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.

The Features of Chaos

2018 ◽  
Author(s):  
Ray Huffaker ◽  
Marco Bittelli ◽  
Rodolfo Rosa
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Correlation Dimension ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Average Rate ◽  
Quantitative Measure ◽  
Fractal Dimensions ◽  
Recurrence Plot ◽  
Sensitivity To Initial Conditions

In this chapter we introduce the features of Chaotic systems. We describe “sensitivity to initial conditions” and its quantitative measure, the Lyapunov exponent, which reflect the average rate of divergence (if any) between two neighboring trajectories. We describe the dynamic “strangeness” of the system. Which has its counterpart in the “strangeness” of the attractor's geometry and concerns with the texture woven by the system in phase space. Fractal dimensions are measures of such strange geometries and they are here described. The concept of recurrence is introduced and the recurrence plot is described, and code provided to generate it. The correlation dimension is addressed and the R code to compute is listed and detailed. Poincare map is introduced and applied to the study of the damped, driven pendulum.

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.

ENERGY-LIKE FUNCTIONS FOR SOME DISSIPATIVE CHAOTIC SYSTEMS

2005 ◽  
Vol 15 (08) ◽  
pp. 2507-2521 ◽  
Author(s):  
C. SARASOLA ◽  
A. D'ANJOU ◽  
F. J. TORREALDEA ◽  
A. MOUJAHID
Keyword(s):  
Phase Space ◽  
Dynamic Behavior ◽  
Fourth Order ◽  
Chaotic Systems ◽  
Quantitative Measure ◽  
Energy Functions ◽  
Dissipation Of Energy ◽  
Rossler System ◽  
Order Polynomial ◽  
Fourth Order Polynomial

Functions of the phase space variables that can considered as possible energy functions for a given family of dissipative chaotic systems are discussed. This kind of functions are interesting due to their use as an energy-like quantitative measure to characterize different aspects of dynamic behavior of associated chaotic systems. We have calculated quadratic energy-like functions for the cases of Lorenz, Chen, Lü–Chen and Chua, and show the patterns of dissipation of energy on their respective attractors. We also show that in the case of the Rössler system at least a fourth-order polynomial is required to properly represent its energy.

scholarly journals The Study of the Chaotic Behavior in Retailer's Demand Model

2008 ◽  
Vol 2008 ◽  
pp. 1-12 ◽  
Author(s):  
Junhai Ma ◽  
Yun Feng
Keyword(s):  
Lyapunov Exponent ◽  
Systems Theory ◽  
Bifurcation Diagram ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Deterministic Chaos ◽  
Model Parameters ◽  
Initial Condition ◽  
Maximal Lyapunov Exponent ◽  
Demand Model

Based on the work of domestic and foreign scholars and the application of chaotic systems theory, this paper presents an investigation simulation of retailer's demand and stock. In simulation of the interaction, the behavior of the system exhibits deterministic chaos with consideration of system constraints. By the method of space's reconstruction, the maximal Lyapunov exponent of retailer's demand model was calculated. The result shows the model is chaotic. By the results of bifurcation diagram of model parameters , and changing initial condition, the system can be led to chaos.

scholarly journals Chaotic Saddles in a Generalized Lorenz Model of Magnetoconvection

2020 ◽  
Vol 30 (12) ◽  
pp. 2030034
Author(s):  
Francis F. Franco ◽  
Erico L. Rempel
Keyword(s):  
Nonlinear Dynamics ◽  
Fractal Dimension ◽  
Phase Space ◽  
Lyapunov Exponent ◽  
Chaotic Attractors ◽  
Maximum Lyapunov Exponent ◽  
Lorenz Model ◽  
Fractal Basin Boundaries ◽  
Chaotic Transients ◽  
Basin Boundaries

The nonlinear dynamics of a recently derived generalized Lorenz model [ Macek & Strumik, 2010 ] of magnetoconvection is studied. A bifurcation diagram is constructed as a function of the Rayleigh number where attractors and nonattracting chaotic sets coexist inside a periodic window. The nonattracting chaotic sets, also called chaotic saddles, are responsible for fractal basin boundaries with a fractal dimension near the dimension of the phase space, which causes the presence of very long chaotic transients. It is shown that the chaotic saddles can be used to infer properties of chaotic attractors outside the periodic window, such as their maximum Lyapunov exponent.

scholarly journals Time Recurrence Analysis of a Near Singular Billiard

2019 ◽  
Vol 24 (2) ◽  
pp. 50 ◽  
Author(s):  
Rodrigo Simile Baroni ◽  
Ricardo Egydio de Carvalho ◽  
Bruno Castaldi ◽  
Bruno Furlanetto
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Singular Limit ◽  
Largest Lyapunov Exponent ◽  
Classical Dynamics ◽  
Sharp Drop ◽  
Nonlinear Part ◽  
Return Times

Billiards exhibit rich dynamical behavior, typical of Hamiltonian systems. In the present study, we investigate the classical dynamics of particles in the eccentric annular billiard, which has a mixed phase space, in the limit that the scatterer is point-like. We call this configuration the near singular, in which a single initial condition (IC) densely fills the phase space with straight lines. To characterize the orbits, two techniques were applied: (i) Finite-time Lyapunov exponent (FTLE) and (ii) time recurrence. The largest Lyapunov exponent λ was calculated using the FTLE method, which for conservative systems, λ > 0 indicates chaotic behavior and λ = 0 indicates regularity. The recurrence of orbits in the phase space was investigated through recurrence plots. Chaotic orbits show many different return times and, according to Slater’s theorem, quasi-periodic orbits have at most three different return times, the bigger one being the sum of the other two. We show that during the transition to the near singular limit, a typical orbit in the billiard exhibits a sharp drop in the value of λ, suggesting some change in the dynamical behavior of the system. Many different recurrence times are observed in the near singular limit, also indicating that the orbit is chaotic. The patterns in the recurrence plot reveal that this chaotic orbit is composed of quasi-periodic segments. We also conclude that reducing the magnitude of the nonlinear part of the system did not prevent chaotic behavior.

Simple Chaotic Hyperjerk System

2016 ◽  
Vol 26 (11) ◽  
pp. 1650189 ◽  
Author(s):  
Fatma Yildirim Dalkiran ◽  
J. C. Sprott
Keyword(s):  
Dynamical System ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Nonlinear Function ◽  
Nonlinear Functions ◽  
Third Order ◽  
Experimental Realization ◽  
Numerical Studies ◽  
Hyperjerk System ◽  
Jerk System

In literature many chaotic systems, based on third-order jerk equations with different nonlinear functions, are available. A jerk system is taken to be a part of dynamical systems that can exhibit regular and chaotic behavior. By extension, a hyperjerk system can be described as a dynamical system with [Formula: see text]th-order ordinary differential equations where [Formula: see text] is 4 or up to. Hyperjerk systems have been investigated in literature in the last decade. This paper consists of numerical studies and experimental realization on FPAA for fourth-order hyperjerk system with exponential nonlinear function.

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.

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.

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