. 2387-2407 ◽  
Author(s):  
Stefano Lenci ◽  
Giuseppe Rega
Keyword(s):  
Nonlinear Dynamics ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
System Response ◽  
Chaotic Attractors ◽  
Impact System ◽  
Local Bifurcations ◽  
Global Optimal ◽  
Route To Chaos

A method for controlling nonlinear dynamics based on avoiding homoclinic intersection is systematically implemented to perform a numerical analysis of the control induced modifications of the steady attractors and bifurcation scenario of a two-well impact system. The work is divided into two parts. This paper (Part I) deals with the analysis of the harmonic (reference) and global optimal excitations, which are both symmetric. The bifurcation diagrams obtained for increasing values of the excitation amplitude show there exist a "basic" attractor and other "complementary" solutions. The range of stability of the principal complementary attractors is numerically established, and the mechanisms leading to their disapperance are identified. The role of classical and nonclassical local bifurcations in determining the system response is emphasized. Chaotic attractors are seen to appear and disappear both by classical period doubling route to chaos and by sudden changes. Subductions, boundary and interior crises are repeatedly observed. By comparison of the system response under different excitations we obtain information on the performances of global control, which furnishes relatively low gain in terms of regularization but succeeds in controlling the whole phase space.

Bifurcation in a New Fractional Model of Cerebral Aneurysm at the Circle of Willis

2021 ◽  
Vol 31 (09) ◽  
pp. 2150135
Author(s):  
Zhoujin Cui ◽  
Min Shi ◽  
Zaihua Wang
Keyword(s):  
Blood Flow ◽  
Cerebral Aneurysm ◽  
Fractional Order ◽  
Period Doubling ◽  
Circle Of Willis ◽  
Period Doubling Bifurcation ◽  
Damping Term ◽  
Dynamic Behaviors ◽  
Fractional Models ◽  
External Power

A fractional-order model is proposed to describe the dynamic behaviors of the velocity of blood flow in cerebral aneurysm at the circle of Willis. The fractional-order derivative is used to model the blood flow damping term that features the viscoelasticity of the blood flow behaving between viscosity and elasticity, unlike the existing fractional models that use fractional-order derivatives to replace the integer-order derivatives as mathematical/logical generalization. A numerical analysis of the nonlinear dynamic behaviors of the model is carried out, and the influence of the damping term and the external power supply on the nonlinear dynamics of the model is investigated. It shows that not only chaos via period-doubling bifurcation is observed, but also two additional small period-doubling-bifurcation-like diagrams isolated from the big one are observed, a phenomenon that needs further investigation.

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