Lyapunov theory

Lyapunov theory is used to make conclusions about trajectories of a system x˙ = f(x) (e.g., G.A.S.) without finding the trajectories (i.e., solving the differential equation) a typical Lyapunov theorem has the form: • if there exists a function V : Rn → R that satisfies some conditions on V and V˙ • then, trajectories of system satisfy some property if such a function V exists we. Lyapunov stability theory was come out of Lyapunov, a Russian mathematician in 1892, and came from his doctoral dissertation. Until now, the theory of Lyapunov stability is still the main theoretical basis of almost all system-controller design (Chen, 1984). 2 Lyapunov's theorem in probability theory is a theorem that establishes very general sufficient conditions for the convergence of the distributions of sums of independent random variables to the normal distribution Lyapunov Stability Theory of Nonsmooth Systems

Lyapunov Stability Theory - an overview ScienceDirect Topic

Lyapunov theorem - Encyclopedia of Mathematic

The method of Lyapunov functions plays a central role in the study of the controllability and stabilizability of control systems. For nonlinear systems, it turns out to be essential to consider nonsmooth Lyapunov functions, even if the underlying control dynamics are themselves smooth Lyapunov's second (or direct) method provides tools for studying (asymp- totic) stability properties of an equilibrium point of a dynamical system (or systems of dif- ferential equations). The intuitive picture is that of a scalar output-function, often thought of as a generalized energy that is bounded below, and decreasing along solutions Lyapunov exponents describe the exponential growth rates of the norms of vectors under successive actions of derivatives of the random diffeomorphisms. The invariant manifold theory is a nonlinear counterpart of the linear theory of Lyapunov exponents. We first give a rough description of this theory

Lyapunov Stability Theory of Nonsmooth System

Lyapunov Theory for 2-D Nonlinear Roesser Models: Application to Asymptotic and Exponential Stability Nima Yeganefar, Nader Yeganefar, Mariem Ghamgui, and Emmanuel Moulay Abstract—This technical note deals with a general class of discrete 2-D possibly nonlinear systems based on the Roesser model Lyapunov Theory Ivan Papusha CDS270-2: Mathematical Methods in Control and System Engineering April 13, 2015 1 / 26. Logistics • hw2 due this Wed, Apr 15 • do an easy problem or CYOA • hw1 solutions posted online • start reading: lmibook, Ch 1-2 • the book is dense, but extremelygood • free online, written in 1994—even more timely now than ever • less important on a first. 3 Lyapunov Stability Analysis - Duration: 1:18:25. Van Sy Tnut 2,308 views. 1:18:25. Thomas Sowell on the Myths of Economic Inequality - Duration: 53:34. Hoover Institution Recommended for you. 53.

Lyapunov 1.1 Introduction Nous avons vu qu'un point singulier pour lequel la matrice du linearis´ e´ a des valeurs propres a partie r` ´eelle n egative est asymptotiquement stable. Si´ certaines des valeurs propres ont des parties r´eelles positives, le point est in-stable. Mais qu'en est-il dans les autres cas? Beaucoup de situations peuvent se produire et il n'existe pas de m. Une formalisation possible de ce lien organique entre les deux champs scientifiques est constituée par la théorie de Lyapunov. Nous nous attachons donc à illustrer les différents aspects que peut recouvrir la relation entre optimisation et théorie de la commande robuste P C Parks, A M Lyapunov's stability theory - 100 years on, IMA J. Math. Control Inform. 9 (4) (1992), 275-303. V V Pavlovskaya, A M Lyapunov's investigations on stability of ellipsoidal figures of equilibrium of a rotating fluid mass (Russian), Analytic methods for investigating nonlinear oscillations (Kiev, 1980), 129-138. N A Pustovoitov, Theory of the stability of motion after Lyapunov. Lyapunov stability of a point relative to the family of mappings is equivalent to the continuity at this point of the mapping $ x \mapsto x ( \cdot ) $ of a neighbourhood of this point into the set of functions $ x ( \cdot ) $ defined by the formula $ x ( t) = f _ {t} ( x) $, equipped with the topology of uniform convergence on $ G ^ {+} $. Lyapunov stability of a point relative to a mapping. Noté /5. Retrouvez Lyapunov Exponents and Smooth Ergodic Theory et des millions de livres en stock sur Amazon.fr. Achetez neuf ou d'occasio

Vector Lyapunov functions and stability analysis of nonlinear systems (1991) Lyapunov Exponents (1991) Theory and application of Liapunov's direct method (1963) Stability of motion (1963) Auteurs en relation avec ce thème (17 ressources dans data.bnf.fr) Auteur du texte (17). Lyapunov Theory is a powerful tool in systems theory which can be used to analyze the stability of linear and nonlinear systems without explicitly computing their trajectories. General Lyapunov Theory. A function of the system is a candidate Lyapunov function of the system if. Note that we talk of a Lyapunov candidate function because the Lyapunov might or might not prove stability for the. Control: A Lyapunov Approach Wenqi Cui and Baosen Zhang Abstract—The increase in penetration of inverter-based re-sources provide us with more flexibility in frequency regulation of power systems in addition to conventional linear droop con-trollers. Because of the fast power electronic interfaces, inverter- based resources can be used to realize complex control functions and potentially. Linear quadratic Lyapunov theory 13-11. generalization: if A stable, Q ≥ 0, and (Q,A) observable, then P > 0 to see this, the Lyapunov integral shows P ≥ 0 if Pz = 0, then 0 = zTPz = zT Z ∞ 0 etA T QetA dt z = Z ∞ 0 Q1/2etAz 2 dt so we conclude Q1/2etAz = 0 for all t ≥ 0 this implies that Qz = 0, QAz = 0, . . . , QAn−1z = 0, contradicting (Q,A) observable Linear quadratic. Lyapunov theory 1. Lyapunov Theory Sistem Non Linear Aditya A. Purnama / 1410501023 Dosen Pembimbing : R. Suryoto Edy Raharjo, S.T., M.Eng 2. Table of content • Stability • Positive definite functions • Lyapunov stability theorems • Converse Lyapunov theorems • Finding Lyapunov functions 3. Stability • Stability of solutions of differential equations and of trajectories of.

Control-Lyapunov function - Wikipedi

  1. There are 79 lyapunov's theory-related words in total, with the top 5 most semantically related being input-to-state stability, state space representation, stable manifold, stability theory and lyapunov function. You can get the definition(s) of a word in the list below by tapping the question-mark icon next to it
  2. (1)It is known that there exists a fuzzy Lyapunov function if and only if there is a Lyapunov function associated. In the fact is a Lyapunov function associated with system (8). Then, 0 is stable for,and according to Theorem 3, is stable for. (2)According to Theorem 3, is asymptotically stable for if and only if 0 is asymptotically stable for
  3. e stability. Moreover, an estimate of the domain.

Lyapunov (i.s.L.). It is p ossible to ha v e stabilit y in Ly apuno without ha ving asymptotic stabilit y, in whic h case w e refer to the equilibrium p oin t as mar ginal ly stable. Nonlinear systems also exist that satisfy the second requiremen t without b e ing i.s.L., as the follo wing example sho ws. An equilibrium p oin t that is not stable i.s.L. termed unstable. Example 13.1 (Unstable. Lyapunov functions are used extensively in control theory to ensure different forms of system stability. The state of a system at a particular time is often described by a multi-dimensional vector. A Lyapunov function is a scalar measure of this multi-dimensional state. Typically, the function is defined to grow large when the system moves towards undesirable states. System stability is. 4 Lyapunov Stability Theory In this section we review the tools of Lyapunov stability theory. These tools will be used in the next section to analyze the stability properties of a robot controller. We present a survey of the results that we shall need in the sequel, with no proofs

Analysis – Automatic Control Laboratory | ETH Zurich


In this paper, a nonlinear controller using the Lyapunov stability theorem is proposed for quadrotor position and attitude tracking. The quadrotor dynamics is defined for controller design and.. Lyapunov Theory Algorithm Theorem - Linear case Let us consider _x = Ax. The origin is asymptotically stable if and only if all eigenvalues of A have negative real parts. Nicolas Delanoue - Luc Jaulin Attraction domain of a nonlinear system using interval analysis and Lyapunov Theory Our methods deeply exploit the properties of Lyapunov exponents and are based on ergodic and pluripotential theory. Mots clés. Keywords. Dynamique holomorphe, stabilité dynamique, courants positifs, exposants de Lyapunov. Holomorphic dynamics, dynamical stability, positive currents, Lyapunov exponents. Toute la collection. Prix. Adhérent 14 € Non-Adhérent 20 € Quantité-+ Ajouter au.

Video: Stabilité de Liapounov — Wikipédi

IPython Cookbook - 12

Lyapunov theorems  If P > 0, Q > 0, then system is (globally asymptotically) stable. If P > 0, Q  0, and (Q;A) observable, then system is (globally asymptotically) stable. If P > 0, Q  0, then all trajectories of the system are bounded  If Q  0, then the sublevel sets fz j zTPz  ag are invariant. (These are ellipsoids if P > 0. Lyapunov's stability analysis technique is very common and dominant. The main deficiency, which severely limits its utilization, in reality, is the complication linked with the development of the Lyapunov function which is needed by the technique 10 Chaos and Lyapunov exponents 10.1 Chaotic systems Chaotic dynamics exhibit the following properties Trajectories have a nite probability to show aperiodic long-term behaviour. However, a subset of trajectories may still be asymp-totically periodic or quasiperiodic in a chaotic system. System is deterministic, the irregular behavior is due to non- linearity of system and not due to. Lyapunov stability theory is not older than the classical stability theory. See, e.g., the basic stems from state of equilibrium, stationary state, saddle point, roots of the equation or equations,..

Negative Lyapunov exponents are characteristic of dissipative or non-conservative systems (the damped harmonic oscillator for instance). Such systems exhibit asymptotic stability; the more negative the exponent, the greater the stability. Superstable fixed points and superstable periodic points have a Lyapunov exponent of λ = −∞. This is. generalized the concept of Lyapunov stability to irregular trajectories building upon earlier studies of Birkhoff [17], von Neumann [18], Krylov [19]3 and Asonov and Sinai [20]on ergodic theory. Lyapunov exponents quantify exponential sensitivity to initial conditions an In the theory of dynamical systems the stretches of continuum mechanics are called the finite-time Lyapunov or characteristic ex-ponents, (x 0;nˆ;t) = 1 t ln www wwJt ˆn ww ww= 1 2t ln n ˆ>Jt>Jtnˆ : (6.8) They depend on the initial point x 0 and on the direction of the unit vector ˆn, knˆ k= 1 at the initial time. If this vector is aligned along the ith principal stretch, nˆ = u(i.

Lyapunov Function -- from Wolfram MathWorl

  1. aries We will deal with a continuous time, autonomous, time-invariant nonlinear system x_(t) = f(x(t)) (1) For simplicity, suppose that x(t) 2R n, f: Rn!R , and fis continuous. Suppose also that (1) has an equilibrium point at the origin (this is again for simplicity, all the results hold in general), i.e. it holds f(0.
  2. Quantum Lyapunov Spectrum Hrant Gharibyana, Masanori Hanadab, Brian Swinglec and Masaki Tezukad aStanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA b Department of Physics, University of Colorado, Boulder, Colorado 80309, USA cCondensed Matter Theory Center, Maryland Center for Fundamental Physics, Joint Center for Quantum Information and Computer Science.
  3. Lyapunov exponents for some quasi-periodic cocycles - Volume 17 Issue 2 - L.-S. YOUNG Skip to main content Accessibility help We use cookies to distinguish you from other users and to provide you with a better experience on our websites
  4. Experiments show that compared with IEEE 802.15.4, FD-MAC, and MQEB-MAC, the Two-Tier MAC can improve the overall throughput by the Lyapunov theory, reduce power consumption by the time slot isolation scheme, and expend system lifetime by the energy balance. Even in the overloading condition, the QoS performance can be guaranteed thanks to the admission control
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stability of dynamical system via lyapunov theory ----- J'ai un système dynamique qui a la forme suivante : Mon objectif est de trouver les paramètres et via LMI (Inégalité matricielle linéaire) en utilisant la fonction de Lyapunov où est l'état et est une matrice définie positive. Le problème que je n'arrive pas a trouvé une solution faisable avec une seule matrice . J'ai essayé. This book offers a self-contained introduction to the theory of Lyapunov exponents and its applications, mainly in connection with hyperbolicity, ergodic theory and multifractal analysis. It discusses the foundations and some of the main results and main techniques in the area, while also highlighting selected topics of current research interest. With the exception of a few basic results from.

The theory of Lyapunov exponents is not easy. The Scientific American paper gives only a practical recipe for computing them in the case of the logistic formula. Take it as a magic formula: (1) Initialisation x = x0 For i=1 to INIT: x = rx(1-x) (2) Derivation total = 0 of the For i=1 to NLYAP: x = rx(1-x) exponent total = total + Log|r-2rx|/Log 2 exponent = total / NLYAP x0 is arbitrary. Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube

Amir Ali Ahmadi | The Program in Applied & Computational

Lyapunov Exponents Chaos and Time-Series Analysis 10/3/00 Lecture #5 in Physics 505 Comments on Homework #3 (Van der Pol Equation) Some people only took initial conditions inside the attractor; For b < 0 the attractor becomes a repellor (time reverses) The driven system can give limit cycles and toruses but not chaos (?) Can get chaos if you drive the dx/dt equation instead of dy/dt; Review. Moreover, it introduces various types of Lyapunov dimensions of dynamical systems with respect to an invariant set, based on local, global and uniform Lyapunov exponents, and derives analytical formulas for the Lyapunov dimension of the attractors of the Hénon and Lorenz systems. Lastly, the book presents estimates of the topological entropy for general dynamical systems in metric spaces and. Analyisis Lyapunov stability is named after AleksandrLyapunov, a Russian mathematician who published his book The General Problem of Stability of Motion in 1892 Two Methods of Lyapunov Stability First Method: considers the modifications necessary in nonlinear systems to the linear theory of stability based on linearizing near a point of.

Mathematics | Free Full-Text | Birkhoff Normal Forms, KAM

Lyapunov's result was correct and that therefore Darwin's theory was incorrect. In June 1917, Lyapunov left St. Petersburg for Odessa on doctor's orders, since his wife was suffering badly from tuberculosis. However, in Odessa her condition deteriorated, and on 31 October 1918 she died. Lyapunov himself was in poor health and his eyesight was failing. The tragic events of the revolution, the. Découvrez et achetez Lyapunov Methods and Certain Differential Games (Stability and Control : Theory, Methods and Applications). Livraison en Europe à 1 centime seulement The theory of Lyapunov exponents originated over a century ago in the study of the stability of solutions of differential equations. Written by one of the subject's leading authorities, this book is both an account of the classical theory, from a modern view, and an introduction to the significant developments relating the subject to dynamical systems, ergodic theory, mathematical physics and. Découvrez et achetez Lyapunov Exponents. Livraison en Europe à 1 centime seulement HAL Id: tel-00678325 https://tel.archives-ouvertes.fr/tel-00678325 Submitted on 12 Mar 2012 HAL is a multi-disciplinary open access archive for the deposit and.

Lyapunov Exponent - an overview ScienceDirect Topic

Chaos theory is the theory of Dynamic (Non-Linear) Systems. Dynamic systems converge to a state called an Attractor. A fractal is a shape that can be subdivided in parts, each of which is a copy of the whole. This property is called Self-Reference. Self-Referencial Dynamic Systems are represented by a Strange Attractor . I want to start with a few citations of HUMAN DIMENSIONS OF CHAOS THEORY. Lyapunov Theory for Zeno Stability Andrew Lamperski and Aaron D. Ames Abstract—Zeno behavior is a dynamic phenomenon unique to hybrid systems in which an infinite number of discrete transitions occurs in a finite amount of time. This behavior commonly arises in mechanical systems undergoing impacts and optimal control problems, but its characterization for general hybrid systems is not. A Tool to Explore Complex Dynamics, Lyapunov Exponents, Arkady Pikovsky, Antonio Politi, Cambridge University Press. Des milliers de livres avec la livraison chez vous en 1 jour ou en magasin avec -5% de réduction Lyapunov theory to finite times is nontrivial, but some progress has been made to introduce finite time Lyapunov exponents8,9 ~FTLEs!. In contrast to the asymptotic expo-nents, the finite time exponents depend on the initial posi-tions of the trajectories as well as the time of integration of these trajectories. In that sense, they are able to measure the stretching induced by the flow. Lyapunov exponents of arbitrary dynamical systems rarely can be obtained analytically (as a formula), but there are numerical methods that allow them to be calculated with reasonable accuracy. Lyapunov exponents are important in the qualitative theory of dynamical systems. Knowledge of Lyapunov exponents allows making a conclusion about the system behaivor over the time. Quite often it's.

lyapunov theory : définition de lyapunov theory et

  1. Lyapunov, Stabilité de MSC 34D20 (2000) MSC 37B25 (2000) MSC 93D05 (2000) Stabilité au sens de Liapounov Stabilité de Liapounov: L'année : 2000: Notices thématiques en relation (2 ressources dans data.bnf.fr) Termes plus larges (2) Commande, Théorie de la. Stabilité. Documents sur ce thème (20 ressources dans data.bnf.fr) Livres (20) Chaos detection and predictability (2016) Set.
  2. 1. Introduction. For a dynamical system, sensitivity to initial conditions is quantified by the Lyapunov exponents.These measure the rates of expansion or contraction of the principle axes of a phase space.In phase space every parameter of a system is represented as an axis and so a system's evolving state may be ploted as a line (trajectory) from the initial condition to its current state
  3. Lectures on Lyapunov Exponents, VIANA MARCELO, Cambridge University Press. Des milliers de livres avec la livraison chez vous en 1 jour ou en magasin avec -5% de réduction
  4. ant role in the investigation of liveness properties. The objective of this paper is to refocus the discussion on safety by introducing control barrier functions that play a role equivalent to Lyapunov functions in the study of liveness properties. There are two main reasons driving a surge in research related.

NPTEL :: Electrical Engineering - NOC:Nonlinear System

  1. imum-time path planning for drones such as geometric control [14, 19, 32], optimal control [21], flatness [7], stochastic theory [5]. ∗This research has been supported by the French FUI SHARE project (see [3]), supported by a consortium of companies and research labs (Opéra Ergonomie, ONERA, Thales Alénia Space, Eurocopter, Adetel group) and by the European Research.
  2. Abstract: This brief presents a Lyapunov theory-based weight adaptation scheme for a multilayered neural network (MLNN) mainly used to classify a multiple-input-multiple-output (MIMO) problem. Initially, the MLNN system is linearized using Taylor series expansion. Then, the weight adaptation scheme is designed based on the Lyapunov stability theory to iteratively update the weight
  3. Lyapunov contributed to several fields, including differential equations, potential theory, dynamical systems and probability theory.His main preoccupations were the stability of equilibria and the motion of mechanical systems, and the study of particles under the influence of gravity
  4. Les outils théoriques utilisés sont issus de la théorie de Lyapunov et de la séparation topologique. De manière à garantir les performances avec le moins de pessimisme possible, nous proposons de faire appel à des fonctions de Lyapunov dépendant des paramètres. Comme on attache une importance à la mise en oeuvre numérique, des méthodes issues du cadre de la stabilité quadratique.
  5. The paper surveys mathematical tools required for stability and convergence analysis of modern sliding mode control systems. Elements of Filippov theory of differential equations with discontinuous right-hand sides and its recent extensions are discussed. Stability notions (from Lyapunov stability (1982) to fixed-time stability (2012)) are observed
  6. ar - Matilde T. Farinha (Zoom meeting).

Lyapunov Stability Criterion - YouTub

This book is a systematic introduction to smooth ergodic theory. The topics discussed include the general (abstract) theory of Lyapunov exponents and its applications to the stability theory of differential equations, stable manifold theory, absolute continuity, and the ergodic theory of dynamical systems with nonzero Lyapunov exponents (including geodesic flows) Abstract: This paper develops nonsmooth Lyapunov stability theory and LaSalle's invariance principle for a class of nonsmooth Lipschitz continuous Lyapunov functions and absolutely continuous state trajectories. Computable tests based on Filipov's differential inclusion and Clarke's generalized gradient are derived. The primary use of these results is in analyzing the stability of equilibria. Lyapunov's direct method, which is based on the existence of a scalar function of the state that decreases monotonically along trajectories, still serves as the primary tool for establishing stability of nonlinear systems. Since the main challenge in stability analysis based on Lyapunov theory is always to nd a suitable Lyapunov function, weakening the requirements of the Lyapunov function is.

TEL - Thèses en ligne - Théorie de Lyapunov, commande

Lyapunov Theory Arash Rahnama Modzy arash.rahnama@modzy.com Andre T. Nguyen Booz Allen Hamilton nguyenandre@bah.com Edward Raff Booz Allen Hamilton raffedward@bah.com Abstract Deep neural networks (DNNs) are vulnerable to subtle ad-versarial perturbations applied to the input. These adversar-ial perturbations, though imperceptible, can easily mislead the DNN. In this work, we take a control. Mathematical theory of Lyapunov exponents To cite this article: Lai-Sang Young 2013 J. Phys. A: Math. Theor. 46 254001 View the article online for updates and enhancements. Related content Escape rates and conditionally invariant measures Mark F Demers and Lai-Sang Young-Behaviour of the escape rate function in hyperbolic dynamical systems Mark F Demers and Paul Wright-Structural stability and. Lyapunov Theory Non-Autonomous Systems Side 4 af 4 . Barbalat's Lemma (Non-autonomous Systems) Lemma 4.2 s. 123. g is a real function and uniformly continuous for . t ≥ 0. If the limit of the integral ∫ τ τ t 0 g ( )d. eksists for . t →∞ and is a real number, then . limg(t) 0. t = →∞. ---- Remark: If ∈ g L. 2 and &g( t) is limited ⇒ limg(t) 0. t = →∞. ---- Uniform. We then analyze and apply Lyapunov's Direct Method to prove these stability properties, and develop a nonlinear 3-axis attitude pointing control law using Lyapunov theory. Finally, we look at alternate feedback control laws and closed loop dynamics. After this course, you will be able to... * Differentiate between a range of nonlinear stability concepts * Apply Lyapunov's direct method to. general theory of differential equations, there exists a so-calledfundamental matrix At, t ∈Rsuch that v(t) Lyapunov asserts that, under an additional regularity condition, stability re-mains valid for nonlinear perturbations w˙(t)=B(t)·w(t)+F(t,w) with kF(t,w) k ≤constkwk1+ε. That is, the trivial solution w(t)≡0 is still exponentiallyasymptotically stable. The regularity.

Aleksandr Mikhailovich Lyapunov (1857 - 1918) - Biography

The general theory of stability, in addition to stability in the sense of Lyapunov, contains many other concepts and definitions of stable movement. In particular, the concepts of orbital and structural stability are important The theory of Lyapunov exponents and methods from ergodic the-ory have been employed by several authors in order to study persistence prop-erties of dynamical systems generated by ODEs or by maps. Here we derive su-cient conditions for uniform persistence, formulated in the language of Lyapunov exponents, for a large class of dissipative discrete-time dynamical systems on the positive.

Lyapunov stability - Encyclopedia of Mathematic

In particular, Lyapunov exponents have long played a central role in the theory of Anderson localization. These aspects are reviewed here, together with an original application to the transfer matrix. Covariant Lyapunov vectors: theory and applications. CLVs constitute an intrinsic tangent space decomposition into stable and unstable directions. In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories.Quantitatively, two trajectories in phase space with initial separation vector diverge (provided that the divergence can be treated within the linearized approximation) at a rate given b However, it seems to me that passivity theory is simply Lyapunov theory dressed up with fancier language. The core concept are not very different: you have a positive (semi)definite function, which you take the derivative of, and you show that it is negative (semi)definite. The analysis is virtually the same. The only difference is sometimes you have interconnection of passive systems, which. Oliveira [61] on ergodic theory and M. Viana [60] on Lyapunov expo-nents. They cover most of what one needs to know for the rst three chapters of this book. Familiarity with Markov chains is helpful in understanding the approach used in the fourth chapter, and for that, D. Levin and Y. Peres [42, Chapter 1] su ces. Finally, the last chap- ter requires a nontrivial amount of complex and.

Thus, Lyapunov functions allow to determine the stability or instability of a system. The advantage of this method is that we do not need to know the actual solution \(\mathbf{X}\left( t \right).\) In addition, this method allows to study the stability of equilibrium points of non-rough systems, for example, in the case when the equilibrium point is In applied mathematics and dynamical system theory, Lyapunov vectors, named after Aleksandr Lyapunov, describe characteristic expanding and contracting directions of a dynamical system.They have been used in predictability analysis and as initial perturbations for ensemble forecasting in numerical weather prediction. In modern practice they are often replaced by bred vectors for this purpose History. Lyapunov stability is named after Aleksandr Lyapunov, a Russian mathematician who published his book The General Problem of Stability of Motion in 1892. [1] Lyapunov was the first to consider the modifications necessary in nonlinear systems to the linear theory of stability based on linearizing near a point of equilibrium

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