Study on Synchronization of the Centrifugal Flywheel Governor System

2013 ◽  
Vol 433-435 ◽  
pp. 21-29 ◽  
Author(s):  
Jian Kui Peng ◽  
Jian Ning Yu ◽  
Li Zhang ◽  
Ping Hu
Keyword(s):  
Adaptive Control ◽  
Lyapunov Exponents ◽  
External Disturbance ◽  
Global Bifurcation ◽  
Control Methods ◽  
Chaotic Motions ◽  
Poincaré Sections ◽  
Dynamical Behaviors ◽  
System Synchronization ◽  
Poincare Sections

In this paper, the dynamical behaviors of the centrifugal flywheel governor with external disturbance is studied and it has abundant nonlinear behavior.The influence of system parameter is discussed by Lyapunov exponents spectrum and global bifurcation diagram, which accurately portray the partial dynamic behavior of the centrifugal flywheel governor. The routes to chaos are analyzed using Poincaré sections, which are found to be more complex . Periodic and chaotic motions can be clearly distinguished by Poincaré sections, bifurcation diagrams and Lyapunov exponents. Then, the paper proposes coupledfeedback control and adaptive control methods to achieve the chaotic the centrifugal flywheel governor system synchronization, the numerical simulation was provided in order to show the effectiveness of coupled feedback control and adaptive control methods for the synchronization of the chaotic nonautonomous centrifugal flywheel governor system.

Chaotic Motions in Forced Mixed Rayleigh–Liénard Oscillator with External and Parametric Periodic-Excitations

2018 ◽  
Vol 28 (03) ◽  
pp. 1830005 ◽  
Author(s):  
C. H. Miwadinou ◽  
A. V. Monwanou ◽  
A. A. Koukpemedji ◽  
Y. J. F. Kpomahou ◽  
J. B. Chabi Orou
Keyword(s):  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Melnikov Method ◽  
Phase Portraits ◽  
Chaotic Motions ◽  
Poincaré Sections ◽  
Routes To Chaos ◽  
Domain Boundaries ◽  
Bifurcation Structures ◽  
Poincare Sections

This paper addresses the issue of a mixed Rayleigh–Liénard oscillator with external and parametric periodic-excitations. The Melnikov method is utilized to analytically determine the domain boundaries where horseshoe chaos appears. Routes to chaos are investigated through bifurcation structures, Lyapunov exponents, phase portraits and Poincaré sections. The effects of Rayleigh and Liénard parameters are analyzed. Results of analytical investigations are validated and complemented by numerical simulations.

Chaotic Motion of a Parametrically Excited Dielectric Elastomer

2020 ◽  
Vol 12 (03) ◽  
pp. 2050033
Author(s):  
Hamidreza Heidari ◽  
Amin Alibakhshi ◽  
Habib Ramezannejad Azarboni
Keyword(s):  
Time Integration ◽  
Nonlinear Behavior ◽  
Strain Energy Function ◽  
Nonlinear Ordinary Differential Equation ◽  
Dielectric Elastomer ◽  
Dissipation Function ◽  
Bifurcation Diagrams ◽  
Chaotic Motions ◽  
Poincaré Sections ◽  
Poincare Sections

In this paper, an effort is made to study the chaotic motions of a dielectric elastomer (DE). The DE is activated by a time-dependent voltage (AC voltage), which is superimposed on a DC voltage. The Gent strain energy function is employed to model the nonlinear behavior of the elastomeric matter. The nonlinear ordinary differential equation (ODE) in terms of the stretch of the elastomer governing the motion of the system is deduced using the Euler–Lagrange method and the Rayleigh dissipation function. This ODE is solved via the use of a time integration-based solver. The bifurcation diagrams of Poincaré sections are generated to identify the chaotic domains. The largest Lyapunov exponents (LLEs) are illustrated for validation of the results obtained by the bifurcation diagrams. Various types of motion for the system are precisely discussed through the depiction of stretch-time responses, phase-plane diagrams, Poincaré sections and power spectral density (PSD) diagrams. The results reveal that the damping coefficient plays an influential role in suppressing the chaos phenomenon. Besides, the initial stretch of the elastomer could affect the chaotic interval of system parameters.

Studying on the Synchronization of a Mechanical Centrifugal Flywheel Governor System

2008 ◽  
Vol 385-387 ◽  
pp. 309-312
Author(s):  
Yan Dong Chu ◽  
Jian Gang Zhang ◽  
Xian Feng Li ◽  
Ying Xiang Chang
Keyword(s):  
Hopf Bifurcation ◽  
Chaos Synchronization ◽  
System Parameter ◽  
External Disturbance ◽  
Global Bifurcation ◽  
Lyapunov Stability Theory ◽  
Chaotic Synchronization ◽  
Bifurcation And Chaos ◽  
Proper System ◽  
Dynamical Behaviors

In this paper, the dynamical behaviors of the centrifugal flywheel governor with external disturbance are discussed, and the system exhibits exceedingly complicated dynamic behaviors. The influence of system parameter on the chaotic system is discussed through Lyapunov-exponents spectrum and global bifurcation diagram, which accurately portray the partial dynamic behavior of the system. It is chaotic with proper system parameter, and we utilize Poincaré sections to study the Hopf bifurcation and chaos forming of the centrifugal flywheel governor system. Then, we utilize coupled-feedback control and adaptive control to realize the chaotic synchronization and obtain the conditions of chaos synchronization. Finally, we carry on the theory proof using the Lyapunov stability theory to the obtained conditions, the theoretical proof and number simulation shows the effectiveness of these methods.

Stochasticity of Galactic Orbits of Star Clusters

2015 ◽  
Vol 24 (2) ◽  
Author(s):  
L. V. Smirnova
Keyword(s):  
Lyapunov Exponents ◽  
Lyapunov Stability ◽  
Stability Criterion ◽  
Star Clusters ◽  
Open Clusters ◽  
Stability Criteria ◽  
Poincaré Sections ◽  
Lyapunov Stability Criterion ◽  
Poincare Sections

AbstractThe Lyapunov, Lagrange and Poincaré criteria are tested for orbits of 461 open clusters in an axisymmetric potential. Lyapunov exponents and Poincaré sections are computed, and all of the trajectories are found to be stable according to the Lagrange and Poincaré stability criteria. At the same time, some trajectories appear to exhibit minor instability according to the Lyapunov stability criterion.

scholarly journals A Memristive Hyperchaotic System without Equilibrium

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Viet-Thanh Pham ◽  
Christos Volos ◽  
Lucia Valentina Gambuzza
Keyword(s):  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Phase Portraits ◽  
Hyperchaotic System ◽  
Poincaré Sections ◽  
Memristive System ◽  
Poincare Sections

A new memristive system is presented in this paper. The peculiarity of the model is that it does not display any equilibria and exhibits periodic, chaotic, and also hyperchaotic dynamics in a particular range of the parameters space. The behavior of the proposed system is investigated through numerical simulations, such as phase portraits, Lyapunov exponents, and Poincaré sections, and circuital implementation confirmed the hyperchaotic dynamic.

Use of Fractal Dimension in the Characterization of Chaotic Structural Dynamic Sytems

1991 ◽  
Vol 44 (11S) ◽  
pp. S107-S113 ◽  
Author(s):  
E. Hall ◽  
S. Kessler ◽  
S. Hanagud
Keyword(s):  
Fractal Dimension ◽  
Fractal Dimensions ◽  
Specific Class ◽  
Chaotic Motions ◽  
Structural Dynamic ◽  
Poincaré Sections ◽  
Pde Model ◽  
Forcing Function ◽  
Poincare Sections

The purpose of this paper is to investigate the use of fractal dimensions in the characterization of chaotic systems in structural dynamics. The investigation focuses on the example of a simply-supported, Euler-Bernoulli beam which when subjected to a transverse forcing function of a particular amplitude responds chaotically. Three different nonlinear models of the system are studied: a complex partial differential equation (PDE) model, a simplified PDE model, and a Galerkin approximation to the simpler PDE model. The responses of each model are examined through zero velocity Poincare´ sections. To characterize and compare the chaotic trajectories, the box counting fractal dimension of the Poincare´ sections are computed. The results demonstrate that the fractal dimension is a spatial invariant along the length of the beam for the specific class of forcing function studied, and thus it can be used to characterize chaotic motions. In addition, the three models yield different fractal dimensions for the same forcing which indicates that fractal dimensions can also be used to quantify whether a simplification of a chaotic model accurately predicts the chaotic behavior of the full-blown model. Thus the conclusion of the paper is that fractal dimensions may play an important role in the characterization of chaotic structural dynamic systems.

scholarly journals A New No Equilibrium Fractional Order Chaotic System, Dynamical Investigation, Synchronization, and Its Digital Implementation

Inventions ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 49
Author(s):  
Zain-Aldeen S. A. Rahman ◽  
Basil H. Jasim ◽  
Yasir I. A. Al-Yasir ◽  
Raed A. Abd-Alhameed ◽  
Bilal Naji Alhasnawi
Keyword(s):  
Adaptive Control ◽  
Fractional Order ◽  
Chaotic System ◽  
Real World ◽  
Experimental Results ◽  
Chaotic Attractors ◽  
State Variables ◽  
Digital Implementation ◽  
Dynamical Behaviors ◽  
Real World Applications

In this paper, a new fractional order chaotic system without equilibrium is proposed, analytically and numerically investigated, and numerically and experimentally tested. The analytical and numerical investigations were used to describe the system’s dynamical behaviors including the system equilibria, the chaotic attractors, the bifurcation diagrams, and the Lyapunov exponents. Based on the obtained dynamical behaviors, the system can excite hidden chaotic attractors since it has no equilibrium. Then, a synchronization mechanism based on the adaptive control theory was developed between two identical new systems (master and slave). The adaptive control laws are derived based on synchronization error dynamics of the state variables for the master and slave. Consequently, the update laws of the slave parameters are obtained, where the slave parameters are assumed to be uncertain and are estimated corresponding to the master parameters by the synchronization process. Furthermore, Arduino Due boards were used to implement the proposed system in order to demonstrate its practicality in real-world applications. The simulation experimental results were obtained by MATLAB and the Arduino Due boards, respectively, with a good consistency between the simulation results and the experimental results, indicating that the new fractional order chaotic system is capable of being employed in real-world applications.

Hamiltonian Chaos in a Model Alveolus

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
F. S. Henry ◽  
F. E. Laine-Pearson ◽  
A. Tsuda
Keyword(s):  
Reynolds Number ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Fine Particles ◽  
Hamiltonian Dynamics ◽  
Maximal Lyapunov Exponent ◽  
Poincaré Sections ◽  
Pulmonary Acinus ◽  
Hamiltonian Chaos ◽  
Poincare Sections

In the pulmonary acinus, the airflow Reynolds number is usually much less than unity and hence the flow might be expected to be reversible. However, this does not appear to be the case as a significant portion of the fine particles that reach the acinus remains there after exhalation. We believe that this irreversibility is at large a result of chaotic mixing in the alveoli of the acinar airways. To test this hypothesis, we solved numerically the equations for incompressible, pulsatile, flow in a rigid alveolated duct and tracked numerous fluid particles over many breathing cycles. The resulting Poincaré sections exhibit chains of islands on which particles travel. In the region between these chains of islands, some particles move chaotically. The presence of chaos is supported by the results of an estimate of the maximal Lyapunov exponent. It is shown that the streamfunction equation for this flow may be written in the form of a Hamiltonian system and that an expansion of this equation captures all the essential features of the Poincaré sections. Elements of Kolmogorov–Arnol’d–Moser theory, the Poincaré–Birkhoff fixed-point theorem, and associated Hamiltonian dynamics theory are then employed to confirm the existence of chaos in the flow in a rigid alveolated duct.

scholarly journals Nonlinear Dynamics and Chaos in Fractional-Order Hopfield Neural Networks with Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Xia Huang ◽  
Zhen Wang ◽  
Yuxia Li
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Fractional Order ◽  
Hopfield Neural Network ◽  
Hopfield Neural Networks ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Chaotic Motions ◽  
Dynamical Behaviors ◽  
Nonlinear Dynamics And Chaos

A fractional-order two-neuron Hopfield neural network with delay is proposed based on the classic well-known Hopfield neural networks, and further, the complex dynamical behaviors of such a network are investigated. A great variety of interesting dynamical phenomena, including single-periodic, multiple-periodic, and chaotic motions, are found to exist. The existence of chaotic attractors is verified by the bifurcation diagram and phase portraits as well.

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