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.

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.

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.

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.

Nonlinear Dynamic Analysis of a Simplest Fractional-Order Delayed Memristive Chaotic System

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Wei Hu ◽  
Dawei Ding ◽  
Nian Wang
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Chaotic System ◽  
Chaotic Behavior ◽  
Nonlinear Dynamic Analysis ◽  
Period Doubling ◽  
Transversality Condition ◽  
Memristive System ◽  
Route To Chaos ◽  
The Stability

A simplest fractional-order delayed memristive chaotic system is investigated in order to analyze the nonlinear dynamics of the system. The stability and bifurcation behaviors of this system are initially investigated, where time delay is selected as the bifurcation parameter. Some explicit conditions for describing the stability interval and the transversality condition of the emergence for Hopf bifurcation are derived. The period doubling route to chaos behaviors of such a system is discussed by using a bifurcation diagram, a phase diagram, a time-domain diagram, and the largest Lyapunov exponents (LLEs) diagram. Specifically, we study the influence of time delay on the chaotic behavior, and find that when time delay increases, the transitions from one cycle to two cycles, two cycles to four cycles, and four cycles to chaos are observed in this system model. Corresponding critical values of time delay are determined, showing the lowest orders for chaos in the fractional-order delayed memristive system. Finally, numerical simulations are provided to verify the correctness of theoretical analysis using the modified Adams–Bashforth–Moulton method.

INTERMITTENCY AND INTERIOR CRISIS AS ROUTE TO CHAOS IN DYNAMIC WALKING OF TWO BIPED ROBOTS

2012 ◽  
Vol 22 (03) ◽  
pp. 1250056 ◽  
Author(s):  
HASSENE GRITLI ◽  
SAFYA BELGHITH ◽  
NAHLA KHRAEIF
Keyword(s):  
Periodic Orbit ◽  
Chaotic Attractor ◽  
Period Doubling ◽  
Biped Robot ◽  
Type I ◽  
Dynamic Walking ◽  
Unstable Periodic Orbit ◽  
Bifurcation Diagrams ◽  
Biped Robots ◽  
Route To Chaos

Recently, passive and semi-passive dynamic walking has been noticed in researches of biped walking robots. Such biped robots are well-known that they demonstrate only a period-doubling route to chaos while walking down sloped surfaces. In previous researches, such route was shown with respect to a continuous change in some parameter of the biped robot. In this paper, two biped robots are introduced: the passive compass-gait biped robot and the semi-passive torso-driven biped robot. The period-doubling scenario route to chaos is revisited for the first biped as the ground slope changes. Furthermore, we will show through bifurcation diagram that the torso-driven biped exhibits also such route to chaos when the slope angle is varied. For such biped, a modified semi-passive control law is introduced in order to stabilize the torso at some desired position. In this work, we will show through bifurcation diagrams that the dynamic walking of the two biped robots reveals two other routes to chaos namely the intermittency route and the interior crisis route. We will stress that the intermittency is generated via a saddle-node bifurcation where an unstable periodic orbit is created. We will highlight that such event takes place for a Type-I intermittency. However, we will emphasize that the interior crisis event occurs when a collision of the unstable periodic orbit with a weak chaotic attractor happens giving rise to a strong chaotic attractor. In addition, we will explore the intermittent step series induced by the interior crisis and also by the Type-I intermittency. In this study, our analysis on chaos and the routes to chaos will be based, beside bifurcation diagrams, on Lyapunov exponents and fractal (Lyapunov) dimension. These two tools are plotted in the parameter space to classify attractors observed in bifurcation diagrams.

scholarly journals Chaos in a Financial System with Fractional Order and Its Control via Sliding Mode

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Paul Yaovi Dousseh ◽  
Cyrille Ainamon ◽  
Clément Hodévèwan Miwadinou ◽  
Adjimon Vincent Monwanou ◽  
Jean Bio Chabi Orou
Keyword(s):  
Fractional Order ◽  
Chaos Control ◽  
Sliding Mode ◽  
Financial System ◽  
Period Doubling ◽  
Fractional Order Systems ◽  
Uncertain Dynamics ◽  
Dynamical Behaviors ◽  
Order Case ◽  
Route To Chaos

In this paper, the dynamical behaviors and chaos control of a fractional-order financial system are discussed. The lowest fractional order found from which the system generates chaos is 2.49 for the commensurate order case and 2.13 for the incommensurate order case. Also, period-doubling route to chaos was found in this system. The results of this study were validated by the existence of a positive Lyapunov exponent. Besides, in order to control chaos in this fractional-order financial system with uncertain dynamics, a sliding mode controller is derived. The proposed controller stabilizes the commensurate and incommensurate fractional-order systems. Numerical simulations are carried out to verify the analytical results.

Secondary Bifurcations and High Periodic Orbits in Voltage Controlled Buck Converter

1997 ◽  
Vol 07 (12) ◽  
pp. 2755-2771 ◽  
Author(s):  
M. Di Bernardo ◽  
E. Fossas ◽  
G. Olivar ◽  
F. Vasca
Keyword(s):  
Chaotic Attractor ◽  
Invariant Manifolds ◽  
Main Sequence ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Buck Converter ◽  
Bifurcation Diagrams ◽  
Synchronous Switching ◽  
Route To Chaos ◽  
Secondary Bifurcations

Period doubling route to chaos has been shown to occur in the voltage controlled DC/DC buck converter, both experimentally and numerically. A chaotic attractor was found at the end of the sequence, suddenly followed by an increase of its size. In this paper new secondary bifurcations and high periodic phenomena, coexisting with the main sequence are detected and analyzed over the same range of parameters. A(synchronous)-switching and stroboscopic maps, unstable orbits, bifurcation diagrams, invariant manifolds and basins of attraction are outlined. These tools are put together to reveal the dynamical richness of this nonsmooth system.

Chaos in a fractional order logistic map

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Joakim Munkhammar
Keyword(s):  
Fractional Order ◽  
Discrete Time ◽  
Logistic Map ◽  
Period Doubling ◽  
Fractional Dynamics ◽  
Open Problems ◽  
Time Dynamics ◽  
Route To Chaos ◽  
Special Case ◽  
Discrete Time Dynamical Systems

AbstractIn this paper we investigate a fractional order logistic map and its discrete time dynamics. After a brief introduction to the discrete-time dynamical systems and fractional dynamics we show some basic properties of the fractional logistic map. We then move on to prove that the special case α = 1/2 exhibits a period doubling route to chaos. A bifurcation diagram for the special case of α = 1/2 is also included. Finally a discussion concerning the results and open problems is given.

Nonlinear dynamics of fractional order Duffing system

2015 ◽  
Vol 81 ◽  
pp. 111-116 ◽  
Author(s):  
Zengshan Li ◽  
Diyi Chen ◽  
Jianwei Zhu ◽  
Yongjian Liu
Keyword(s):  
Nonlinear Dynamics ◽  
Fractional Order ◽  
Duffing System

scholarly journals Corrigendum to “Chaos in a Financial System with Fractional Order and Its Control via Sliding Mode”

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Paul Yaovi Dousseh ◽  
Cyrille Ainamon ◽  
Clément Hodévèwan Miwadinou ◽  
Adjimon Vincent Monwanou ◽  
Jean Bio Chabi Orou
Keyword(s):  
Fractional Order ◽  
Chaos Control ◽  
Sliding Mode ◽  
Financial System ◽  
Period Doubling ◽  
Fractional Order Systems ◽  
Uncertain Dynamics ◽  
Dynamical Behaviors ◽  
Order Case ◽  
Route To Chaos

In this paper, the dynamical behaviors and chaos control of a fractional-order financial system are discussed. The lowest fractional order found from which the system generates chaos is 2.49 for the commensurate order case and 2.57 for the incommensurate order case. Also, the period-doubling route to chaos was found in this system. The results of this study were validated by the existence of a positive Lyapunov exponent. Besides, in order to control chaos in this fractional-order financial system with uncertain dynamics, a sliding mode controller is derived. The proposed controller stabilizes the commensurate and incommensurate fractional-order systems. Numerical simulations are carried out to verify the analytical results.

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