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.

Non-Linearity and Chaotic Behaviour in Cyprus Stock Market

2015 ◽  
Vol 5 (4) ◽  
Author(s):  
Athina Bougioukou
Keyword(s):  
Lyapunov Exponent ◽  
Stock Market ◽  
Chaos Theory ◽  
Linear Dependence ◽  
Chaotic Behavior ◽  
Efficient Market Hypothesis ◽  
Chaotic Behaviour ◽  
Maximum Lyapunov Exponent ◽  
Efficient Market ◽  
General Index

The intention of this research is to investigate the aspect of non-linearity and chaotic behavior of the Cyprus stock market. For this purpose, we use non-linearity and chaos theory. We perform BDS, Hinich-Bispectral tests and compute Lyapunov exponent of the Cyprus General index. The results show that existence of non-linear dependence and chaotic features as the maximum Lyapunov exponent was found to be positive. This study is important because chaos and efficient market hypothesis are mutually exclusive aspects. The efficient market hypothesis which requires returns to be independent and identically distributed (i.i.d.) cannot be accepted.

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.

Subharmonic and Chaotic Motions of a Hybrid Squeeze-Film Damper-Mounted Rigid Rotor With Active Control

2002 ◽  
Vol 124 (2) ◽  
pp. 198-208 ◽  
Author(s):  
Chieh-Li Chen ◽  
Her-Terng Yau ◽  
Yunhua Li
Keyword(s):  
Lyapunov Exponent ◽  
Active Control ◽  
Chaotic Motion ◽  
Numerical Results ◽  
Operating Conditions ◽  
Squeeze Film ◽  
Speed Ratio ◽  
Maximum Lyapunov Exponent ◽  
Rigid Rotor ◽  
Squeeze Film Damper

The hybrid squeeze-film damper bearing with active control is proposed in this paper. The pressure distribution and the dynamics of a rigid rotor supported by such bearing are studied. A PD (proportional-plus-derivative) controller is used to stabilize the rotor-bearing system. Numerical results show that, due to the nonlinear factors of oil film force, the trajectory of the rotor demonstrates a complex dynamics with rotational speed ratio s. Poincare´ maps, bifurcation diagrams, and power spectra are used to analyze the behavior of the rotor trajectory in the horizontal and vertical directions under different operating conditions. The maximum Lyapunov exponent and fractal dimension concepts are used to determine if the system is in a state of chaotic motion. Numerical results show that the maximum Lyapunov exponent of this system is positive and the dimension of the rotor trajectory is fractal at the nondimensional speed ratio s=3.0, which indicate that the rotor trajectory is chaotic under such operation condition. In order to avoid the nonsynchronous chaotic vibrations, an increased proportional gain is applied to control this system. It is shown that the rotor trajectory will leave chaotic motion to periodic motion in the steady state under control action.

Nonlinear dynamics and chaos

2018 ◽  
Author(s):  
Joseph F. Boudreau ◽  
Eric S. Swanson
Keyword(s):  
Nonlinear Dynamics ◽  
Lyapunov Exponent ◽  
Chaotic Behavior ◽  
Phase Portraits ◽  
Nonlinear Ordinary Differential Equations ◽  
Two Dimensional ◽  
Kam Theorem ◽  
Physical Systems ◽  
Nonlinear Dynamics And Chaos ◽  
Iterative Maps

Simple maps and dynamical systems are used to explore chaos in nature. The discussion starts with a review of the properties of nonlinear ordinary differential equations, including the useful concepts of phase portraits, fixed points, and limit cycles. These notions are developed further in an examination of iterative maps that reveal chaotic behavior. Next, the damped driven oscillator is used to illustrate the Lyapunov exponent that can be used to quantify chaos. The famous KAM theorem on the conditions under which chaotic behavior occurs in physical systems is also presented. The principle is illustrated with the Hénon-Heiles model of a star in a galactic environment and billiard models that describe the motion of balls in closed two-dimensional regions.

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.

Chaos Detection and Optimal Control in a Cannibalistic Prey–Predator System with Harvesting

2020 ◽  
Vol 30 (12) ◽  
pp. 2050171
Author(s):  
Harsha Kharbanda ◽  
Sachin Kumar
Keyword(s):  
Lyapunov Exponent ◽  
Equilibrium Point ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Optimal Level ◽  
Optimal Harvesting ◽  
Predator Population ◽  
Maximum Lyapunov Exponent ◽  
Catch Per Unit Effort ◽  
Chaos Detection

This paper deals with a stage-structured predator–prey system which incorporates cannibalism in the predator population and harvesting in both population. The predator population is categorized into two divisions; adult predator and juvenile predator. The adult predator and prey species are harvested via hypothesis of catch-per-unit-effort, whereas juveniles are safe from being harvested. Mathematically, the dynamic behavior of the system such as existing conditions of equilibria with their stability is studied. The global asymptotic stability of prey-free equilibrium point and nonzero equilibrium point, if they exist, is proved by considering respective Lyapunov functions. The system undergoes transcritical and Hopf–Andronov bifurcations. The impacts of predator harvesting rate and prey harvesting rate on the system are analyzed by taking them as bifurcation parameters. The route to chaos is discussed by showing maximum Lyapunov exponent to be positive with sensitivity dependence on the initial conditions. The chaotic behavior of the system is confirmed by positive maximum Lyapunov exponent and non-integer Kaplan–Yorke dimension. Numerical simulations are executed to probe our theoretic findings. Also, the optimal harvesting policy is studied by applying Pontryagin’s maximum principle. Harvesting effort being an emphatic control instrument is considered to protect prey–predator population, and preserve them also through an optimal level.

scholarly journals The damping term makes the Smale-horseshoe heteroclinic chaotic motion easier

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Huijing Sun ◽  
Hongjun Cao
Keyword(s):  
Chaotic Motion ◽  
Analytical Techniques ◽  
Nonlinear Damping ◽  
Deterministic System ◽  
Maximum Lyapunov Exponent ◽  
Periodic Motions ◽  
Damping Term ◽  
Chaotic Motions ◽  
Parametrically Excited ◽  
Smale Horseshoe

<p style='text-indent:20px;'>The nonlinear Rayleigh damping term that is introduced to the classical parametrically excited pendulum makes the parametrically excited pendulum more complex and interesting. The effect of the nonlinear damping term on the new excitable systems is investigated based on analytical techniques such as Melnikov theory. The threshold conditions for the occurrence of Smale-horseshoe chaos of this deterministic system are obtained. Compared with the existing conclusion, i.e. the smaller the damping term is, the easier the chaotic motions become when the damping term is linear, our analysis, however, finds that the smaller or the larger the damping term is, the easier the Smale-horseshoe heteroclinic chaotic motions become. Moreover, the bifurcation diagram and the patterns of attractors in Poincaré map are studied carefully. The results demonstrate the new system exhibits rich dynamical phenomena: periodic motions, quasi-periodic motions and even chaotic motions. Importantly, according to the property of transitive as well as the fractal layers for a chaotic attractor, we can verify whether a attractor is a quasi-periodic one or a chaotic one when the maximum lyapunov exponent method is difficult to distinguish. Numerical simulations confirm the analytical predictions and show that the transition from regular to chaotic motion.</p>

scholarly journals Chaotic Behavior Analysis of a New Incommensurate Fractional-Order Hopfield Neural Network System

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Nadjette Debbouche ◽  
Adel Ouannas ◽  
Iqbal M. Batiha ◽  
Giuseppe Grassi ◽  
Mohammed K. A. Kaabar ◽  
...  
Keyword(s):  
Neural Network ◽  
Fractional Order ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Hopfield Neural Network ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Neural Network System ◽  
The Stability ◽  
Arbitrary Location

This study intends to examine different dynamics of the chaotic incommensurate fractional-order Hopfield neural network model. The stability of the proposed incommensurate-order model is analyzed numerically by continuously varying the values of the fractional-order derivative and the values of the system parameters. It turned out that the formulated system using the Caputo differential operator exhibits many rich complex dynamics, including symmetry, bistability, and coexisting chaotic attractors. On the other hand, it has been detected that by adapting the corresponding controlled constants, such systems possess the so-called offset boosting of three variables. Besides, the resultant periodic and chaotic attractors can be scattered in several forms, including 1D line, 2D lattice, and 3D grid, and even in an arbitrary location of the phase space. Several numerical simulations are implemented, and the obtained findings are illustrated through constructing bifurcation diagrams, computing Lyapunov exponents, calculating Lyapunov dimensions, and sketching the phase portraits in 2D and 3D projections.

scholarly journals Forecast of Chaotic Series in a Horizon Superior to the Inverse of the Maximum Lyapunov Exponent

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Miguel Alfaro ◽  
Guillermo Fuertes ◽  
Manuel Vargas ◽  
Juan Sepúlveda ◽  
Matias Veloso-Poblete
Keyword(s):  
Lyapunov Exponent ◽  
Dynamic Systems ◽  
Chaotic Behavior ◽  
Chaotic Dynamic ◽  
Salt Lake ◽  
Global Dimension ◽  
Support Vector ◽  
Maximum Lyapunov Exponent ◽  
Forecast Models ◽  
Temporal Horizon

In this article, two models of the forecast of time series obtained from the chaotic dynamic systems are presented: the Lorenz system, the manufacture system, and the volume of the Great Salt Lake of Utah. The theory of the nonlinear dynamic systems indicates the capacity of making good-quality predictions of series coming from dynamic systems with chaotic behavior up to a temporal horizon determined by the inverse of the major Lyapunov exponent. The analysis of the Fourier power spectrum and the calculation of the maximum Lyapunov exponent allow confirming the origin of the series from a chaotic dynamic system. The delay time and the global dimension are employed as parameters in the models of forecast of artificial neuronal networks (ANN) and support vector machine (SVM). This research demonstrates how forecast models built with ANN and SVM have the capacity of making forecasts of good quality, in a superior temporal horizon at the determined interval by the inverse of the maximum Lyapunov exponent or theoretical forecast frontier before deteriorating exponentially.

scholarly journals Chaotic Motion in Forced Duffing System Subject to Linear and Nonlinear Damping

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Tai-Ping Chang
Keyword(s):  
Lyapunov Exponent ◽  
Nonlinear System ◽  
Chaotic Motion ◽  
Chaotic Behavior ◽  
Duffing Oscillator ◽  
Nonlinear Damping ◽  
Critical Value ◽  
Duffing System ◽  
Linear And Nonlinear

This paper investigates the chaotic motion in forced Duffing oscillator due to linear and nonlinear damping by using Melnikov technique. In particular, the critical value of the forcing amplitude of the nonlinear system is calculated by Melnikov technique. Further, the top Lyapunov exponent of the nonlinear system is evaluated by Wolf’s algorithm to determine whether the chaotic phenomenon of the nonlinear system actually occurs. It is concluded that the chaotic motion of the nonlinear system occurs when the forcing amplitude exceeds the critical value, and the linear and nonlinear damping can generate pronounced effects on the chaotic behavior of the forced Duffing oscillator.

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