APPROACHING NONLINEAR DYNAMICS BY STUDYING THE MOTION OF A PENDULUM II: ANALYZING CHAOTIC MOTION

1994 ◽  
Vol 04 (04) ◽  
pp. 761-771 ◽  
Author(s):  
R. DOERNER ◽  
B. HÜBINGER ◽  
H. HENG ◽  
W. MARTIENSSEN
Keyword(s):  
Nonlinear Dynamics ◽  
Lyapunov Exponents ◽  
Time Scale ◽  
Chaotic Motion ◽  
Deterministic Chaos ◽  
Fractal Dimensions ◽  
Equation Of Motion ◽  
Unstable Periodic Orbits ◽  
Initial State ◽  
Sensitive Dependence

Using a driven damped pendulum as a demonstration model we illustrate some fundamental concepts of nonlinear dynamics. We find deterministic chaos in the motion of the pendulum by observing its sensitive dependence on the initial state. We calculate the corresponding Lyapunov exponents from the equation of motion. The largest exponent gives the average predictability time scale. We estimate fractal dimensions of the attractor by determining the Kaplan Yorke dimension. Further we investigate the organization of unstable periodic orbits embedded in the attractor of the pendulum.

scholarly journals Nonlinear Phenomena in Cournot Duopoly Model

Systems ◽  
2018 ◽  
Vol 6 (3) ◽  
pp. 30
Author(s):  
Pavel Pražák ◽  
Jaroslav Kovárník
Keyword(s):  
Demand Function ◽  
Deterministic Chaos ◽  
Linear Functions ◽  
Nonlinear Phenomena ◽  
Cournot Duopoly ◽  
Limit Values ◽  
Economic Agents ◽  
Duopoly Model ◽  
Sensitive Dependence ◽  
Stackelberg Duopoly

The economic world is very dynamic, and most phenomena appearing in this world are mutually interconnected. These connections may result in the emergence of nonlinear relationships among economic agents. Research discussions about different markets’ structures cannot be considered as finished yet. Even such a well-known concept as oligopoly can be described with different models applying diverse assumptions and using various values of parameters; for example, the Cournot duopoly game, Bertrand duopoly game or Stackelberg duopoly game can be and are used. These models usually assume linear functions and make analyses of the behavior of the two companies. The aim of this paper is to consider a nonlinear inverse demand function in the Cournot duopoly model. Supposing there is a sufficiently large proportion among the costs of the two companies, we can possibly detect nonlinear phenomena such as bifurcation of limit values of production or deterministic chaos. To prove a sensitive dependence on the initial condition, which accompanies deterministic chaos, the concept of Lyapunov exponents is used. We also point out the fact that even though some particular values of parameters are irrelevant for the above-mentioned nonlinear phenomena, it is worth being aware of their existence.

scholarly journals Conditional Entropy: A Tool to Explore the Phase Space

1999 ◽  
Vol 172 ◽  
pp. 195-209
Author(s):  
P. Cincotta ◽  
C. Simó
Keyword(s):  
Phase Space ◽  
Chaotic Motion ◽  
Conditional Entropy ◽  
Characteristic Number ◽  
Random Variable ◽  
Arc Length ◽  
Unstable Periodic Orbits ◽  
Long Time ◽  
Stable Periodic ◽  
Lyapunov Characteristic Number

AbstractIn this paper we show that the Conditional Entropy of nearby orbits may be a useful tool to explore the phase space associated to a given Hamiltonian. The arc length parameter along the orbits, instead of the time, is used as a random variable to compute the entropy. In the first part of this work we summarise the main analytical results to support this tool while, in the second part, we present numerical evidence that this technique is able to localise (stable) periodic and quasiperiodic orbits, ‘aperiodic’ orbits (chaotic motion) and unstable periodic orbits (the ‘source’ of chaotic motion). Besides, we show that this technique provides a measure of chaos which is similar to that given by the largest Lyapunov Characteristic Number. It is important to remark that this method is very simple to compute and does not require long time integrations, just realistic physical times.

Three Conditions Lyapunov Exponents Should Satisfy

2003 ◽  
Author(s):  
Jingjun Lou ◽  
Shijian Zhu
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Chaotic Motion ◽  
The Other ◽  
Cubic Nonlinearity ◽  
3 Dimensional ◽  
Duffing System ◽  
Dissipative Dynamical System

In contrast to the unilateral claim in some papers that a positive Lyapunov exponent means chaos, it was claimed in this paper that this is just one of the three conditions that Lyapunov exponent should satisfy in a dissipative dynamical system when the chaotic motion appears. The other two conditions, any continuous dynamical system without a fixed point has at least one zero exponent, and any dissipative dynamical system has at least one negative exponent and the sum of all of the 1-dimensional Lyapunov exponents id negative, are also discussed. In order to verify the conclusion, a MATLAB scheme was developed for the computation of the 1-dimensional and 3-dimensional Lyapunov exponents of the Duffing system with square and cubic nonlinearity.

Topological characterization of deterministic chaos: enforcing orientation preservation

2007 ◽  
Vol 366 (1865) ◽  
pp. 559-567 ◽  
Author(s):  
Marc Lefranc ◽  
Pierre-Emmanuel Morant ◽  
Michel Nizette
Keyword(s):  
Periodic Orbits ◽  
Topological Analysis ◽  
Three Dimensional ◽  
Deterministic Chaos ◽  
Three Dimensions ◽  
Unstable Periodic Orbits ◽  
Topological Characterization ◽  
Triangulated Surfaces ◽  
Theoretic Characterization

The determinism principle, which states that dynamical state completely determines future time evolution, is a keystone of nonlinear dynamics and chaos theory. Since it precludes that two state space trajectories intersect, it is a core ingredient of a topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits embedded in a strange attractor. However, knot theory can be applied only to three-dimensional systems. Still, determinism applies in any dimension. We propose an alternative framework in which this principle is enforced by constructing an orientation-preserving dynamics on triangulated surfaces and find that in three dimensions our approach numerically predicts the correct topological entropies for periodic orbits of the horseshoe map.

USING EMPIRICAL MODE DECOMPOSITION WEIGHTING FUNCTIONALS AS TIME SERIES FILTERING COEFFICIENTS FOR PHYSIOLOGICAL RECORDINGS

2012 ◽  
Vol 04 (01n02) ◽  
pp. 1250015 ◽  
Author(s):  
JOHN L. AVEN ◽  
ARNOLD J. MANDELL ◽  
RICHARD COPPOLA
Keyword(s):  
Time Series ◽  
Lyapunov Exponents ◽  
Empirical Mode Decomposition ◽  
Temporal Resolution ◽  
Fractal Dimensions ◽  
Weighting Scheme ◽  
Data Smoothing ◽  
Filtering Method ◽  
Smoothing Methods ◽  
Mode Decomposition

We present a method for enhancing signals possessing nonlinear and nonstationary characteristics, which we call weighting functional-empirical mode decomposition (WF-EMD). The filtering method is based upon the empirical mode decomposition (EMD) and utilizes an energy-based weighting scheme to recombine the decomposed modes into a single cleansed version of the signal. The filter has been developed in such a way that no restrictive assumptions about the data are required. Furthermore, the temporal resolution of the data is left unaltered, as it would occur in many common data-smoothing methods. The design of this filter has been influenced by improving the calculation accuracy of dynamical measures, such as fractal dimensions and Lyapunov exponents, of neurodynamical recordings such as those obtained through electroencephalography (EEG) or magnetoencephalography (MEG).

Photon emission as a probe of chaotic processes accompanying fracture

1989 ◽  
Vol 4 (5) ◽  
pp. 1272-1279 ◽  
Author(s):  
S. C. Langford ◽  
Ma Zhenyi ◽  
J. T. Dickinson
Keyword(s):  
Fracture Surface ◽  
Photon Emission ◽  
Deterministic Chaos ◽  
Fractal Dimensions ◽  
Significant Degree ◽  
Box Dimension ◽  
Dynamic Crack ◽  
Emission Data ◽  
Correlation Integral ◽  
The Fourier Transform

Photon emission accompanying the fracture of an epoxy and single crystal MgO is examined for evidence of deterministic chaos by means of the autocorrelation function, the Fourier transform, the correlation integral of Grassberger and Procaccia, and the fractal box dimension. A positive Lyapunov exponent is also obtained from the epoxy phE data. Each of these measures is consistent with a significant degree of deterministic chaos associated with attractors of relatively low dimension. A typical epoxy fracture surface was analyzed for fractal character by means of the slit island technique, yielding a fractal dimension of 1.32 ± 0.03. The fractal dimensions of the fracture surface and the photon emission data (box dimension) of the epoxy are in good agreement. These observations suggest that fluctuations in photon emission intensity during fracture reflect the production of fractal surface features as they are being produced and thus provide important information on the process of dynamic crack growth.

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