
	Where results make sense

Topic: Lyapunov exponent

Related Topics

Lyapunov test

Lyapunov stability

Lyapunov function

Dynamical system

Linear theory

Aleksandr Lyapunov

Phase space

Lyapunov fractal

Chaos theory

Special relativity for beginners

Attractor

Fractal dimension

Logistic map

Recurrence plot

Strange attractor

In the News (Tue 5 Jun 12)

EXECUTIVE POWER

Iran

ANALYSIS

Pakistan

Family compact

	4.3 Lyapunov Exponent
		This number, called the Lyapunov exponent "λ" [lambda], is useful for distinguishing among the various types of orbits.
		A Lyapunov exponent of zero indicates that the system is in some sort of steady state mode.
		The logistic equation is superstable at this point, which makes the Lyapunov exponent equal to negative infinity (the limit of the log function as the variable approaches zero).
	hypertextbook.com /chaos/43.shtml (1075 words)

	Lyapunov exponent - Wikipedia, the free encyclopedia
		The Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories.
		The Lyapunov exponents describe the behavior of vectors in the tangent space of the phase space and are defined from the Jacobian matrix
		The set of Lyapunov exponents will be the same for almost all starting points of an ergodic component of the dynamical system.
	en.wikipedia.org /wiki/Lyapunov_exponent (517 words)

	David M. Barnett, Abstracts
		It shows the Lyapunov exponent to be a function of the time integral of the correlation function for fluctuations in the second derivative of the inter-particle potential (approximately a power 1/3 law).
		It shows the Lyapunov exponent to be a function of the time integral of the correlation function for the second derivative of the interparticle potential (approximately a power 1/3 law).
		It shows the Lyapunov exponent to be a function of the time integral of the two time auto-correlation function for the second derivative of the interparticle potential (approximately a power 1/3 law).
	www.users.zetnet.co.uk /david_barnett/writings/Abstracts.html (901 words)

	Lyapunov Page
		Readers who demand precision can more accurately estimate the lyapunov exponent by increasing the number of iterations and at the end of the procedure, by dividing the sum logarithms by the number of iterations.
		From a Lyapunov exponent of zero down to minus infinity, the shade ranges continuously from light to dark.
		If the Lyapunov exponent is plotted for a succession of initial x values, it may take on a specific value, say 0.015 for a number of these initial values.
	rawilson.net /chaos/lyapunov/index.html (1745 words)

	Random Attractors - Found using Lyapunov Exponents
		It is this average rate of divergence (or convergence) that the Lyapunov exponent captures.
		The magnitude of the Lyapunov exponent is a measure of the sensitivity to initial conditions, the primary characteristic of a chaotic system.
		Lyapunov exponents and chaotic attractors in Choas and fractals - new frontiers of science.
	astronomy.swin.edu.au /~pbourke/fractals/lyapunov (683 words)

	Lyapunov exponent: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-20)
		The inverse of the largest Lyapunov exponent is sometimes referred in literature as Lyapunov time, EHandler: no quick summary.
		The lyapunov exponent or lyapunov characteristic exponent of a dynamical system is a measure that determines for each point of phase space, how quickly...
		While there is a whole spectrum of Lyapunov exponents (their number is equal to the dimension of the phase space), EHandler: no quick summary.
	www.absoluteastronomy.com /encyclopedia/l/ly/lyapunov_exponent.htm (1122 words)

	FRACTINT Lyapunov Fractals (Site not responding. Last check: 2007-10-20)
		Negative Lyapunov exponents indicate a stable, periodic behavior and are plotted in color.
		When two parts of a Lyapunov overlap, which spike overlaps which is strongly dependent on the initial value of the population model.
		Like the Bifurcation model, the Lyapunov allow you to set the number of cycles that will be run to allow the model to approach equilibrium before the lyapunov exponent calculation is begun.
	spanky.triumf.ca /www/fractint/lyapunov_type.html (469 words)

	Introduction to Markus-Lyapunov fractals
		In a nutshell, the pictures are nothing but a mapping of Lyapunov exponents relative to a simple extension of the logistic formula (this extension is due to Mario Markus, hence the name Markus-Lyapunov).
		Lyapunov was a Russian mathematician who lived around the turn of the century, long before computers were invented.
		The stability or the chaos can be analysed by computing the Lyapunov exponent, through the above recipe, where the only change is that r is now forced to follow the prescribed periodic pattern.
	perso.wanadoo.fr /charles.vassallo/en/lyap_art/lyapdoc.html (1257 words)

	Lyapunov Characteristic Exponent
		The notion of a Lyapunov exponent is a generalization of the idea of an eigenvalue as a measure of the stability of a fixed point or a characteristic exponent [1] as the measure of the stability of a periodic orbit.
		That is, the Lyapunov characteristic exponent measures the average growth rate of vectors in the tangent manifold.
		From a physical point of view, the Lyapunov exponent is a very useful indicator distinguishing a chaotic from a nonchaotic trajectory [16].
	www.drchaos.net /drchaos/Book/node132.html (433 words)

	Energy Time Series and Chaos
		Lyapunov exponent set up us the “power” of prediction and its reversed value quotes the predictability.
		A positive Lyapunov exponent indicates chaos and it sets the time scale which makes the state of prediction possible.
		The calculation of Hurst exponent or fractal dimension enables us to evaluate how chaotic the time series is. The calculation of Lyapunov exponent or predictability enables us to evaluate the reliability of prediction.
	www.iqnet.cz /dostal/CHA2.htm (413 words)

	Lyapunov fractals (Site not responding. Last check: 2007-10-20)
		There are as many lyapunov exponents as there are dimensions in the state space of the system, but the largest is usually the most important.
		A17c: For the common periodic forcing pictures, the lyapunov exponent is: lambda = limit as N->infinity of 1/N times sum from n=1 to N of log2(abs(dx sub n+1 over dx sub n)) In other words, at each point in the sequence, the derivative of the iterated equation is evaluated.
		The Lyapunov exponent is the average value of the log of the derivative.
	www.faqs.org /faqs/fractal-faq/section-17.html (405 words)

	Random Attractors - Found using Lyapunov Exponents
		The magnitude of the Lyapunov exponent is a measure of the sensitivity to initial conditions, the primary characteristic of a chaotic system.
		If the Lyapunov exponent is less than zero then the system attracts to a fixed point or stable periodic orbit.
		Lyapunov exponents and chaotic attractors in Choas and fractals - new frontiers of science.
	local.wasp.uwa.edu.au /~pbourke/fractals/lyapunov (691 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		As a consequence, the Lyapunov exponent is almost everywhere positive and the absolutely continuous spectrum is almost surely empty.
		Note that the Lyapunov exponents for the two families are the same and that the half-line operators cannot have absolutely continuous spectrum if the whole-line operators do not have any.
		It follows from \cite{k,s} that the Lyapunov exponent for $\{H_\omega\}_{\omega \in \Omega}$ is almost everywhere positive and $\sigma_{{\rm ac}}(H_\omega)$ is empty for almost every $\omega \in \Omega$ with respect to the $(\frac{1}{2},\frac{1}{2})$-Bernoulli measure on $\Omega$.
	www.ma.utexas.edu /mp_arc/papers/04-166 (1114 words)

	technocosm.org - Strange Attractors
		The signature of chaos is that at least one of the Lyapunov exponents is positive.
		Furthermore, the magnitude of the negative exponent has to be greater than the positive one so that initial conditions scattered throughout the basin of attraction contract onto an attractor that occupies a negligible portion of the plane.
		The largest Lyapunov exponent of a system can be computed by taking the difference of the current iteration of x(n) and the current iteration of x(n) plus an epsilon, dx(n), and dividing by the difference of the previous iteration of x(n-1) and previous iteration of x(n-1) + dx(n-1) and taking the natural logarithm.
	technocosm.org /chaos/attr-part2.html (860 words)

	efg's Fractals And Chaos -- Lyapunov Exponents Lab Report
		The purpose of this project is to describe how to compute and display the Lyapunov exponents of the logistic map with periodic forcing (and to compute and display some very interesting graphics images).
		The "Raw Image" of the Lyapunov map is formed by storing raw IEEE floating-point "single" values (trunctated from the computations that used double-precision values) as the pixel data in a pf32bit bitmap.
		In this experiment of displaying floating point values, the floating-point exponent is ignored, and the floating-point fraction becomes the 24-bit RGB value that is displayed.
	www.efg2.com /Lab/FractalsAndChaos/Lyapunov.htm (1023 words)

	MA1251 Chaos & Fractals: September 2004 Examination: Lab 3R
		It is clear that the Lyapunov exponent allows us to distinguish between different types of orbits in dynamical systems.
		Therefore, it is a useful to compute the Lyapunov exponent whenever exploring the behaviour of a dynamical system.
		Obviously, estimating the Lyapunov exponent with the method described in Q1 is far from efficient.
	www.math.le.ac.uk /people/rbrownlee/ma1251/resit/lab3r.html (731 words)

	The Lyapunov exponent
		A quantitative measure of the sensitive dependence on the initial conditions is the Lyapunov exponent
		Lyapunov exponents can be defined also for other non-wandering sets like fixed points etc. How can we calculate the Lyapunov exponent for fixed points?
		For the driven pendula, one Lyapunov exponent is always zero and at least one is always negative.
	monet.physik.unibas.ch /~elmer/pendulum/lyapexp.htm (209 words)

	AMCA: Stability of Dual-spin Spacecraft via Maximal Lyapunov Exponent Based Approach by Samuel F. Asokanthan (Site not responding. Last check: 2007-10-20)
		In the present paper, a technique based on evaluation of top Lyapunov exponents is applied to study the attitude stability behaviour of an asymmetric dual-spin spacecraft system which incorporates a flexible joint.
		It is well known that Lyapunov exponents characterise the exponential rates of change of the response of dynamical systems.
		A numerical scheme for calculating the maximal Lyapunov exponent is developed to examine the stability of solutions of the equations of motion.
	at.yorku.ca /c/a/a/v/21.htm (408 words)

	LCS Tutorial: How the FTLE is used
		The seminal works of Lyapunov (1992), Perron (1930), and Oseledets (1968), were very important in laying the theory of Lyapunov exponent, although the manuscript by Barreira and Pensin (2002) contains a good modern and comprehensive treatment of the subject.
		Because of its asymptotic nature, the classical Lyapunov exponent is not suited for analyzing time-dependent dynamical systems or those that are only defined on a finite time interval, so its value is quite limited for practical analyses.
		Nonetheless, the Finite-Time Lyapunov Exponent is applicable to many time-dependent applications that are perhaps only defined by a discrete data set.
	www.cds.caltech.edu /~shawn/LCS-tutorial/FTLE-interp.html (821 words)

	[No title]
		Calculate LYAPUNOV exponents for the dyadic map and the quadratic map (the latter computationally).
		In more than one dimension, there may be different LYAPUNOV exponents in different directions, and this fact is still true for all of them, by much the same argument.
		Then the LYAPUNOV exponents are [[lambda]]j, and the characteristic directions v are the eigenvectors of A. If the dynamical system is discrete, with mapping F = [[phi]]1 = B linear with real eigenvalues uj and a basis of eigenvectors, then the LYAPUNOV exponents are [[lambda]]j = lnuj.
	www.mathphysics.com /pde/dynam/ch6c.html (1637 words)

	Integer Lyapunov Exponent
		The meaning of the exponent has also significantly changed from that of the real domain Lyapunov exponent, in part because all finite field iterators are necessarily periodic.
		Empirically it was found that for most common hash functions, the integer Lyapunov exponent is a function of table size when evaluated with m=N iterations.
		The integer Lyapunov exponent, however, does preserve one important characteristic of the real Lyapunov exponent; it serves as a measure of the average distance that very close values will be separated by an average iteration.
	www.ece.unm.edu /faculty/heileman/hash/node7.html (319 words)

	Lyapunov fractal - Wikipedia, the free encyclopedia
		Standard Lyapunov logistic fractal with iteration sequence AB Generalized Lyapunov logistic fractal with iteration sequence AABAB
		In mathematics Lyapunov fractals (also known as Markus-Lyapunov fractals) are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population, r, periodically switches between two values a and b.
		A Lyapunov fractal is constructed by mapping the regions of stability and chaotic behaviour (measured using the Lyapunov exponent) in the a-b plane for a given periodic sequence of as and bs.
	en.wikipedia.org /wiki/Lyapunov_fractal (123 words)

	CHARACTERIZATION OF CHAOTIC PARTICLE DYNAMICS IN THE EARTH’S MAGNETOTAIL
		In particular, we calculate the Lyapunov exponent, a Benettin and Strelcyn, in which the divergence of two numerical algorithms.
		One should be careful in the interpretation of the results, since the Lyapunov exponent is defined as a time asymptotic quantity and we are dealing with a chaotic scattering system where the particles have a finite residence time.
		Resonnace surfaces have higher average stochastic Lyapunov exponents, as is expected because stochastic particles are trapped in the system longer and so have more interations with the current sheet.
	www.phy.ilstu.edu /CRISP/rappa.html (486 words)

	Mandelbrot Interior
		Red indicates a large positive (divergent) exponent, moving through yellow, green and blue to fl, which indicates a small or negative (attractive) exponent.
		Lyapunov Exponent of iterates, this time showing the attractive strength.
		In fact, the third graphic appears to show that some areas of attractive Lyapunov exponent lie outside of the basin of attraction.
	www.linas.org /art-gallery/mandel/mandel.html (846 words)

	Multiplicative noise, alpha = 0.01 (Site not responding. Last check: 2007-10-20)
		Each parameter combination (a,b) is associated the numerical approximation of the Lyapunov exponent of a typical sample path.
		The figure clearly indicates that the Lyapunov exponent becomes zero for ab = 1.
		In the upper half of the figure there is a curve of parameters for which the Lyapunov exponent is zero.
	www.iew.unizh.ch /home/klaus/logistic/mult.le.01.html (377 words)

	Lyapunov Exponent and Dimension of the Lorenz Attractor (Site not responding. Last check: 2007-10-20)
		For reasons unknown, published calculations of the largest Lyapunov exponent for the Lorenz attractor [see for example, A. Wolf, J. Swift, H. Swinney, and J. Vasano, Physica 16D, 285 (1985)] have usually used the values p = 16, r = 45.92, and b = 4 for which the Lyapunov exponents (base-2) are (2.16, 0, -32.4).
		It is more natural to express the exponents for a flow in base-e, in which case the values are (1.50, 0, -22.46).
		For a flow, one of the exponents must be zero and the sum of the exponents is -p - 1 - b = -21, which is approximately satisfied by the quoted results.
	sprott.physics.wisc.edu /chaos/lorenzle.htm (393 words)

	Lyapunov Exponent (Site not responding. Last check: 2007-10-20)
		The exponent of equation (8) represents the mean exponential rate of divergence or contraction between two nearby orbits.
		A positive Lyapunov exponent indicates error growth, which means that the iterator being measured is sensitive to initial conditions.
		A zero or negative Lyapunov exponent indicates either no dependence on initial conditions, or a contractive iterator where small errors are damped with each successive iteration.
	www.jea.acm.org /TURING/Vol2Nbr3/node6.html (164 words)

	Extracting beauty from chaos
		Alexander M. Lyapunov, 1857-1918) is used to measure the extent to which ever-smaller "infinitesimal" errors are amplified.
		The Lyapunov Exponent is defined to be the limiting value of the above quantity as the initial error E[0] is made ever smaller-and-smaller, and the number of iterations n is sent to infinity.
		The sharply-defined curves which sometimes appear in these pictures correspond to very large negative Lyapunov exponents (meaning that the system is very stable, in fact "super"-stable, and errors tend to vanish away).
	plus.maths.org /issue9/features/lyapunov (2546 words)

Try your search on: Qwika (all wikis)