Chaotic Behavior in the Double Pendulum Under Parametric Resonance

2016 ◽  
Author(s):  
Rafael H. Avanço ◽  
Helio A. Navarro ◽  
Airton Nabarrete ◽  
José M. Balthazar ◽  
Angelo Marcelo Tusset
Keyword(s):  
Friction Coefficient ◽  
Parametric Resonance ◽  
Chaotic Behavior ◽  
Excitation Frequency ◽  
Harmonic Excitation ◽  
Phase Portraits ◽  
Double Pendulum ◽  
Bifurcation Diagrams ◽  
Harmonic Motion ◽  
Parametric Pendulum

In literature, the classic parametrically excited pendulum is vastly studied. It consists of a pendulum vertically displaced with a harmonic motion in the support while it oscillates. The chaos in this mechanism may appear depending on the frequency and amplitude of excitation in superharmonic and subharmonic resonance. The double pendulum is also well analyzed in literature, but not under parametric excitation. Therefore, this is the novelty in the present paper. The present analysis considers a double pendulum under a harmonic excitation following the same idea performed previously for a single pendulum. The results are obtained based on methods, such as, phase portraits, Poincaré sections and bifurcation diagrams. The 0–1 tests analyze the presence of chaos while the parameters are varied. The dimensionless parameters take into account the excitation frequency and amplitude as mentioned for the classic parametric pendulum. In this case, we have the particular characteristic that the two pendulums have the same length, the same mass and the same friction coefficient in the joints. The types of motion observed include fixed points, oscillations, rotations and chaos. Results also demonstrated that there was a self-synchronization between these pendulums in ideal excitation.

Subharmonic Bifurcations and Transition to Chaos in a Pipe Conveying Fluid under Harmonic Excitation

2013 ◽  
Vol 444-445 ◽  
pp. 791-795
Author(s):  
Yi Xiang Geng ◽  
Han Ze Liu
Keyword(s):  
Chaotic Behavior ◽  
Melnikov Method ◽  
Harmonic Excitation ◽  
Phase Portraits ◽  
Pipe Conveying Fluid ◽  
Bifurcation Diagrams ◽  
Dynamical Behaviors ◽  
Transition To Chaos ◽  
Subharmonic Bifurcations ◽  
Conveying Fluid

The subharmonic and chaotic behavior of a two end-fixed fluid conveying pipe whose base is subjected to a harmonic excitation are investigated. Melnikov method is applied for the system, and Melnikov criterions for subharmonic and homoclinic bifurcations are obtained analytically. The numerical simulations (including bifurcation diagrams, maximal Lyapunov exponents, phase portraits and Poincare map) confirm the analytical predictions and exhibit the complicated dynamical behaviors.

scholarly journals Identification and Machine Learning Prediction of Nonlinear Behavior in a Robotic Arm System

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1445
Author(s):  
Cheng-Chi Wang ◽  
Yong-Quan Zhu
Keyword(s):  
Neural Network ◽  
Lyapunov Exponent ◽  
Chaotic Motion ◽  
Chaotic Behavior ◽  
Phase Portraits ◽  
Robotic Arm ◽  
Maximum Lyapunov Exponent ◽  
Double Pendulum ◽  
Chaotic Motions ◽  
Robotic Arms

In this study, the subject of investigation was the dynamic double pendulum crank mechanism used in a robotic arm. The arm is driven by a DC motor though the crank system and connected to a fixed side with a mount that includes a single spring and damping. Robotic arms are now widely used in industry, and the requirements for accuracy are stringent. There are many factors that can cause the induction of nonlinear or asymmetric behavior and even excite chaotic motion. In this study, bifurcation diagrams were used to analyze the dynamic response, including stable symmetric orbits and periodic and chaotic motions of the system under different damping and stiffness parameters. Behavior under different parameters was analyzed and verified by phase portraits, the maximum Lyapunov exponent, and Poincaré mapping. Firstly, to distinguish instability in the system, phase portraits and Poincaré maps were used for the identification of individual images, and the maximum Lyapunov exponents were used for prediction. GoogLeNet and ResNet-50 were used for image identification, and the results were compared using a convolutional neural network (CNN). This widens the convolutional layer and expands pooling to reduce network training time and thickening of the image; this deepens the network and strengthens performance. Secondly, the maximum Lyapunov exponent was used as the key index for the indication of chaos. Gaussian process regression (GPR) and the back propagation neural network (BPNN) were used with different amounts of data to quickly predict the maximum Lyapunov exponent under different parameters. The main finding of this study was that chaotic behavior occurs in the robotic arm system and can be more efficiently identified by ResNet-50 than by GoogLeNet; this was especially true for Poincaré map diagnosis. The results of GPR and BPNN model training on the three types of data show that GPR had a smaller error value, and the GPR-21 × 21 model was similar to the BPNN-51 × 51 model in terms of error and determination coefficient, showing that GPR prediction was better than that of BPNN. The results of this study allow the formation of a highly accurate prediction and identification model system for nonlinear and chaotic motion in robotic arms.

scholarly journals Controlling Chaos through Period-Doubling Bifurcations in Attitude Dynamics for Power Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Shun-Chang Chang
Keyword(s):  
Power Systems ◽  
Control Method ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Lyapunov Stability Theory ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Frequency Spectra ◽  
Controlling Chaos ◽  
Period Doubling Bifurcations

This paper addresses the complex nonlinear dynamics involved in controlling chaos in power systems using bifurcation diagrams, time responses, phase portraits, Poincaré maps, and frequency spectra. Our results revealed that nonlinearities in power systems produce period-doubling bifurcations, which can lead to chaotic motion. Analysis based on the Lyapunov exponent and Lyapunov dimension was used to identify the onset of chaotic behavior. We also developed a continuous feedback control method based on synchronization characteristics for suppressing of chaotic oscillations. The results of our simulation support the feasibility of using the proposed method. The robustness of parametric perturbations on a power system with synchronization control was analyzed using bifurcation diagrams and Lyapunov stability theory.

scholarly journals Chaotic Behavior in Gear System. Adaptation of Bifurcation Diagrams and Method of Statistical Mechanics.

1995 ◽  
Vol 61 (587) ◽  
pp. 3108-3115
Author(s):  
Keijin Sato ◽  
Sumio Yamamoto ◽  
Kazutaka Yokota ◽  
Toshihiro Aoki ◽  
Shu Karube
Keyword(s):  
Statistical Mechanics ◽  
Chaotic Behavior ◽  
Gear System ◽  
Bifurcation Diagrams ◽  
System Adaptation

scholarly journals Bifurcations of Oscillatory and Rotational Solutions of Double Pendulum with Parametric Vertical Excitation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Michal Marszal ◽  
Krzysztof Jankowski ◽  
Przemyslaw Perlikowski ◽  
Tomasz Kapitaniak
Keyword(s):  
Periodic Solutions ◽  
Parameter Space ◽  
Double Pendulum ◽  
Bifurcation Diagrams ◽  
Kinematic Excitation ◽  
Vertical Excitation ◽  
Two Parameter ◽  
Domain Of Periodic Solutions

This paper investigates dynamics of double pendulum subjected to vertical parametric kinematic excitation. It includes detailed bifurcation diagrams in two-parameter space (excitation’s frequency and amplitude) for both oscillations and rotations in the domain of periodic solutions.

Analyzing Bifurcation, Stability and Chaos for a Passive Walking Biped Model with a Sole Foot

2018 ◽  
Vol 28 (09) ◽  
pp. 1850113 ◽  
Author(s):  
Maysam Fathizadeh ◽  
Sajjad Taghvaei ◽  
Hossein Mohammadi
Keyword(s):  
Dynamic Models ◽  
Chaotic Behavior ◽  
Bifurcation Diagrams ◽  
Behavior Simulation ◽  
Passive Walking ◽  
Low Energy Consumption ◽  
Nonlinear Dynamic Models ◽  
Soles Of The Feet ◽  
The Stability ◽  
Intrinsic Characteristic

Human walking is an action with low energy consumption. Passive walking models (PWMs) can present this intrinsic characteristic. Simplicity in the biped helps to decrease the energy loss of the system. On the other hand, sufficient parts should be considered to increase the similarity of the model’s behavior to the original action. In this paper, the dynamic model for passive walking biped with unidirectional fixed flat soles of the feet is presented, which consists of two inverted pendulums with L-shaped bodies. This model can capture the effects of sole foot in walking. By adding the sole foot, the number of phases of a gait increases to two. The nonlinear dynamic models for each phase and the transition rules are determined, and the stable and unstable periodic motions are calculated. The stability situations are obtained for different conditions of walking. Finally, the bifurcation diagrams are presented for studying the effects of the sole foot. Poincaré section, Lyapunov exponents, and bifurcation diagrams are used to analyze stability and chaotic behavior. Simulation results indicate that the sole foot has such a significant impression on the dynamic behavior of the system that it should be considered in the simple PWMs.

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 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.

Bifurcation Diagrams and Quotient Topological Spaces Under the Action of the Affine Group of a Family of Planar Quadratic Vector Fields

2015 ◽  
Vol 25 (11) ◽  
pp. 1550150 ◽  
Author(s):  
Oxana Cerba Diaconescu ◽  
Dana Schlomiuk ◽  
Nicolae Vulpe
Keyword(s):  
Parameter Space ◽  
Group Action ◽  
Sufficient Conditions ◽  
Phase Portraits ◽  
Topological Spaces ◽  
Canonical Forms ◽  
Affine Transformations ◽  
Bifurcation Diagrams ◽  
Dimensional Parameter ◽  
Quotient Spaces

In this article, we consider the class [Formula: see text] of all real quadratic differential systems [Formula: see text], [Formula: see text] with gcd (p, q) = 1, having invariant lines of total multiplicity four and two complex and one real infinite singularities. We first construct compactified canonical forms for the class [Formula: see text] so as to include limit points in the 12-dimensional parameter space of this class. We next construct the bifurcation diagrams for these compactified canonical forms. These diagrams contain many repetitions of phase portraits and we show that these are due to many symmetries under the group action. To retain the essence of the dynamics we finally construct the quotient spaces under the action of the group G = Aff(2, ℝ) × ℝ* of affine transformations and time homotheties and we place the phase portraits in these quotient spaces. The final diagrams retain only the necessary information to capture the dynamics under the motion in the parameter space as well as under this group action. We also present here necessary and sufficient conditions for an affine line to be invariant of multiplicity k for a quadratic system.

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