scholarly journals Chaotic Vibrations of Nonlinearly Supported Tubes in Crossflow

1993 ◽  
Vol 115 (2) ◽  
pp. 128-134 ◽  
Author(s):  
Y. Cai ◽  
S. S. Chen
Keyword(s):  
Mathematical Model ◽  
Instability Region ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Chaotic Motions ◽  
Support Plate ◽  
Chaotic Vibrations ◽  
Power Spectral ◽  
Fluid Damping ◽  
Inactive Mode

Chaotic vibrations associated with the fluidelastic instability of nonlinearly supported tubes in a crossflow is studied theoretically on the basis of the unsteady-flow theory and a bilinear mathematical model. Effective tools, including phase portraits, power spectral density, Poincare maps, Lyapunov exponent, fractal dimension, and bifurcation diagrams, are utilized to distinguish periodic and chaotic motions when the tubes vibrate in the instability region. The results show periodic and chaotic motions in the region corresponding to fluid-damping-controlled instability. Nonlinear supports, with symmetric or asymmetric gaps, significantly affect the distribution of periodic, quasi-periodic, and chaotic motions of a tube exposed to various flow velocities in the instability region of the tube-support-plate-inactive mode.

Bifurcations in a Seasonally Forced Predator–Prey Model with Generalized Holling Type IV Functional Response

2016 ◽  
Vol 26 (12) ◽  
pp. 1650203 ◽  
Author(s):  
Jingli Ren ◽  
Xueping Li
Keyword(s):  
Periodic Solutions ◽  
Functional Response ◽  
Complex Dynamics ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Chaotic Motions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Numerical Bifurcation

A seasonally forced predator–prey system with generalized Holling type IV functional response is considered in this paper. The influence of seasonal forcing on the system is investigated via numerical bifurcation analysis. Bifurcation diagrams for periodic solutions of periods one and two, containing bifurcation curves of codimension one and bifurcation points of codimension two, are obtained by means of a continuation technique, corresponding to different bifurcation cases of the unforced system illustrated in five bifurcation diagrams. The seasonally forced model exhibits more complex dynamics than the unforced one, such as stable and unstable periodic solutions of various periods, stable and unstable quasiperiodic solutions, and chaotic motions through torus destruction or cascade of period doublings. Finally, some phase portraits and corresponding Poincaré map portraits are given to illustrate these different types of solutions.

Experiment of Chaotic Vibration of Loosely Supported Tube Rows in Cross-Flow

1995 ◽  
Vol 117 (3) ◽  
pp. 204-212 ◽  
Author(s):  
S. S. Chen ◽  
S. Zhu ◽  
Y. Cai
Keyword(s):  
Experimental Data ◽  
Experimental Study ◽  
Unsteady Flow ◽  
Cross Flow ◽  
Mean Square ◽  
Support Plate ◽  
Chaotic Vibration ◽  
Power Spectral ◽  
Power Spectral Densities ◽  
Inactive Mode

A tube array supported by baffle plates in cross-flow may be subjected to fluidelastic instability in the tube-support-plate-inactive mode. An experimental study is presented to characterize the tube motion. Three series of tests were performed to measure tube displacements as a function of flow velocity for differing clearances. The motion was examined by root-mean-square values of tube displacements, power spectral densities, phase planes, Poincare´ maps, and Lyapunov exponents. The experimental data agree reasonably well with an analytical model based on the unsteady flow theory.

NUMERICAL TREATMENT OF EDUCATIONAL CHAOS OSCILLATOR

2007 ◽  
Vol 17 (10) ◽  
pp. 3657-3661 ◽  
Author(s):  
ARÜNAS TAMAŠEVIČIUS ◽  
TATJANA PYRAGIENĖ ◽  
KȨSTUTIS PYRAGAS ◽  
SKAIDRA BUMELIENĖ ◽  
MANTAS MEŠKAUSKAS
Keyword(s):  
Mathematical Model ◽  
Lyapunov Exponents ◽  
Power Spectra ◽  
Numerical Results ◽  
Phase Portraits ◽  
Numerical Treatment ◽  
Bifurcation Diagrams

A mathematical model of a recently suggested chaos oscillator for educational purposes is treated and numerical results are presented. Bifurcation diagrams, phase portraits, power spectra, Lyapunov exponents are simulated. In addition, the Feigenbaum number is estimated.

scholarly journals Nonlinear and Dual-Parameter Chaotic Vibrations of Lumped Parameter Model in Blisk under Combined Aerodynamic Force and Varying Rotating Speed

2021 ◽  
Author(s):  
W. Zhang ◽  
L. Ma ◽  
Y. F. Zhang ◽  
K. Behdinan
Keyword(s):  
Multiple Scales ◽  
Aerodynamic Force ◽  
Lumped Parameter Model ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Mode Localization ◽  
Rotating Speed ◽  
Chaotic Vibrations ◽  
Lumped Parameter ◽  
Parametric And External Excitations

Abstract In this paper, the nonlinear and dual-parameter chaotic vibrations are investigated for the blisk structure with the lumped parameter model under combined the aerodynamic force and varying rotating speed. The varying rotating speed and aerodynamic force are respectively simplified to the parametric and external excitations. The nonlinear governing equations of motion for the rotating blisk are established by using Hamilton’s principle. The free vibration and mode localization phenomena are studied for the tuning and mistuning blisks. Due to the mistuning, the periodic characteristics of the blisk structure are destroyed and uniform distribution of the energy is broken. It is found that there is a positive correlation between the mistuning variable and mode localization factor to exhibit the large vibration of the blisk in a certain region. The method of multiple scales is applied to derive four-dimensional averaged equations of the blisk under 1:1 internal and principal parametric resonances. The amplitude-frequency response curves of the blisk are studied, which illustrate the influence of different parameters on the bandwidth and vibration amplitudes of the blisk. Lyapunov exponent, bifurcation diagrams, phase portraits, waveforms and Poincare maps are depicted. The dual-parameter Lyapunov exponents and bifurcation diagrams of the blisk reveal the paths leading to the chaos. The influences of different parameters on the bifurcation and chaotic vibrations are analyzed. The numerical results demonstrate that the parametric and external excitations, rotating speed and damping determine the occurrence of the chaotic vibrations and paths leading to the chaotic vibrations in the blisk.

Hopf Bifurcation and Chaos Synchronization for a Class of Mechanical Centrifugal Flywheel Governor System

2007 ◽  
Vol 348-349 ◽  
pp. 349-352
Author(s):  
Yan Dong Chu ◽  
Jian Gang Zhang ◽  
Xian Feng Li ◽  
Ying Xiang Chang
Keyword(s):  
Hopf Bifurcation ◽  
Lyapunov Exponent ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Dynamical Equation ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Chaotic Motions ◽  
Control Laws ◽  
Synchronization Scheme

Chaos and chaos synchronization of the centrifugal flywheel governor system are studied in this paper. By mechanics analyzing, the dynamical equation of the centrifugal flywheel governor system are established. Because of the non-linear terms of the system, the system exhibit both regular and chaotic motions. The evolution from Hopf bifurcation to chaos is shown by the bifurcation diagrams and a series of Poincaré sections under different sets of system parameters, and the bifurcation diagrams are verified by the related Lyapunov exponent spectra. This paper addresses control for the chaos synchronization of feedback control laws in two coupled non-autonomous chaotic systems with different coupling terms, which is demonstrated and verified by Lyapunov exponent spectra and phase portraits. Finally, numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme.

scholarly journals Coexisting infinitely many attractors in a new chaotic system with a curve of equilibria: Its extreme multi-stability and Kolmogorov–Sinai entropy computation

2019 ◽  
Vol 11 (11) ◽  
pp. 168781401988804
Author(s):  
Atefeh Ahmadi ◽  
Xiong Wang ◽  
Fahimeh Nazarimehr ◽  
Fawaz E Alsaadi ◽  
Fuad E Alsaadi ◽  
...  
Keyword(s):  
Dynamical System ◽  
Mathematical Model ◽  
Lyapunov Exponent ◽  
Chaotic System ◽  
Phase Portraits ◽  
Initial Condition ◽  
Bifurcation Diagrams ◽  
Hidden Attractors ◽  
New Chaotic System ◽  
The Mathematical Model

A new five-dimensional chaotic system with extreme multi-stability is introduced in this article. The mathematical model is established, and numerical simulations are done. This dynamical system complicates incident of extreme multi-stability. Most significantly, relied on the mathematical model, the recently proposed system has a curve of equilibria that ends in the occurrence of hidden attractors. We examine the initial-condition-dependent dynamics of this system. We inspect that there is an unrestricted number of coexistent attractors, which signifies the occurrence of extreme multi-stability strictly. In addition, the extreme multi-stability according to initial condition is investigated consuming the Lyapunov exponent spectra and bifurcation diagrams. The existence of coexisting infinitely many attractors is displayed with phase portraits. In the end, we calculate and debate Kolmogorov–Sinai entropy in the chaotic system. We direct trying the Kolmogorov–Sinai technique of entropic inspection for the dynamics of the system.

Nonlinear Dynamics and Application of the Four Dimensional Autonomous Hyper-Chaotic System

2014 ◽  
Author(s):  
Ge Kai ◽  
Wei Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Simulation ◽  
Differential Equations ◽  
Chaotic System ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Phase Portraits ◽  
State Variables ◽  
Chaotic Motions ◽  
Finance System

In this paper, we establish a dynamic model of the hyper-chaotic finance system which is composed of four sub-blocks: production, money, stock and labor force. We use four first-order differential equations to describe the time variations of four state variables which are the interest rate, the investment demand, the price exponent and the average profit margin. The hyper-chaotic finance system has simplified the system of four dimensional autonomous differential equations. According to four dimensional differential equations, numerical simulations are carried out to find the nonlinear dynamics characteristic of the system. From numerical simulation, we obtain the three dimensional phase portraits that show the nonlinear response of the hyper-chaotic finance system. From the results of numerical simulation, it is found that there exist periodic motions and chaotic motions under specific conditions. In addition, it is observed that the parameter of the saving has significant influence on the nonlinear dynamical behavior of the four dimensional autonomous hyper-chaotic system.

scholarly journals Chaos and Control in Coronary Artery System

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Yanxiang Shi
Keyword(s):  
Coronary Artery ◽  
Feedback Control ◽  
Numerical Simulations ◽  
Melnikov Method ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Nonlinear Dynamical ◽  
The Road ◽  
Two Systems ◽  
And Control

Two types of coronary artery system N-type and S-type, are investigated. The threshold conditions for the occurrence of Smale horseshoe chaos are obtained by using Melnikov method. Numerical simulations including phase portraits, potential diagram, homoclinic bifurcation curve diagrams, bifurcation diagrams, and Poincaré maps not only prove the correctness of theoretical analysis but also show the interesting bifurcation diagrams and the more new complex dynamical behaviors. Numerical simulations are used to investigate the nonlinear dynamical characteristics and complexity of the two systems, revealing bifurcation forms and the road leading to chaotic motion. Finally the chaotic states of the two systems are effectively controlled by two control methods: variable feedback control and coupled feedback control.

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.

scholarly journals Fear effect in discrete prey-predator model incorporating square root functional response

2021 ◽  
Vol 2 (2) ◽  
pp. 51-57
Author(s):  
P.K. Santra
Keyword(s):  
Functional Response ◽  
Stability Condition ◽  
Discrete Model ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Square Root ◽  
Bifurcation Diagrams ◽  
Fear Effect ◽  
Hypothetical Data ◽  
Predator Model

In this work, an interaction between prey and its predator involving the effect of fear in presence of the predator and the square root functional response is investigated. Fixed points and their stability condition are calculated. The conditions for the occurrence of some phenomena namely Neimark-Sacker, Flip, and Fold bifurcations are given. Base on some hypothetical data, the numerical simulations consist of phase portraits and bifurcation diagrams are demonstrated to picturise the dynamical behavior. It is also shown numerically that rich dynamics are obtained by the discrete model as the effect of fear.

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