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.

HOMOCLINIC BIFURCATION AND CHAOS IN Φ6-RAYLEIGH OSCILLATOR WITH THREE WELLS DRIVEN BY AN AMPLITUDE MODULATED FORCE

2011 ◽  
Vol 21 (06) ◽  
pp. 1583-1593 ◽  
Author(s):  
M. SIEWE SIEWE ◽  
C. TCHAWOUA ◽  
S. RAJASEKAR
Keyword(s):  
Lyapunov Exponent ◽  
Bifurcation Diagram ◽  
Poincaré Map ◽  
Chaotic Behavior ◽  
Homoclinic Bifurcation ◽  
Maximal Lyapunov Exponent ◽  
Poincare Map ◽  
Bifurcation And Chaos

With amplitude modulated excitation, the effect on chaotic behavior of Φ6-Rayleigh oscillator with three wells is investigated in this paper. The Melnikov theorem is used to detect the conditions for possible occurrence of chaos. The results show that the domain of the appearance of chaos is enlarged as both amplitudes of modulated and unmodulated forces increase. The effect of these two amplitudes, when both frequencies of modulated and unmodulated forces are different, on bifurcation diagram and Poincaré map is also investigated, in addition to the surface of Maximal Lyapunov exponent versus modulated and unmodulated parameters for suppressing chaos being shown.

scholarly journals Maximal Lyapunov exponent variations of volcanic tremor recorded during explosive and effusive activity at Mt Semeru volcano, Indonesia

2013 ◽  
Vol 20 (6) ◽  
pp. 1137-1145 ◽  
Author(s):  
K. I. Konstantinou ◽  
C. A. Perwita ◽  
S. Maryanto ◽  
A. Budianto ◽  
M. Hendrasto ◽  
...  
Keyword(s):  
Lyapunov Exponent ◽  
Chaotic Behavior ◽  
Fluid Pressure ◽  
Volcanic Tremor ◽  
Period Doubling ◽  
Generation Model ◽  
Maximal Lyapunov Exponent ◽  
Effusive Activity ◽  
Unsteady Fluid Flow ◽  
Unsteady Fluid

Abstract. We analyze 25 episodes of volcanic tremor recorded from 22 November until 31 December 2009 at Mt Semeru volcano in order to investigate their spectral and dynamical properties. The overtone frequencies for most of the tremor events indicate a pattern of period-doubling, which is one possible route that can lead a system to chaotic behavior. Exponential divergence of the phase space orbits is a strong indicator of chaos and was quantified by estimating the maximal Lyapunov exponent (MLE) for all tremor events. MLEs were found to vary linearly with the number of frequency overtones present in the tremor signals. This implies that the tremor source at Semeru fluctuates between a quasi-periodic state with few overtone frequencies (2–3) and small MLEs (~0.013), and a chaotic one with more overtones (up to 8) and larger MLEs (up to 0.039). These results agree well with the tremor generation model suggested previously by Julian (1994), which describes wall oscillations of a crack excited by unsteady fluid flow. In this model, as fluid pressure increases, a period-doubling cascade leads to numerous new frequencies and a chaotic tremor signal. The temporal variation of MLEs exhibited significant fluctuations from 23 until 31 December when the eruptive activity shifted from explosive to effusive. Such a situation may reflect variable fluid pressure conditions inside the conduit, where at first magma is accumulated and subsequently is erupted, releasing the buildup of pressure. Our results give further evidence for the role of nonlinear deterministic processes in generating volcanic tremor and call for similar investigations to be conducted in other volcanoes.

scholarly journals Chaotic planar piecewise smooth vector fields with non-trivial minimal sets

2014 ◽  
Vol 36 (2) ◽  
pp. 458-469 ◽  
Author(s):  
CLAUDIO A. BUZZI ◽  
TIAGO DE CARVALHO ◽  
RODRIGO D. EUZÉBIO
Keyword(s):  
Vector Fields ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Deterministic Chaos ◽  
Piecewise Smooth ◽  
Smooth Vector ◽  
Minimal Sets ◽  
Piecewise Smooth Vector Fields

In this paper some aspects on chaotic behavior and minimality in planar piecewise smooth vector fields theory are treated. The occurrence of non-deterministic chaos is observed and the concept of orientable minimality is introduced. Some relations between minimality and orientable minimality are also investigated and the existence of new kinds of non-trivial minimal sets in chaotic systems is observed. The approach is geometrical and involves the ordinary techniques of non-smooth systems.

scholarly journals Effect of parameter calculation in direct estimation of the Lyapunov exponent in short time series

2002 ◽  
Vol 7 (1) ◽  
pp. 41-52 ◽  
Author(s):  
A. M. López Jiménez ◽  
C. Camacho Martínez Vara de Rey ◽  
A. R. García Torres
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Logistic Map ◽  
Direct Methods ◽  
Short Time Series ◽  
Direct Estimation ◽  
Non Linear ◽  
Short Time

The literature about non-linear dynamics offers a few recommendations, which sometimes are divergent, about the criteria to be used in order to select the optimal calculus parameters in the estimation of Lyapunov exponents by direct methods. These few recommendations are circumscribed to the analysis of chaotic systems. We have found no recommendation for the estimation ofλstarting from the time series of classic systems. The reason for this is the interest in distinguishing variability due to a chaotic behavior of determinist dynamic systems of variability caused by white noise or linear stochastic processes, and less in the identification of non-linear terms from the analysis of time series. In this study we have centered in the dependence of the Lyapunov exponent, obtained by means of direct estimation, of the initial distance and the time evolution. We have used generated series of chaotic systems and generated series of classic systems with varying complexity. To generate the series we have used the logistic map.

scholarly journals Parallel kd-Tree Based Approach for Computing the Prediction Horizon Using Wolf’s Method

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
J. J. Águila ◽  
E. Arias ◽  
M. M. Artigao ◽  
J. J. Miralles
Keyword(s):  
Dynamical System ◽  
Time Series ◽  
Lyapunov Exponent ◽  
Measurement Data ◽  
Original System ◽  
Model Parameters ◽  
Maximal Lyapunov Exponent ◽  
Science And Engineering ◽  
Prediction Horizon ◽  
Means Of Measurement

In different fields of science and engineering, a model of a given underlying dynamical system can be obtained by means of measurement data records called time series. This model becomes very important to understand the original system behaviour and to predict the future values of that system. From the model, parameters such as the prediction horizon can be computed to obtain the point where the prediction becomes useless. In this work, a new parallel kd-tree based approach for computing the prediction horizon is presented. The parallel approach uses the maximal Lyapunov exponent, which is computed by Wolf’s method, as an estimator of the prediction horizon.

A Simple Guide for Plotting a Proper Bifurcation Diagram

2021 ◽  
Vol 31 (01) ◽  
pp. 2150011
Author(s):  
Ali Jafari ◽  
Iqtadar Hussain ◽  
Fahimeh Nazarimehr ◽  
Seyed Mohammad Reza Hashemi Golpayegani ◽  
Sajad Jafari
Keyword(s):  
Bifurcation Diagram ◽  
Chaotic Systems ◽  
Critical Slowing Down ◽  
Initial Condition ◽  
Bifurcation Diagrams ◽  
Transient Time ◽  
Slowing Down ◽  
Critical Issues

In this paper, we propose a guideline for plotting the bifurcation diagrams of chaotic systems. We discuss numerical and mathematical facts in order to obtain more accurate and more elegant bifurcation diagrams. The importance of transient time and the phenomena of critical slowing down are investigated. Some critical issues related to multistability are discussed. Finally, a solution for fast obtaining an accurate sketch of the bifurcation diagram is presented. The solution is based on running the system for only one sample in each parameter value and using the system’s state in the previous value of the parameter as the initial condition.

scholarly journals Dynamical properties of a novel one dimensional chaotic map

2022 ◽  
Vol 19 (3) ◽  
pp. 2489-2505
Author(s):  
Amit Kumar ◽  
◽  
Jehad Alzabut ◽  
Sudesh Kumari ◽  
Mamta Rani ◽  
...  
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Chaotic Behavior ◽  
Chaotic Map ◽  
Logistic Map ◽  
Maximal Lyapunov Exponent ◽  
Bifurcation Diagrams ◽  
One Dimensional ◽  
Dynamical Properties ◽  
Chaotic Logistic Map

<abstract><p>In this paper, a novel one dimensional chaotic map $ K(x) = \frac{\mu x(1\, -x)}{1+ x} $, $ x\in [0, 1], \mu &gt; 0 $ is proposed. Some dynamical properties including fixed points, attracting points, repelling points, stability and chaotic behavior of this map are analyzed. To prove the main result, various dynamical techniques like cobweb representation, bifurcation diagrams, maximal Lyapunov exponent, and time series analysis are adopted. Further, the entropy and probability distribution of this newly introduced map are computed which are compared with traditional one-dimensional chaotic logistic map. Moreover, with the help of bifurcation diagrams, we prove that the range of stability and chaos of this map is larger than that of existing one dimensional logistic map. Therefore, this map might be used to achieve better results in all the fields where logistic map has been used so far.</p></abstract>

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.

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.

Dynamics, Synchronization and Electronic Implementations of a New Lorenz-like Chaotic System with Nonhyperbolic Equilibria

2019 ◽  
Vol 29 (14) ◽  
pp. 1950197 ◽  
Author(s):  
P. D. Kamdem Kuate ◽  
Qiang Lai ◽  
Hilaire Fotsin
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Adaptive Synchronization ◽  
The Novel ◽  
Algebraic Equations ◽  
The Lorenz System

The Lorenz system has attracted increasing attention on the issue of its simplification in order to produce the simplest three-dimensional chaotic systems suitable for secure information processing. Meanwhile, Sprott’s work on elegant chaos has revealed a set of 19 chaotic systems all described by simple algebraic equations. This paper presents a new piecewise-linear chaotic system emerging from the simplification of the Lorenz system combined with the elegance of Sprott systems. Unlike the majority, the new system is a non-Shilnikov chaotic system with two nonhyperbolic equilibria. It is multiplier-free, variable-boostable and exclusively based on absolute value and signum nonlinearities. The use of familiar tools such as Lyapunov exponents spectra, bifurcation diagrams, frequency power spectra as well as Poincaré map help to demonstrate its chaotic behavior. The novel system exhibits inverse period doubling bifurcations and multistability. It has only five terms, one bifurcation parameter and a total amplitude controller. These features allow a simple and low cost electronic implementation. The adaptive synchronization of the novel system is investigated and the corresponding electronic circuit is presented to confirm its feasibility.

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