. . "\u041F\u043E\u043A\u0430\u0437\u0430\u0442\u0435\u043B\u044C \u041B\u044F\u043F\u0443\u043D\u043E\u0432\u0430 \u0434\u0438\u043D\u0430\u043C\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u0441\u0438\u0441\u0442\u0435\u043C\u044B \u2014 \u0432\u0435\u043B\u0438\u0447\u0438\u043D\u0430, \u0445\u0430\u0440\u0430\u043A\u0442\u0435\u0440\u0438\u0437\u0443\u044E\u0449\u0430\u044F \u0441\u043A\u043E\u0440\u043E\u0441\u0442\u044C \u0443\u0434\u0430\u043B\u0435\u043D\u0438\u044F \u0434\u0440\u0443\u0433 \u043E\u0442 \u0434\u0440\u0443\u0433\u0430 \u0442\u0440\u0430\u0435\u043A\u0442\u043E\u0440\u0438\u0439. \u041F\u043E\u043B\u043E\u0436\u0438\u0442\u0435\u043B\u044C\u043D\u043E\u0441\u0442\u044C \u043F\u043E\u043A\u0430\u0437\u0430\u0442\u0435\u043B\u044F \u041B\u044F\u043F\u0443\u043D\u043E\u0432\u0430 \u043E\u0431\u044B\u0447\u043D\u043E \u0441\u0432\u0438\u0434\u0435\u0442\u0435\u043B\u044C\u0441\u0442\u0432\u0443\u0435\u0442 \u043E \u0445\u0430\u043E\u0442\u0438\u0447\u0435\u0441\u043A\u043E\u043C \u043F\u043E\u0432\u0435\u0434\u0435\u043D\u0438\u0438 \u0441\u0438\u0441\u0442\u0435\u043C\u044B. \u041D\u0430\u0437\u0432\u0430\u043D \u0432 \u0447\u0435\u0441\u0442\u044C \u0410\u043B\u0435\u043A\u0441\u0430\u043D\u0434\u0440\u0430 \u041C\u0438\u0445\u0430\u0439\u043B\u043E\u0432\u0438\u0447\u0430 \u041B\u044F\u043F\u0443\u043D\u043E\u0432\u0430."@ru . . . . "Dans l'analyse d'un syst\u00E8me dynamique, l'exposant de Liapounov permet de quantifier la stabilit\u00E9 ou l'instabilit\u00E9 de ses mouvements. Un exposant de Liapounov peut \u00EAtre soit un nombre r\u00E9el fini, soit+\u221E ou \u2013\u221E.Un mouvement instable a un exposant de Liapounov positif, un mouvement stable correspond \u00E0 un exposant de Liapounov n\u00E9gatif. Les mouvements born\u00E9s d'un syst\u00E8me lin\u00E9aire ont un exposant de Liapounov n\u00E9gatif ou nul.L'exposant de Liapounov peut servir \u00E0 \u00E9tudier la stabilit\u00E9 (ou l'instabilit\u00E9) des points d'\u00E9quilibre des syst\u00E8mes non lin\u00E9aires. Lorsqu'on lin\u00E9arise un tel syst\u00E8me au voisinage d'un point d'\u00E9quilibre, si le syst\u00E8me non lin\u00E9aire est non autonome, le syst\u00E8me lin\u00E9aire obtenu est \u00E0 coefficients variables ; chacun de ses mouvements a son propre exposant de Liapounov. \n* Si chacun d'eux est n\u00E9gatif et si le syst\u00E8me lin\u00E9aire est \u00AB r\u00E9gulier \u00BB (notion que nous d\u00E9taillerons plus loin), alors le point d'\u00E9quilibre est (localement) asymptotiquement stable pour le syst\u00E8me non lin\u00E9aire. \n* Si l'un de ces exposants de Liapounov est positif et si le syst\u00E8me lin\u00E9aire est r\u00E9gulier, alors le point d'\u00E9quilibre est instable pour le syst\u00E8me non lin\u00E9aire. Dans ce cas, le comportement du syst\u00E8me est extr\u00EAmement \u00AB sensible aux conditions initiales \u00BB, dans le sens o\u00F9 une incertitude sur celles-ci entra\u00EEne une incertitude sur le mouvement qui grandit de mani\u00E8re exponentielle au cours du temps. Ce ph\u00E9nom\u00E8ne est parfois assimil\u00E9, \u00E0 tort (du moins en g\u00E9n\u00E9ral), \u00E0 un comportement chaotique ; il en est n\u00E9anmoins une condition n\u00E9cessaire. L'inverse du plus grand exposant de Liapounov est un temps caract\u00E9ristique du syst\u00E8me, appel\u00E9 parfois horizon de Liapounov. Le caract\u00E8re pr\u00E9dictible de l'\u00E9volution du syst\u00E8me ne subsiste que pour les dur\u00E9es tr\u00E8s inf\u00E9rieures \u00E0 cet horizon, pendant lesquelles l'erreur sur le point courant de la trajectoire garde une taille comparable \u00E0 l'erreur sur les conditions initiales. En revanche, pour les temps sup\u00E9rieurs, toute pr\u00E9diction devient pratiquement impossible, m\u00EAme si le th\u00E9or\u00E8me de Cauchy-Lipschitz, qui suppose la connaissance parfaite des conditions initiales, reste valide."@fr . "Wyk\u0142adnik Lapunowa"@pl . . . . . . . "250"^^ . . . "Nella teoria dei sistemi dinamici, un esponente di Ljapunov di un sistema dinamico (deterministico) in un punto nello spazio delle fasi fornisce una misura di quanto sensibilmente le orbite del sistema sono dipendenti dai dati iniziali, caratterizzando la presenza di dinamiche caotiche. Gli esponenti di Ljapunov misurano in particolare la velocit\u00E0 media di allontanamento di due orbite infinitesimamente vicine per tempi sufficientemente lunghi. Ad un punto nello spazio delle fasi sono associati un numero di esponenti di Ljapunov pari alla dimensione dello spazio; se l'esponente di Ljapunov massimo \u00E8 , e se la distanza tra le orbite \u00E8 abbastanza piccola, allora il vettore ha un'evoluzione nel tempo (tasso di separazione delle due orbite) che per tempi grandi \u00E8 data approssimativamente da: Se \u00E8 positivo allora il sistema presenta una dipendenza sensibile dai dati iniziali (in modo esponenziale), ed \u00E8 quindi un sistema caotico. Il momento in cui un sistema diventa caotico \u00E8 dato dal reciproco di , ed \u00E8 detto tempo caratteristico o tempo di Ljapunov del sistema. Esso rappresenta il limite di predicibilit\u00E0 del sistema."@it . "vertical"@en . . . . . . "166441"^^ . . . . . "\u5728\u6570\u5B66\u9886\u57DF\u4E2D\uFF0C\u674E\u4E9A\u666E\u8BFA\u592B\u6307\u6570\uFF08Lyapunov exponent\uFF09\u6216\u674E\u4E9A\u666E\u8BFA\u592B\u7279\u5F81\u6307\u6570\uFF08Lyapunov characteristic exponent\uFF09\u7528\u4E8E\u91CF\u5316\u52A8\u529B\u7CFB\u7EDF\u4E2D\u65E0\u9650\u63A5\u8FD1\u7684\u8F68\u8FF9\u4E4B\u95F4\u7684\u5206\u79BB\u7387\u3002\u5177\u4F53\u800C\u8A00\uFF0C\u76F8\u7A7A\u95F4\u4E2D\u521D\u59CB\u95F4\u9694\u7684\u4E24\u6761\u8F68\u8FF9\u7684\u5206\u79BB\u7387\u4E3A\uFF08\u5047\u5B9A\u5206\u79BB\u53EF\u6309\u7EBF\u6027\u8FD1\u4F3C\u6765\u5904\u7406\uFF09 \u5176\u4E2D\u5373\u4E3A\u674E\u4E9A\u666E\u8BFA\u592B\u6307\u6570\u3002 \u5F53\u521D\u59CB\u5206\u79BB\u5411\u91CF\u7684\u65B9\u5411\u4E0D\u540C\u65F6\uFF0C\u5176\u5206\u79BB\u7387\u4E5F\u4E0D\u540C\u3002\u56E0\u800C\u5B58\u5728\u674E\u4E9A\u666E\u8BFA\u592B\u6307\u6570\u8C31\uFF08spectrum of Lyapunov exponents\uFF09\uFF0C\u5176\u6570\u91CF\u4E0E\u76F8\u7A7A\u95F4\u7684\u7EF4\u5EA6\u76F8\u540C\u3002\u901A\u5E38\u5C06\u5176\u4E2D\u6700\u5927\u7684\u79F0\u4E3A\u6700\u5927\u674E\u4E9A\u666E\u8BFA\u592B\u6307\u6570\uFF08Maximal Lyapunov exponent\uFF0C\u7B80\u79F0MLE\uFF09\uFF0C\u56E0\u4E3A\u5B83\u51B3\u5B9A\u4E86\u52A8\u529B\u7CFB\u7EDF\u7684\u53EF\u9884\u6D4B\u6027\u3002\u6B63\u7684MLE\u901A\u5E38\u8868\u660E\u7CFB\u7EDF\u662F\u6DF7\u6C8C\u7684\uFF08\u5047\u5B9A\u5176\u4ED6\u6761\u4EF6\u6EE1\u8DB3\uFF0C\u5982\u76F8\u7A7A\u95F4\u7684\u7D27\u81F4\u6027\uFF09\u3002\u9700\u8981\u6CE8\u610F\u7684\u662F\uFF0C\u4EFB\u610F\u521D\u59CB\u5206\u79BB\u5411\u91CF\u4E00\u822C\u5305\u62EC\u4E86MLE\u6240\u5728\u65B9\u5411\u7684\u90E8\u5206\u5206\u91CF\uFF0C\u7531\u4E8E\u5176\u968F\u6307\u6570\u589E\u957F\u7684\u7279\u5F81\uFF0C\u5176\u4ED6\u5206\u91CF\u7684\u6548\u679C\u968F\u7740\u65F6\u95F4\u6700\u7EC8\u4F1A\u88AB\u63A9\u76D6\u3002 \u674E\u4E9A\u666E\u8BFA\u592B\u6307\u6570\u662F\u4EE5\u4FC4\u7F57\u65AF\u6570\u5B66\u5BB6\u4E9A\u5386\u5C71\u5927\u00B7\u674E\u4E9A\u666E\u8BFA\u592B\u7684\u540D\u5B57\u547D\u540D\u7684\u3002"@zh . . . . . . . . . "Der Ljapunow-Exponent eines dynamischen Systems (nach Alexander Michailowitsch Ljapunow) beschreibt die Geschwindigkeit, mit der sich zwei (nahe beieinanderliegende) Punkte im Phasenraum voneinander entfernen oder ann\u00E4hern (je nach Vorzeichen). Pro Dimension des Phasenraums gibt es einen Ljapunow-Exponenten, die zusammen das sogenannte Ljapunow-Spektrum bilden. H\u00E4ufig betrachtet man allerdings nur den gr\u00F6\u00DFten Ljapunow-Exponenten, da dieser in der Regel das gesamte Systemverhalten bestimmt. Betrachtet man allgemeiner Trajektorieverl\u00E4ufe im Phasenraum, dann liefern die Exponenten ein Ma\u00DF f\u00FCr die Rate an Separation von einer Ursprungstrajektorie . In Bezug auf eine zeitkontinuierliche Betrachtung eines dynamischen Systems l\u00E4sst sich dieser Zusammenhang formal allgemein darstellen als: , wobei die Linearisierung der Trajektorie zum Zeitpunkt darstellt."@de . "Esponente di Ljapunov"@it . . "O expoente de Lyapunov de um sistema din\u00E2mico, ep\u00F4nimo de Aleksandr Lyapunov, descreve a velocidade de fase com a qual dois pontos pr\u00F3ximos no espa\u00E7o f\u00E1sico aproximam-se ou afastam-se. Para uma dimens\u00E3o do espa\u00E7o de fase existe um expoente de Lyapunov que forma o espectro de Lyapunov. Frequentemente interessa observar apenas o maior expoente de Lyapunov, pois este determina o comportamento geral do sistema. No espa\u00E7o unidimensional o expoente de Lyapunov \u00E9 uma transforma\u00E7\u00E3o iterada como definida a seguir: ."@pt . . . . . . . . . . . . "Wyk\u0142adnik Lapunowa, wsp\u00F3\u0142czynnik Lapunowa uk\u0142adu dynamicznego \u2013 miara, kt\u00F3ra charakteryzuje tempo separacji infinitezymalnie (niesko\u0144czenie) bliskich trajektorii. Pozwala on te\u017C ustali\u0107 zachowanie si\u0119 uk\u0142adu dynamicznego dla okre\u015Blonych zmiennych (parametr\u00F3w). Og\u00F3lnie s\u0142u\u017Cy do badania uk\u0142ad\u00F3w dynamicznych. Podstawy matematycznej teorii stabilno\u015Bci ruchu stworzy\u0142 A.M.Lapunow, kt\u00F3ry rozpatrywa\u0142, jak szybko wzrasta w czasie ewolucji odleg\u0142o\u015B\u0107 pomi\u0119dzy dwiema bliskimi trajektoriami. Je\u017Celi uk\u0142ad dynamiczny jest chaotyczny, odleg\u0142o\u015B\u0107 taka ro\u015Bnie w czasie t wyk\u0142adniczo jak gdzie wsp\u00F3\u0142czynnik zwany wyk\u0142adnikiem Lapunowa jest dodatni. Wyk\u0142adniki Lapunowa umo\u017Cliwiaj\u0105 ocen\u0119 zjawiska chaotycznego w tzw. przestrzeni fazowej. Przestrze\u0144 fazowa to inny spos\u00F3b obrazowania wielowymiarowych zjawisk dynam"@pl . . . "El Exponente Lyapunov o Exponente caracter\u00EDstico Lyapunov de un sistema din\u00E1mico es una cantidad que caracteriza el grado de separaci\u00F3n de dos trayectorias infinitesimalmente cercanas. Cuantitativamente, dos trayectorias en el espacio-fase con separaci\u00F3n inicial diverge El radio de separaci\u00F3n puede ser distinto para diferentes orientaciones del vector de separaci\u00F3n inicial. Aunque, hay un completo espectro del exponente Lyapunov; el n\u00FAmero de ellos es igual al n\u00FAmero de dimensiones del espacio-fase. Es com\u00FAn referirse s\u00F3lo a la m\u00E1s grande, porque determina la predictibilidad de un sistema."@es . "25947"^^ . . . . "Nella teoria dei sistemi dinamici, un esponente di Ljapunov di un sistema dinamico (deterministico) in un punto nello spazio delle fasi fornisce una misura di quanto sensibilmente le orbite del sistema sono dipendenti dai dati iniziali, caratterizzando la presenza di dinamiche caotiche. Gli esponenti di Ljapunov misurano in particolare la velocit\u00E0 media di allontanamento di due orbite infinitesimamente vicine per tempi sufficientemente lunghi."@it . . . . . . "\u041F\u043E\u043A\u0430\u0437\u0430\u0442\u0435\u043B\u044C \u041B\u044F\u043F\u0443\u043D\u043E\u0432\u0430 \u0434\u0438\u043D\u0430\u043C\u0438\u0447\u0435\u0441\u043A\u043E\u0439 \u0441\u0438\u0441\u0442\u0435\u043C\u044B \u2014 \u0432\u0435\u043B\u0438\u0447\u0438\u043D\u0430, \u0445\u0430\u0440\u0430\u043A\u0442\u0435\u0440\u0438\u0437\u0443\u044E\u0449\u0430\u044F \u0441\u043A\u043E\u0440\u043E\u0441\u0442\u044C \u0443\u0434\u0430\u043B\u0435\u043D\u0438\u044F \u0434\u0440\u0443\u0433 \u043E\u0442 \u0434\u0440\u0443\u0433\u0430 \u0442\u0440\u0430\u0435\u043A\u0442\u043E\u0440\u0438\u0439. \u041F\u043E\u043B\u043E\u0436\u0438\u0442\u0435\u043B\u044C\u043D\u043E\u0441\u0442\u044C \u043F\u043E\u043A\u0430\u0437\u0430\u0442\u0435\u043B\u044F \u041B\u044F\u043F\u0443\u043D\u043E\u0432\u0430 \u043E\u0431\u044B\u0447\u043D\u043E \u0441\u0432\u0438\u0434\u0435\u0442\u0435\u043B\u044C\u0441\u0442\u0432\u0443\u0435\u0442 \u043E \u0445\u0430\u043E\u0442\u0438\u0447\u0435\u0441\u043A\u043E\u043C \u043F\u043E\u0432\u0435\u0434\u0435\u043D\u0438\u0438 \u0441\u0438\u0441\u0442\u0435\u043C\u044B. \u041D\u0430\u0437\u0432\u0430\u043D \u0432 \u0447\u0435\u0441\u0442\u044C \u0410\u043B\u0435\u043A\u0441\u0430\u043D\u0434\u0440\u0430 \u041C\u0438\u0445\u0430\u0439\u043B\u043E\u0432\u0438\u0447\u0430 \u041B\u044F\u043F\u0443\u043D\u043E\u0432\u0430."@ru . . "Explanations of the Lyapunov exponent"@en . . . . "Exposant de Liapounov"@fr . . "\u041F\u043E\u043A\u0430\u0437\u0430\u0442\u0435\u043B\u044C \u041B\u044F\u043F\u0443\u043D\u043E\u0432\u0430"@ru . . . "\u674E\u4E9A\u666E\u8BFA\u592B\u6307\u6570"@zh . "1117726193"^^ . . . . . "Expoente de Lyapunov"@pt . . "\u30EA\u30A2\u30D7\u30CE\u30D5\u6307\u6570"@ja . "\u30EA\u30A2\u30D7\u30CE\u30D5\u6307\u6570\uFF08\u30EA\u30A2\u30D7\u30CE\u30D5\u3057\u3059\u3046\u3001\u82F1: Lyapunov exponent\uFF09\u3068\u306F\u3001\u529B\u5B66\u7CFB\u306B\u304A\u3044\u3066\u3054\u304F\u63A5\u8FD1\u3057\u305F\u8ECC\u9053\u304C\u96E2\u308C\u3066\u3044\u304F\u5EA6\u5408\u3044\u3092\u8868\u3059\u91CF\u3067\u3042\u308B\u3002\u30EA\u30E3\u30D7\u30CE\u30D5\u6307\u6570\u3068\u3082\u8868\u8A18\u3055\u308C\u308B\u3002\u30ED\u30B7\u30A2\u4EBA\u79D1\u5B66\u8005 \u0410\u043B\u0435\u043A\u0441\u0430\u0301\u043D\u0434\u0440 \u041B\u044F\u043F\u0443\u043D\u043E\u0301\u0432\uFF08\u30A2\u30EC\u30AF\u30B5\u30F3\u30C9\u30EB\u30FB\u30EA\u30D7\u30CE\u30FC\u30D5\u3001Aleksandr Lyapunov\uFF09\u306B\u305D\u306E\u540D\u3092\u3061\u306A\u3080\u3002 \u7CFB\u306E\u76F8\u7A7A\u9593\u4E0A\u306E2\u3064\u306E\u8ECC\u9053\u306B\u3064\u3044\u3066\u8003\u3048\u308B\u30022\u3064\u306E\u8ECC\u9053\u4E0A\u306E\u6642\u523B t \u306B\u304A\u3051\u308B\u70B9\u306E\u8DDD\u96E2\u3092\u30D9\u30AF\u30C8\u30EB \u03B4(t) \u3068\u3057\u3066\u3001\u521D\u671F\u72B6\u614B t = 0 \u306B\u306F\u3001\u3053\u308C\u3089\u306E\u8ECC\u9053\u306F\u8DDD\u96E2 \u03B4(0) \u3060\u3051\u96E2\u308C\u3066\u3044\u308B\u3068\u3059\u308B\u3002\u03B4(t) \u3092\u8FD1\u4F3C\u7684\u306B\u6B21\u306E\u3088\u3046\u306B\u8868\u3059\u3002 \u3053\u3053\u3067 \u306F\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u30CE\u30EB\u30E0\u3092\u610F\u5473\u3059\u308B\u3002\u4E0A\u5F0F\u3067 \u03BB > 0 \u306E\u5834\u5408\u306F\u8ECC\u9053\u306F\u96E2\u308C\u3066\u3044\u304D\u3001 \u03BB < 0 \u306E\u5834\u5408\u306F\u8ECC\u9053\u306F\u8FD1\u3065\u3044\u3066\u3044\u304F\u3002\u3088\u3063\u3066\u3001\u8ECC\u9053\u304C\u96E2\u308C\u3066\u3044\u304F\u5EA6\u5408\u3044\u306F \u03BB \u306E\u5024\u306B\u3088\u308A\u6C7A\u5B9A\u3055\u308C\u308B\u3002\u3053\u306E \u03BB \u304C\u30EA\u30A2\u30D7\u30CE\u30D5\u6307\u6570\u3067\u3042\u308B\u3002\u8ECC\u9053\u304C\u30AB\u30AA\u30B9\u7684\u3067\u3042\u308B\u3068\u304D\u3001\u4E0A\u5F0F\u306E\u3088\u3046\u306B\u8ECC\u9053\u306F\u6307\u6570\u95A2\u6570\u7684\u306B\u96E2\u308C\u3066\u3044\u304F\u3002\u3059\u306A\u308F\u3061\u3001\u30EA\u30A2\u30D7\u30CE\u30D5\u6307\u6570\u304C\u6B63\u3067\u3042\u308B\u3053\u3068\u304C\u8ECC\u9053\u304C\u30AB\u30AA\u30B9\u7684\u3067\u3042\u308B\u3053\u3068\u306E1\u3064\u306E\u5B9A\u7FA9\u3068\u3055\u308C\u308B\u3002"@ja . . . . "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 by where is the Lyapunov exponent. The rate of separation can be different for different orientations of initial separation vector. Thus, there is a spectrum of Lyapunov exponents\u2014equal in number to the dimensionality of the phase space. It is common to refer to the largest one as the maximal Lyapunov exponent (MLE), because it determines a notion of predictability for a dynamical system. A positive MLE is usually taken as an indication that the system is chaotic (provided some other conditions are met, e.g., phase space compactness). Note that an arbitrary initial separation vector will typically contain some component in the direction associated with the MLE, and because of the exponential growth rate, the effect of the other exponents will be obliterated over time. The exponent is named after Aleksandr Lyapunov."@en . . . . "Wyk\u0142adnik Lapunowa, wsp\u00F3\u0142czynnik Lapunowa uk\u0142adu dynamicznego \u2013 miara, kt\u00F3ra charakteryzuje tempo separacji infinitezymalnie (niesko\u0144czenie) bliskich trajektorii. Pozwala on te\u017C ustali\u0107 zachowanie si\u0119 uk\u0142adu dynamicznego dla okre\u015Blonych zmiennych (parametr\u00F3w). Og\u00F3lnie s\u0142u\u017Cy do badania uk\u0142ad\u00F3w dynamicznych. Podstawy matematycznej teorii stabilno\u015Bci ruchu stworzy\u0142 A.M.Lapunow, kt\u00F3ry rozpatrywa\u0142, jak szybko wzrasta w czasie ewolucji odleg\u0142o\u015B\u0107 pomi\u0119dzy dwiema bliskimi trajektoriami. Je\u017Celi uk\u0142ad dynamiczny jest chaotyczny, odleg\u0142o\u015B\u0107 taka ro\u015Bnie w czasie t wyk\u0142adniczo jak gdzie wsp\u00F3\u0142czynnik zwany wyk\u0142adnikiem Lapunowa jest dodatni. Wyk\u0142adniki Lapunowa umo\u017Cliwiaj\u0105 ocen\u0119 zjawiska chaotycznego w tzw. przestrzeni fazowej. Przestrze\u0144 fazowa to inny spos\u00F3b obrazowania wielowymiarowych zjawisk dynamicznych. W zwyk\u0142ym przebiegu czasowym o\u015B pozioma wykresu obrazuje czas, natomiast o\u015B pionowa odpowiada za stan zjawiska w danej chwili (np. pr\u0119dko\u015B\u0107). W przestrzeni fazowej mo\u017Cemy oceni\u0107 wszystkie mo\u017Cliwe stany systemu w ka\u017Cdej chwili czasowej. Trajektoria to obraz wszystkich mo\u017Cliwych stan\u00F3w, kt\u00F3re przyjmuje uk\u0142ad w kolejnych chwilach czasu. Je\u015Bli przyj\u0105\u0107 sko\u0144czone i odpowiednio ma\u0142e odcinki czasu, to ewolucj\u0119 uk\u0142adu dynamicznego mo\u017Cna opisa\u0107 rekurencyjnym r\u00F3wnaniem algebraicznym: gdzie: \u2013 przyjmuje kolejne warto\u015Bci ca\u0142kowite, kt\u00F3re mo\u017Cna uzna\u0107 za kolejne interwa\u0142y czasowe, \u2013 zmienna opisuj\u0105ca stan uk\u0142adu dynamicznego w chwili \u2013 to stan uk\u0142adu dynamicznego w chwili Stan uk\u0142adu w chwili otrzymuje si\u0119 przez przekszta\u0142cenie stanu za pomoc\u0105 odpowiedniej funkcji Otrzymuje si\u0119 w\u00F3wczas ci\u0105g o postaci: Taki spos\u00F3b przekszta\u0142cania nazywa si\u0119 iteracj\u0105. Ci\u0105g kolejnych iteracji tworzy orbit\u0119 odwzorowania. Je\u017Celi rozpatrzy\u0107 dwa stany pocz\u0105tkowe r\u00F3\u017Cni\u0105ce si\u0119 w niewielkim stopniu o e, to po up\u0142ywie czasu (lub inaczej po iteracjach) otrzymuje si\u0119 warto\u015Bci: gdzie oznacza n-t\u0105 iteracj\u0119 na punkcie Stany mog\u0105 by\u0107 od siebie oddalone w r\u00F3\u017Cnym stopniu. Miar\u0105 oddalania/zbli\u017Cania si\u0119 jest wyk\u0142adnik Lapunowa definiowany w postaci: Wsp\u00F3\u0142czynnik ten jest r\u00F3wnie\u017C miar\u0105 utraty informacji o uk\u0142adzie w jednym przekszta\u0142ceniu. Mog\u0105 zaistnie\u0107 trzy mo\u017Cliwo\u015Bci: \n* \u2013 orbita zmierza do stabilnego punktu lub staje si\u0119 orbit\u0105 periodyczn\u0105. Ujemny wyk\u0142adnik Lapunowa charakteryzuje uk\u0142ady dyssypatywne np. t\u0142umione wahad\u0142o. \n* \u2013 orbita zmierza do neutralnego, sta\u0142ego punktu. Warto\u015B\u0107 ta oznacza, \u017Ce system znajduje si\u0119 w najbardziej stabilnym stadium rozwoju. \n* \u2013 orbita jest niestabilna i chaotyczna. Dwa bliskie stany pocz\u0105tkowe oddalaj\u0105 si\u0119 wyk\u0142adniczo od siebie z up\u0142ywem czasu. Obliczanie wyk\u0142adnik\u00F3w Lapunowa dla uk\u0142ad\u00F3w wielowymiarowych jest bardzo z\u0142o\u017Cone. Dodatnia warto\u015B\u0107 najwi\u0119kszego wyk\u0142adnika oznacza, \u017Ce uk\u0142ad jest chaotyczny. Wynika to z natury zjawiska, to znaczy im wi\u0119ksza jest warto\u015B\u0107 wyk\u0142adnika tym szybciej rozbiega si\u0119 analizowane zjawisko. Odleg\u0142o\u015B\u0107 mi\u0119dzy pocz\u0105tkowo s\u0105siednimi dwoma punktami zwi\u0119ksza si\u0119 w\u0142a\u015Bnie w spos\u00F3b wyk\u0142adniczy. Wyk\u0142adniki Lapunowa opisuj\u0105 jak szybko poszczeg\u00F3lne punkty oddalaj\u0105 si\u0119 od siebie (lub zbli\u017Caj\u0105 je\u015Bli wyk\u0142adnik jest ujemny). Dla ka\u017Cdego wymiaru wyst\u0119puje osobny wyk\u0142adnik. Zatem mo\u017Ce zdarzy\u0107 si\u0119, \u017Ce oddalanie nast\u0119puje wy\u0142\u0105cznie wzd\u0142u\u017C niekt\u00F3rych wymiar\u00F3w. Poniewa\u017C oddalanie nast\u0119puje znacznie szybciej ni\u017C zbli\u017Canie si\u0119, jeden dodatni wyk\u0142adnik Lapunowa odpowiedzialny za oddalanie si\u0119 powoduje zwi\u0119kszanie si\u0119 globalnej odleg\u0142o\u015Bci, nawet je\u015Bli we wszystkich pozosta\u0142ych wymiarach zjawisko charakteryzuje si\u0119 ujemnymi wyk\u0142adnikami."@pl . "Lyapunov exponent"@en . "Lyapunov-exponent.svg"@en . "Orbital instability .png"@en . "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 by where is the Lyapunov exponent. The exponent is named after Aleksandr Lyapunov."@en . . "\u5728\u6570\u5B66\u9886\u57DF\u4E2D\uFF0C\u674E\u4E9A\u666E\u8BFA\u592B\u6307\u6570\uFF08Lyapunov exponent\uFF09\u6216\u674E\u4E9A\u666E\u8BFA\u592B\u7279\u5F81\u6307\u6570\uFF08Lyapunov characteristic exponent\uFF09\u7528\u4E8E\u91CF\u5316\u52A8\u529B\u7CFB\u7EDF\u4E2D\u65E0\u9650\u63A5\u8FD1\u7684\u8F68\u8FF9\u4E4B\u95F4\u7684\u5206\u79BB\u7387\u3002\u5177\u4F53\u800C\u8A00\uFF0C\u76F8\u7A7A\u95F4\u4E2D\u521D\u59CB\u95F4\u9694\u7684\u4E24\u6761\u8F68\u8FF9\u7684\u5206\u79BB\u7387\u4E3A\uFF08\u5047\u5B9A\u5206\u79BB\u53EF\u6309\u7EBF\u6027\u8FD1\u4F3C\u6765\u5904\u7406\uFF09 \u5176\u4E2D\u5373\u4E3A\u674E\u4E9A\u666E\u8BFA\u592B\u6307\u6570\u3002 \u5F53\u521D\u59CB\u5206\u79BB\u5411\u91CF\u7684\u65B9\u5411\u4E0D\u540C\u65F6\uFF0C\u5176\u5206\u79BB\u7387\u4E5F\u4E0D\u540C\u3002\u56E0\u800C\u5B58\u5728\u674E\u4E9A\u666E\u8BFA\u592B\u6307\u6570\u8C31\uFF08spectrum of Lyapunov exponents\uFF09\uFF0C\u5176\u6570\u91CF\u4E0E\u76F8\u7A7A\u95F4\u7684\u7EF4\u5EA6\u76F8\u540C\u3002\u901A\u5E38\u5C06\u5176\u4E2D\u6700\u5927\u7684\u79F0\u4E3A\u6700\u5927\u674E\u4E9A\u666E\u8BFA\u592B\u6307\u6570\uFF08Maximal Lyapunov exponent\uFF0C\u7B80\u79F0MLE\uFF09\uFF0C\u56E0\u4E3A\u5B83\u51B3\u5B9A\u4E86\u52A8\u529B\u7CFB\u7EDF\u7684\u53EF\u9884\u6D4B\u6027\u3002\u6B63\u7684MLE\u901A\u5E38\u8868\u660E\u7CFB\u7EDF\u662F\u6DF7\u6C8C\u7684\uFF08\u5047\u5B9A\u5176\u4ED6\u6761\u4EF6\u6EE1\u8DB3\uFF0C\u5982\u76F8\u7A7A\u95F4\u7684\u7D27\u81F4\u6027\uFF09\u3002\u9700\u8981\u6CE8\u610F\u7684\u662F\uFF0C\u4EFB\u610F\u521D\u59CB\u5206\u79BB\u5411\u91CF\u4E00\u822C\u5305\u62EC\u4E86MLE\u6240\u5728\u65B9\u5411\u7684\u90E8\u5206\u5206\u91CF\uFF0C\u7531\u4E8E\u5176\u968F\u6307\u6570\u589E\u957F\u7684\u7279\u5F81\uFF0C\u5176\u4ED6\u5206\u91CF\u7684\u6548\u679C\u968F\u7740\u65F6\u95F4\u6700\u7EC8\u4F1A\u88AB\u63A9\u76D6\u3002 \u674E\u4E9A\u666E\u8BFA\u592B\u6307\u6570\u662F\u4EE5\u4FC4\u7F57\u65AF\u6570\u5B66\u5BB6\u4E9A\u5386\u5C71\u5927\u00B7\u674E\u4E9A\u666E\u8BFA\u592B\u7684\u540D\u5B57\u547D\u540D\u7684\u3002"@zh . . "Ljapunow-Exponent"@de . . "Dans l'analyse d'un syst\u00E8me dynamique, l'exposant de Liapounov permet de quantifier la stabilit\u00E9 ou l'instabilit\u00E9 de ses mouvements. Un exposant de Liapounov peut \u00EAtre soit un nombre r\u00E9el fini, soit+\u221E ou \u2013\u221E.Un mouvement instable a un exposant de Liapounov positif, un mouvement stable correspond \u00E0 un exposant de Liapounov n\u00E9gatif. Les mouvements born\u00E9s d'un syst\u00E8me lin\u00E9aire ont un exposant de Liapounov n\u00E9gatif ou nul.L'exposant de Liapounov peut servir \u00E0 \u00E9tudier la stabilit\u00E9 (ou l'instabilit\u00E9) des points d'\u00E9quilibre des syst\u00E8mes non lin\u00E9aires."@fr . . . . "Exponente de Lyapunov"@es . "\u30EA\u30A2\u30D7\u30CE\u30D5\u6307\u6570\uFF08\u30EA\u30A2\u30D7\u30CE\u30D5\u3057\u3059\u3046\u3001\u82F1: Lyapunov exponent\uFF09\u3068\u306F\u3001\u529B\u5B66\u7CFB\u306B\u304A\u3044\u3066\u3054\u304F\u63A5\u8FD1\u3057\u305F\u8ECC\u9053\u304C\u96E2\u308C\u3066\u3044\u304F\u5EA6\u5408\u3044\u3092\u8868\u3059\u91CF\u3067\u3042\u308B\u3002\u30EA\u30E3\u30D7\u30CE\u30D5\u6307\u6570\u3068\u3082\u8868\u8A18\u3055\u308C\u308B\u3002\u30ED\u30B7\u30A2\u4EBA\u79D1\u5B66\u8005 \u0410\u043B\u0435\u043A\u0441\u0430\u0301\u043D\u0434\u0440 \u041B\u044F\u043F\u0443\u043D\u043E\u0301\u0432\uFF08\u30A2\u30EC\u30AF\u30B5\u30F3\u30C9\u30EB\u30FB\u30EA\u30D7\u30CE\u30FC\u30D5\u3001Aleksandr Lyapunov\uFF09\u306B\u305D\u306E\u540D\u3092\u3061\u306A\u3080\u3002 \u7CFB\u306E\u76F8\u7A7A\u9593\u4E0A\u306E2\u3064\u306E\u8ECC\u9053\u306B\u3064\u3044\u3066\u8003\u3048\u308B\u30022\u3064\u306E\u8ECC\u9053\u4E0A\u306E\u6642\u523B t \u306B\u304A\u3051\u308B\u70B9\u306E\u8DDD\u96E2\u3092\u30D9\u30AF\u30C8\u30EB \u03B4(t) \u3068\u3057\u3066\u3001\u521D\u671F\u72B6\u614B t = 0 \u306B\u306F\u3001\u3053\u308C\u3089\u306E\u8ECC\u9053\u306F\u8DDD\u96E2 \u03B4(0) \u3060\u3051\u96E2\u308C\u3066\u3044\u308B\u3068\u3059\u308B\u3002\u03B4(t) \u3092\u8FD1\u4F3C\u7684\u306B\u6B21\u306E\u3088\u3046\u306B\u8868\u3059\u3002 \u3053\u3053\u3067 \u306F\u30E6\u30FC\u30AF\u30EA\u30C3\u30C9\u30CE\u30EB\u30E0\u3092\u610F\u5473\u3059\u308B\u3002\u4E0A\u5F0F\u3067 \u03BB > 0 \u306E\u5834\u5408\u306F\u8ECC\u9053\u306F\u96E2\u308C\u3066\u3044\u304D\u3001 \u03BB < 0 \u306E\u5834\u5408\u306F\u8ECC\u9053\u306F\u8FD1\u3065\u3044\u3066\u3044\u304F\u3002\u3088\u3063\u3066\u3001\u8ECC\u9053\u304C\u96E2\u308C\u3066\u3044\u304F\u5EA6\u5408\u3044\u306F \u03BB \u306E\u5024\u306B\u3088\u308A\u6C7A\u5B9A\u3055\u308C\u308B\u3002\u3053\u306E \u03BB \u304C\u30EA\u30A2\u30D7\u30CE\u30D5\u6307\u6570\u3067\u3042\u308B\u3002\u8ECC\u9053\u304C\u30AB\u30AA\u30B9\u7684\u3067\u3042\u308B\u3068\u304D\u3001\u4E0A\u5F0F\u306E\u3088\u3046\u306B\u8ECC\u9053\u306F\u6307\u6570\u95A2\u6570\u7684\u306B\u96E2\u308C\u3066\u3044\u304F\u3002\u3059\u306A\u308F\u3061\u3001\u30EA\u30A2\u30D7\u30CE\u30D5\u6307\u6570\u304C\u6B63\u3067\u3042\u308B\u3053\u3068\u304C\u8ECC\u9053\u304C\u30AB\u30AA\u30B9\u7684\u3067\u3042\u308B\u3053\u3068\u306E1\u3064\u306E\u5B9A\u7FA9\u3068\u3055\u308C\u308B\u3002 \u3088\u308A\u8A73\u7D30\u306B\u306F\u3001\u7CFB\u306E\u72B6\u614B\u5909\u6570\u304C k \u500B\uFF08k > 1\uFF09\u306E\u5834\u5408\u3001\u3059\u306A\u308F\u3061\u76F8\u7A7A\u9593\u304C k \u6B21\u5143\u3067\u3042\u308B\u5834\u5408\u306F\u5404\u6B21\u5143\u3054\u3068\u306B\u56FA\u6709\u306E\u30EA\u30A2\u30D7\u30CE\u30D5\u6307\u6570\u3092\u6301\u3064\u3002\u3053\u308C\u3089\u306E\u30EA\u30A2\u30D7\u30CE\u30D5\u6307\u6570\u306E\u7D44\u3092\u30EA\u30A2\u30D7\u30CE\u30D5\u30B9\u30DA\u30AF\u30C8\u30E9\u30E0\u3068\u547C\u3073\u3001\u305D\u306E\u3046\u3061\u306E\u6700\u5927\u306E\u30EA\u30A2\u30D7\u30CE\u30D5\u6307\u6570\u3092\u6700\u5927\u30EA\u30A2\u30D7\u30CE\u30D5\u6307\u6570\u3068\u547C\u3076\u3002\u5404\u3005\u306E\u30EA\u30A2\u30D7\u30CE\u30D5\u6307\u6570\u3092\u898B\u308C\u3070\u6B63\u3067\u3042\u3063\u305F\u308A\u8CA0\u3067\u3042\u3063\u305F\u308A\u3059\u308B\u304C\u3001\u6700\u5927\u30EA\u30A2\u30D7\u30CE\u30D5\u6307\u6570\u304C\u6B63\u3067\u3042\u308C\u3070\u3001\u305D\u306E\u7CFB\u306F\u30AB\u30AA\u30B9\u306E\u7279\u5FB4\u306E1\u3064\u3067\u3042\u308B\u521D\u671F\u5024\u92ED\u654F\u6027\u3092\u6301\u3064\u3068\u3044\u3048\u308B\u3002"@ja . "Der Ljapunow-Exponent eines dynamischen Systems (nach Alexander Michailowitsch Ljapunow) beschreibt die Geschwindigkeit, mit der sich zwei (nahe beieinanderliegende) Punkte im Phasenraum voneinander entfernen oder ann\u00E4hern (je nach Vorzeichen). Pro Dimension des Phasenraums gibt es einen Ljapunow-Exponenten, die zusammen das sogenannte Ljapunow-Spektrum bilden. H\u00E4ufig betrachtet man allerdings nur den gr\u00F6\u00DFten Ljapunow-Exponenten, da dieser in der Regel das gesamte Systemverhalten bestimmt."@de . "O expoente de Lyapunov de um sistema din\u00E2mico, ep\u00F4nimo de Aleksandr Lyapunov, descreve a velocidade de fase com a qual dois pontos pr\u00F3ximos no espa\u00E7o f\u00E1sico aproximam-se ou afastam-se. Para uma dimens\u00E3o do espa\u00E7o de fase existe um expoente de Lyapunov que forma o espectro de Lyapunov. Frequentemente interessa observar apenas o maior expoente de Lyapunov, pois este determina o comportamento geral do sistema. No espa\u00E7o unidimensional o expoente de Lyapunov \u00E9 uma transforma\u00E7\u00E3o iterada como definida a seguir: ."@pt . . . . . . . . . . "El Exponente Lyapunov o Exponente caracter\u00EDstico Lyapunov de un sistema din\u00E1mico es una cantidad que caracteriza el grado de separaci\u00F3n de dos trayectorias infinitesimalmente cercanas. Cuantitativamente, dos trayectorias en el espacio-fase con separaci\u00F3n inicial diverge El radio de separaci\u00F3n puede ser distinto para diferentes orientaciones del vector de separaci\u00F3n inicial. Aunque, hay un completo espectro del exponente Lyapunov; el n\u00FAmero de ellos es igual al n\u00FAmero de dimensiones del espacio-fase. Es com\u00FAn referirse s\u00F3lo a la m\u00E1s grande, porque determina la predictibilidad de un sistema."@es . . .