scholarly journals Experimental Evidence of Chaos Generated by a Minimal Universal Oscillator Model

2021 ◽  
Vol 31 (12) ◽  
pp. 2150205
Author(s):  
Leonardo Ricci ◽  
Alessio Perinelli ◽  
Michele Castelluzzo ◽  
Stefano Euzzor ◽  
Riccardo Meucci
Keyword(s):  
Lyapunov Exponent ◽  
Chaotic Behavior ◽  
Maximum Lyapunov Exponent ◽  
Universal Model ◽  
Numerical Characterization ◽  
Theoretical Predictions ◽  
Experimental Time Series ◽  
Reproducible Analysis ◽  
Electronic Implementation ◽  
Spiking Activity

Detection of chaos in experimental data is a crucial issue in nonlinear science. Historically, one of the first evidences of a chaotic behavior in experimental recordings came from laser physics. In a recent work, a Minimal Universal Model of chaos was developed by revisiting the model of laser with feedback, and a first electronic implementation was discussed. Here, we propose an upgraded electronic implementation of the Minimal Universal Model, which allows for a precise and reproducible analysis of the model’s parameters space. As a marker of a possible chaotic behavior the variability of the spiking activity that characterizes one of the system’s coordinates was used. Relying on a numerical characterization of the relationship between spiking activity and maximum Lyapunov exponent at different parameter combinations, several potentially chaotic settings were selected. The analysis via divergence exponent method of experimental time series acquired by using those settings confirmed a robust chaotic behavior and provided values of the maximum Lyapunov exponent that are in very good agreement with the theoretical predictions. The results of this work further uphold the reliability of the Minimal Universal Model. In addition, the upgraded electronic implementation provides an easily controllable setup that allows for further developments aiming at coupling multiple chaotic systems and investigating synchronization processes.

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.

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 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 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.

Analysis and Identification of Chaotic Dynamics in a Flying Vibratory System

2004 ◽  
Author(s):  
Jin-Wei Liang ◽  
Shy-Leh Chen ◽  
Ching-Ming Yen
Keyword(s):  
Time Series ◽  
Correlation Dimension ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Random Noise ◽  
Maximum Lyapunov Exponent ◽  
Embedding Dimension ◽  
Vibratory System ◽  
Experimental Time Series ◽  
Acceleration Signals

This paper aims at determining whether chaotic dynamics exist in a flying vibratory system. It is important to identify chaotic behavior in a flying system since it may jeopardize the structure of the flying object and cause instability subsequently. It can also cause uncomfortable experience for passengers in a passenger airplane or inaccurate targeting for a missile. Identification of chaotic dynamics from experimental time series is a nontrivial task, since the data is likely to be contaminated with random noise that possesses similar properties to chaos. In this work, acceleration signals were measured at nine different locations or orientations of the flying object during a test fly. Steady-state acceleration signals were extracted and analyzed. The analysis is based on the pseudo phase-space trajectories reconstructed from the experimental time series using the method of delays. Two indices, the correlation dimension and the maximum Lyapunov exponent, are employed to identify the chaotic behavior and to distinguish it from random noise. In general, the correlation dimension calculated from the pseudo trajectory depends on the embedding dimension. It is found in three of the nine-channel signals that the correlation dimension saturates when the embedding dimension is larger than a critical value. The critical embedding dimension is the minimum dimension required for fully un-stretching the phase-space trajectories. This phenomenon indicates a possible existence of chaotic dynamics. It is also found that the maximum Lyapunov exponents calculated from the same acceleration signals are all positive, which further verifies the possibility of the existence of chaotic motion. In addition, some computational issues regarding the embedding dimension, correlation dimension, and maximum Lyapunov exponent are discussed in this paper.

scholarly journals Experimental Realization of a Multiscroll Chaotic Oscillator with Optimal Maximum Lyapunov Exponent

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Esteban Tlelo-Cuautle ◽  
Ana Dalia Pano-Azucena ◽  
Victor Hugo Carbajal-Gomez ◽  
Mauro Sanchez-Sanchez
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Mathematical Description ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Quantitative Measure ◽  
Operational Amplifiers ◽  
Chaotic Oscillator ◽  
Maximum Lyapunov Exponent ◽  
Experimental Realization

Nowadays, different kinds of experimental realizations of chaotic oscillators have been already presented in the literature. However, those realizations do not consider the value of the maximum Lyapunov exponent, which gives a quantitative measure of the grade of unpredictability of chaotic systems. That way, this paper shows the experimental realization of an optimized multiscroll chaotic oscillator based on saturated function series. First, from the mathematical description having four coefficients (a, b, c, d1), an optimization evolutionary algorithm varies them to maximize the value of the positive Lyapunov exponent. Second, a realization of those optimized coefficients using operational amplifiers is given. Hereina, b, c, d1are implemented with precision potentiometers to tune up to four decimals of the coefficients having the range between 0.0001 and 1.0000. Finally, experimental results of the phase-space portraits for generating from 2 to 10 scrolls are listed to show that their associated value for the optimal maximum Lyapunov exponent increases by increasing the number of scrolls, thus guaranteeing a more complex chaotic behavior.

scholarly journals Inductorless Chua's Circuit: Experimental Time Series Analysis

2007 ◽  
Vol 2007 ◽  
pp. 1-16 ◽  
Author(s):  
R. M. Rubinger ◽  
A. W. M. Nascimento ◽  
L. F. Mello ◽  
C. P. L. Rubinger ◽  
N. Manzanares Filho ◽  
...  
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Time Series Analysis ◽  
Operational Amplifier ◽  
Control Parameter ◽  
Chaotic Behavior ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Experimental Time ◽  
Experimental Time Series

We have implemented an operational amplifier inductorless realization of the Chua's circuit. We have registered time series from its dynamical variables with the resistorRas the control parameter and varying from 1300Ωto 2000Ω. Experimental time series at fixedRwere used to reconstruct attractors by the delay vector technique. The flow attractors and their Poincaré maps considering parameters such as the Lyapunov spectrum, its subproduct the Kaplan-Yorke dimension, and the information dimension are also analyzed here. The results for a typical double scroll attractor indicate a chaotic behavior characterized by a positive Lyapunov exponent and with a Kaplan-Yorke dimension of 2.14. The occurrence of chaos was also investigated through numerical simulations of the Chua's circuit set of differential equations.

scholarly journals Dynamic Response and Chaotic Behavior of a Controllable Flexible Robot

2021 ◽  
Author(s):  
Caixia Ban ◽  
Ganwei Cai ◽  
Wei Wei ◽  
Sixu Peng
Keyword(s):  
Phase Diagram ◽  
Lyapunov Exponent ◽  
Poincaré Map ◽  
Chaotic Behavior ◽  
Angular Speed ◽  
Normal Operation ◽  
Maximum Lyapunov Exponent ◽  
Poincare Map ◽  
Flexible Robot ◽  
Vibration Time

Abstract Flexible robots with controllable mechanisms have advantages over common tandem robots in vibration magnitude, residual vibration time, working speed, and efficiency. However, abnormal vibration can sometimes occur during their use, affecting their normal operation. In order to better understand the causes of this abnormal vibration, our work takes a controllable flexible robot as a research object, and uses a combination of Lagrangian and finite element methods to establish its nonlinear elastic dynamics. The effectiveness of the model is verified by comparing the frequency of the numerical calculation and the test. The time-domain diagram, phase diagram, Poincaré map, and maximum Lyapunov exponent of the elastic motion of the robot wrist are studied, and the chaotic phenomena in the system are identified through the phase diagram, Poincaré map, and the maximum Lyapunov exponent. The relationship between the parameters of the robot motion and the maximum Lyapunov exponent is discussed, including trajectory angular speed and radius. The results show that chaotic behavior exists in the controllable flexible robot, and that trajectory angular speed and radius all have an influence on the chaotic motion, which provides a theoretical basis for further research on the control and optimal design of the mechanism.

Study on State Recognition of ASCE Benchmark Based on Lyapunov Exponent Spectrum Entropy

2011 ◽  
Vol 243-249 ◽  
pp. 5435-5439 ◽  
Author(s):  
Jian Xi Yang ◽  
Jian Ting Zhou ◽  
Yue Chen
Keyword(s):  
Lyapunov Exponent ◽  
Time Sequence ◽  
Maximum Lyapunov Exponent ◽  
Entropy Analysis ◽  
Structural System ◽  
Time Duration ◽  
State Recognition ◽  
Chaotic Characteristic ◽  
Lyapunov Exponent Spectrum ◽  
Maximum Lyapunov Exponents

The paper has made a maximum Lyapunov exponent and Lyapunov exponent spectrum entropy analysis of ASCE Benchmark using non-linear theory and chaos time sequence. The maximum Lyapunov exponents in the two kinds of structural monitored data are both over zero, indicating that in the structural system chaos phenomenon has appeared. And, experiments have shown that the maximum Lyapunov exponent is sensitive of the amount of samples and the time delay. So, to compute the chaos index, the amount of samples and the time duration are of importance. Meanwhile, the Lyapunov exponent spectrum entropy is effective to measure the chaotic characteristic of the system, but ,the entropy is less sensitive to state recognition more than the max Lyapunov exponent.

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