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	Aleksandr Lyapunov - Wikipedia, the free encyclopedia
		Lyapunov had already begun to study this stability in his previous two-years attempts at solving the task.
		Lyapunov had lectured already from 1880 at the Department of Mechanics and this had taken him a lot of time.
		Lyapunov lectured at the university on themes from theoretical mechanics, integrals of differential equations and the theory of probability.
	en.wikipedia.org /wiki/Aleksandr_Lyapunov (1642 words)

	Aleksandr Lyapunov -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		Aleksandr Mikhailovich Lyapunov (Александр Михайлович Ляпунов) (June 6 1857 – November 3 1918, all (additional info and facts about new style) new style) was a (A native or inhabitant of Russia) Russian (A person skilled in mathematics) mathematician, (additional info and facts about mechanician) mechanician and (A scientist trained in physics) physicist.
		Lyapunov had already begun to study this (The quality or attribute of being firm and steadfast) stability in his previous two-years attempts at solving the task.
		Lyapunov lectured at the university on themes from theoretical mechanics, (The result of a mathematical integration; F(x) is the integral of f(x) if dF/dx = f(x)) integrals of (An equation containing differentials of a function) differential equations and the (additional info and facts about theory of probability) theory of probability.
	www.absoluteastronomy.com /encyclopedia/a/al/aleksandr_lyapunov.htm (2064 words)

	Aleksandr Lyapunov - FreeEncyclopedia (Site not responding. Last check: 2007-10-21)
		Aleksandr Mikhailovich Lyapunov (Александр Михайлович Ляпунов) (June 6, 1857 - November 3, 1918) was a Russian mathematician, mechanician and physicist.
		Lyapunov wrote his first independent scientific works under the guidance of professor of mechanics, D. Bobilev.
		Lyapunov had lectured already from 1880 at the faculty for mechanics and this had taken him a lot of time.
	openproxy.ath.cx /al/Aleksandr_Mikhailovich_Lyapunov.html (1609 words)

	Strange Attractors
		Equations that exhibit chaos have solutions that are unstable but bounded; the solution never settles down to a fixed value or even to a repeating pattern but neither does it move off to infinity.
		An equation such as the logistic equation that predicts the next value of a quantity from the previous value is called an iterated map because it is like a road map in which each point on the earth is mapped to a corresponding point on a piece of paper.
		Three-dimensional cubic equations have sixty coefficients quartic equations have 105 coefficients and quintic equations have 168 coefficients.
	sprott.physics.wisc.edu /fractals/booktext/sabook.htm (18265 words)

	[No title]
		On the contrary of the results of Hofer and Tibken where the optimal solution has been obtained in terms of a sequence of the differential Riccati equations, in the presented method only solutions of a sequence of the differential Lyapunov equations are required.
		For such equations we give results on the monotonic dependecne of the solutions of the coefficients and initial values as well as results on convergence of solutions.
		The control laws determined by the use of this method are in time invariant, feedback form and approximately minimize a performance index which is the integral of a positive definite function of the state plus a quadratic function of the control.
	www.ee.byu.edu /~beard/papers/library/refs.bib (5355 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		In general there is a Lyapunov exponent for each degree of freedom in the dynamical valuables, and the sorted set of such Lyapunov exponents, known as the Lyapunov spectrum, has been the subject of study in many-particle systems.
		Under the assumption that the random interactions are expressed by a Gaussian white randomness, the master equation is simply attributed to a Fokker-Planck equation, and leads to a direct connection between the Lyapunov exponents and the time correlation of the matrix specifying the particle interactions.
		An "anti-Fokker-Planck equation" to describe the time-reversed tangent vector dynamics is introduced and used to calculate the negative Lyapunov exponents under the assumption of Gaussian white particle interactions.
	www.mpipks-dresden.mpg.de /~chaotran/abstracts/taniguchi/taniguchi.html (396 words)

	4.4 Lyapunov Space
		The differential equation for the damped harmonic oscillator was driven by the addition of a time-dependent force.
		In a similar manner, a time-dependent population modifier could be tacked on to the left side of the logistic equation.
		As long as the growth factor is within the limits of realistic behavior, the population may not exceed the carrying capacity of the environment.
	www.hypertextbook.com /chaos/44.shtml (1323 words)

	Citations: Lyapunov stability of ground states of nonlinear dispersive evolution equations - Weinstein (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		The Nonlinear Schrödinger Equation as Both a PDE and a..
		Lyapunov stability of ground states of nonlinear dispersive evolution equations.
		....numerous other examples of equations for which the persistence of coherent structures under perturbations is linked to the perturbation respecting a certain symmetry of the unperturbed problem.
	citeseer.ist.psu.edu /context/873547/0 (1165 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		R wrote: > "To derive a stable adaptation law, we first observe that A is > asymptotically stable and there exists a matrix P such that > > A(T)P + PA = -I > > Where A(T) denotes the transpose of A, I is the identity matrix of same > order as A".
		Dear Ashok, your equation is referred to as a Lyapunov equation.
		dlyap.m converts the discrete Lyapunov equation to a continous Lyapunov equation, using lyap.m for solving it.
	www.math.niu.edu /~rusin/known-math/99/lyapunov (658 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).
		For each parameter pair, (a,b), the Lyapunov exponent is computed using double-precision floating-point arithmetic.
		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.
	www.efg2.com /Lab/FractalsAndChaos/Lyapunov.htm (1023 words)

	Сибирский Математический Журнал, Том 45 (2004), Номер 6, с. 1271-1284 (Site not responding. Last check: 2007-10-21)
		We consider a quasilinear system of differential equations with periodic coefficients in the linear terms.
		We obtain estimates for the attraction domain of the zero solution and establish estimates for the decay rate of solutions at infinity.
		The results are stated in terms of the integrals of the norm of a periodic solution to the Lyapunov differential equation.
	www.univie.ac.at /EMIS/journals/SMZ/2004/06/1271.htm (83 words)

	Lyapunov Exponent (Site not responding. Last check: 2007-10-21)
		Lyapunov was a Russian Mathematician who lived between 1857 and 1918.
		The average rates of expansion or contraction along the principal axes are the Lyapunov exponents.
		The Lyapunov exponents are taken to be the averages of those obtained over many segments of the central trajectory.
	members.fortunecity.com /arberk/lyapu.html (552 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		A_nn) > of dimension n^2, and the same for the unknown X. > Then the equation turns into > > A = Q X > > with Q being the n x n x n x n matrix > of block form > > R + 1 R_11 1 R_12 1 R_13...
		Lyapunov's theorem states that matrix A has all its eigenvalues in the left half-plane if and only if there exists a unique symmetric positive definite matrix P (i.e.
		I+ATA'-T = 0 is the discrete Lyapunov equation, readily solved by for instance matlab (see dlyap.m).
	math.niu.edu /~rusin/known-math/99/lyapunov (658 words)

	RECSY - High Performance Sylvester-Type Equation Solver Library
		The Sylvester-type matrix equations considered are a set of linear matrix equations that appear and are commonly used in different control theory applications.
		For example, the coefficient matrices A and B in the Sylvester equation AX - XB = C are in upper triangular or upper quasi-triangular form.
		Triangular matrix equations also appear naturally in estimating the condition numbers of matrix equations and different eigenspace computations, including block-diagonalization of matrices and matrix pairs and computation of functions of matrices.
	www.cs.umu.se /research/parallel/recsy (2533 words)

	ECE 722 Spring 2004 (Site not responding. Last check: 2007-10-21)
		Practice the solver of the Lyapunov equation on Matlab: Choose three systems, one with a 2 x 2 matrix F, one with a 5 x 5 matrix F and one with a 10 x 10 matrix F. You design or select these matrices.
		Solve (using Matlab) the matrix Lyapunov equation for the steady state matrix P: Check whether P is symmetric positive definite in each case.
		Assuming that the system is sampled with the sampling period = 1 second, compute the discrete-time transition matrices corresponding to the three matrices F above and solve the discrete-time matrix Lyapunov equation (shown on page 97).
	bass.gmu.edu /ececourses/inft841/hw/ECE722hw5.html (169 words)

	Lyapunov equation - Encyclopedia, History, Geography and Biography (Site not responding. Last check: 2007-10-21)
		Lyapunov equation - Encyclopedia, History, Geography and Biography
		In control theory, the discrete Lyapunov equation is a system of the form
		The discrete Lyapunov equations can, by using Schur complements, be written as
	www.arikah.net /encyclopedia/Lyapunov_equation (126 words)

	T. Penzl : A cyclic low rank Smith method for large, sparse Lyapunov equations with applications in model reduction and ...
		We present a new method for the computation of low rank approximations to the solution of large, sparse, stable Lyapunov equations.
		Besides the efficient solution of the matrix equation itself, a thorough integration of the method into several control algorithms can improve their performance to a high degree.
		This heuristic, which is based on a pair of Arnoldi processes, does not require any a priori knowledge on the spectrum of the coefficient matrix of the Lyapunov equation.
	www.mathematik.tu-chemnitz.de /preprint/1998/SFB393_06.html (272 words)

	Lyapunov Equation and Mathematical Population Genetics - Abstract
		Lyapunov Equation and Mathematical Population Genetics - Abstract
		This is the Lyapunov differential equation, known in the control theory.
		The methodology developed can also be applied to model coevolution of disease and marker loci, of further use for linkage disequilibrium mapping of disease genes.
	www.icm.edu.pl /wydarzenia/seminaria/sem_abst_kimmel.php (176 words)

	Ralph Byers' Preprint and Reprint Archive
		We derive a differential equation for the left factor, develop Euler-like and extrapolated Euler-like numerical methods for approximating an analytic SVD and prove that the Euler-like method converges.
		The sensitivity of the solution of Riccati equations to data perturbations is measured by the condition number.
		This conjecture is related to the numerical stability of methods for solving the matrix Lyapunov equation.
	www.math.ku.edu /~byers/archive/index.html (4687 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The bounds are given in terms of only three quantities: the solution \(X\) of the corresponding Lyapunov equation \(A^X + XA = -I\) and the upper and lower bound of the spectrum of \(A + A^\).
		Our estimate is a consequence of a stronger, local estimate on \(e^{At}\psi\) which nicely depends on the quantity \((X\psi,\psi)\) showing that the spectral geometry of the Lyapunov solution \(X\) --- which is always Hermitian and positive definite --- replaces, at least partly the possibly poor spectral geometry of the generator \(A\).
		A set of examples illustrates the power of our decay estimate for finite matrices; our bound appears to be never drastically worse and is not seldom drastically better than the existing estimates.
	130.39.90.26 /evolve/papers/20030327-3.txt (207 words)

	Lyapunov equation - Wikipedia, the free encyclopedia
		It occurs in many branches of control theory, such as stability analysis and optimal control.
		This and related equations are named after the Russian mathematician Aleksandr Lyapunov.
		This page was last modified 12:47, 14 September 2004.
	en.wikipedia.org /wiki/Lyapunov_equation (78 words)

	P. Benner; Enrique S. Quintana-Orti : Solving stable generalized Lyapunov equations with the matrix sign function
		We investigate the numerical solution of the stable generalized Lyapunov equation via the sign function method.
		This approach has already been proposed to solve standard Lyapunov equations in several publications.
		We consider some modifications and discuss how to solve generalized Lyapunov equations with semidefinite constant term for the Cholesky factor.
	www.mathematik.tu-chemnitz.de /preprint/1997/SFB393_23.html (200 words)

	Oxford University Press
		The text employs Lyapunov's stability theory as a unifying medium for different types of dynamical systems, using it--with its variants--to analyze dynamical system models.
		Specifically, optimal, fuzzy, sliding mode, and chaotic controllers are all constructed with the aid of the Lyapunov method and its extensions.
		It requires previous knowledge of basic differential equations and elements of linear algebra and provides a review of the necessary mathematical techniques and terminology.
	www.oup.com /ca/isbn/0-19-515011-2 (337 words)

	Lyapunov Exponents for the 2-Dimensional Logistic Equation (Site not responding. Last check: 2007-10-21)
		This program, which is included with the PNGwriter source (in examples/), will output an image characterizing the lyapunov exponents of the logistic equation when the parameter R is not constant but a periodic series of 2 values.
		Take a simple equation, and iterate it (take an initial value, put it into the equation, get what it spits out, and plug it in again to the equation, etc, etc).
		You can calculate how chaotic this setup is for these fixed values of A and B. Take A and B as your x-y coordinates, and the lyapunov exponent as the colour value: chaotic is red, super-stable is deep blue, and in between is fl.
	pngwriter.sourceforge.net /examples/lyapunov (756 words)

	Physics at Minnesota:
		solves the special and general forms of the Lyapunov matrix equation.
		Lyapunov equations arise in several areas of control, including stability theory and the study of the RMS behavior of systems.
		The continuous Lyapunov equation has a (unique) solution if the eigenvalues
	www.physics.umn.edu /support/doc/matlab/toolbox/control/lyap.html?printer=yes& (140 words)

	Report
		We discuss the numerical solution and perturbation theory for the generalized continuous-time Lyapunov equation $E^XA+A^XE=-G$ with a singular matrix $E$.
		We generalize a Bartels-Stewart method and a Hammarling method to compute a partial solution of the generalized Lyapunov equation with a special right-hand side.
		A spectral condition number is introduced and perturbation bounds for such an equation are presented.
	www.math.tu-berlin.de /preprints/abstracts/T.Stykel_MonJun3162943.rdf.html (100 words)

	Aleksandr Mikhailovich Lyapunov - Information
		Looking For aleksandr mikhailovich lyapunov - Find aleksandr mikhailovich lyapunov and more at Lycos Search.
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		See the original editable 'Aleksandr Mikhailovich Lyapunov' article.
	www.logicjungle.com /wiki/Aleksandr_Mikhailovich_Lyapunov (1691 words)

	Numerical Analysis Seminar Talk - Feb 11, 2001 (Site not responding. Last check: 2007-10-21)
		The nonlinear generalized diffusion equations called Swift-Hohenberg equation (SH) is considered for which a Lyapunov functional is know.
		We prove that the scheme allows strict (independent of the truncation error) implementation of a discrete approximation of the Lyapunov functional.
		The new scheme is used to investigate the pattern formation from random initial conditions and spatially chaotic states are found.
	www.math.tamu.edu /research/numerical_analysis/seminar/past_seminars/s021102.html (106 words)

	Matrix Reference Manual: Matrix Equations
		In all the equations below, x, y, z, X, Y and Z are the unknown vectors or matrices.
		This is a special case of the Sylvester equation.
		The lyapunov equation is a special case of the Sylvester equation with B=A
	www.ee.ic.ac.uk /hp/staff/dmb/matrix/equation.html (930 words)

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