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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Definition_of_the_maximal_Lyapunov_exponent" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition_of_the_maximal_Lyapunov_exponent"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definition of the maximal Lyapunov exponent</span> </div> </a> <ul id="toc-Definition_of_the_maximal_Lyapunov_exponent-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition_of_the_Lyapunov_spectrum" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition_of_the_Lyapunov_spectrum"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definition of the Lyapunov spectrum</span> </div> </a> <ul id="toc-Definition_of_the_Lyapunov_spectrum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lyapunov_exponent_for_time-varying_linearization" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lyapunov_exponent_for_time-varying_linearization"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Lyapunov exponent for time-varying linearization</span> </div> </a> <button aria-controls="toc-Lyapunov_exponent_for_time-varying_linearization-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Lyapunov exponent for time-varying linearization subsection</span> </button> <ul id="toc-Lyapunov_exponent_for_time-varying_linearization-sublist" class="vector-toc-list"> <li id="toc-Perron_effects_of_largest_Lyapunov_exponent_sign_inversion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Perron_effects_of_largest_Lyapunov_exponent_sign_inversion"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Perron effects of largest Lyapunov exponent sign inversion</span> </div> </a> <ul id="toc-Perron_effects_of_largest_Lyapunov_exponent_sign_inversion-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Basic_properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Basic_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Basic properties</span> </div> </a> <ul id="toc-Basic_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Significance_of_the_Lyapunov_spectrum" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Significance_of_the_Lyapunov_spectrum"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Significance of the Lyapunov spectrum</span> </div> </a> <ul id="toc-Significance_of_the_Lyapunov_spectrum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numerical_calculation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Numerical_calculation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Numerical calculation</span> </div> </a> <ul id="toc-Numerical_calculation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Local_Lyapunov_exponent" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Local_Lyapunov_exponent"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Local Lyapunov exponent</span> </div> </a> <ul id="toc-Local_Lyapunov_exponent-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conditional_Lyapunov_exponent" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conditional_Lyapunov_exponent"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Conditional Lyapunov exponent</span> </div> </a> <ul id="toc-Conditional_Lyapunov_exponent-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Further reading</span> </div> </a> <button aria-controls="toc-Further_reading-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Further reading subsection</span> </button> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> <li id="toc-Software" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Software"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>Software</span> </div> </a> <ul id="toc-Software-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" 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class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Exponent_de_Liapunov" title="Exponent de Liapunov – Catalan" lang="ca" hreflang="ca" data-title="Exponent de Liapunov" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ljapunow-Exponent" title="Ljapunow-Exponent – German" lang="de" hreflang="de" data-title="Ljapunow-Exponent" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Exponente_de_Lyapunov" title="Exponente de Lyapunov – Spanish" lang="es" hreflang="es" data-title="Exponente de Lyapunov" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Exposant_de_Liapounov" title="Exposant de Liapounov – French" lang="fr" hreflang="fr" data-title="Exposant de Liapounov" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Esponente_di_Ljapunov" title="Esponente di Ljapunov – Italian" lang="it" hreflang="it" data-title="Esponente di Ljapunov" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%A7%D7%A1%D7%A4%D7%95%D7%A0%D7%A0%D7%98_%D7%9C%D7%99%D7%90%D7%A4%D7%95%D7%A0%D7%95%D7%91" 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href="https://pt.wikipedia.org/wiki/Expoente_de_Lyapunov" title="Expoente de Lyapunov – Portuguese" lang="pt" hreflang="pt" data-title="Expoente de Lyapunov" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BA%D0%B0%D0%B7%D0%B0%D1%82%D0%B5%D0%BB%D1%8C_%D0%9B%D1%8F%D0%BF%D1%83%D0%BD%D0%BE%D0%B2%D0%B0" title="Показатель Ляпунова – Russian" lang="ru" hreflang="ru" data-title="Показатель Ляпунова" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BA%D0%B0%D0%B7%D0%BD%D0%B8%D0%BA_%D0%9B%D1%8F%D0%BF%D1%83%D0%BD%D0%BE%D0%B2%D0%B0" title="Показник Ляпунова – Ukrainian" lang="uk" hreflang="uk" 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.trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:242px;max-width:242px"><div class="trow"><div class="tsingle" style="width:240px;max-width:240px"><div class="thumbimage" style="height:197px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Lyapunov-exponent.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Lyapunov-exponent.svg/238px-Lyapunov-exponent.svg.png" decoding="async" width="238" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Lyapunov-exponent.svg/357px-Lyapunov-exponent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Lyapunov-exponent.svg/476px-Lyapunov-exponent.svg.png 2x" data-file-width="213" data-file-height="177" /></a></span></div></div></div><div class="trow"><div class="tsingle" style="width:240px;max-width:240px"><div class="thumbimage" style="height:178px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Orbital_instability_(Lyapunov_exponent).png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Orbital_instability_%28Lyapunov_exponent%29.png/238px-Orbital_instability_%28Lyapunov_exponent%29.png" decoding="async" width="238" height="179" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Orbital_instability_%28Lyapunov_exponent%29.png/357px-Orbital_instability_%28Lyapunov_exponent%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Orbital_instability_%28Lyapunov_exponent%29.png/476px-Orbital_instability_%28Lyapunov_exponent%29.png 2x" data-file-width="945" data-file-height="709" /></a></span></div><div class="thumbcaption">Explanations of the Lyapunov exponent</div></div></div></div></div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>Lyapunov exponent</b> or <b>Lyapunov characteristic exponent</b> of a <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical system</a> is a quantity that characterizes the rate of separation of infinitesimally close <a href="/wiki/Trajectory" title="Trajectory">trajectories</a>. Quantitatively, two trajectories in <a href="/wiki/Phase_space" title="Phase space">phase space</a> with initial separation vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \mathbf {Z} _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \mathbf {Z} _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0197de2ff71c4f6e2610fd4b16975a847836980c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.737ex; height:2.676ex;" alt="{\displaystyle \delta \mathbf {Z} _{0}}"></span> diverge (provided that the divergence can be treated within the linearized approximation) at a rate given by </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\delta \mathbf {Z} (t)|\approx e^{\lambda t}|\delta \mathbf {Z} _{0}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≈</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>t</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\delta \mathbf {Z} (t)|\approx e^{\lambda t}|\delta \mathbf {Z} _{0}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f56076031e9fca916bb1f8c9ccf05c26c7826cfa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.622ex; height:3.176ex;" alt="{\displaystyle |\delta \mathbf {Z} (t)|\approx e^{\lambda t}|\delta \mathbf {Z} _{0}|}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> is the Lyapunov exponent. </p><p>The rate of separation can be different for different orientations of initial separation vector. Thus, there is a <b>spectrum of Lyapunov exponents</b>—equal in number to the dimensionality of the phase space. It is common to refer to the largest one as the <b>maximal Lyapunov exponent</b> (MLE), because it determines a notion of <a href="/wiki/Predictability" title="Predictability">predictability</a> for a dynamical system. A positive MLE is usually taken as an indication that the system is <a href="/wiki/Chaos_theory" title="Chaos theory">chaotic</a> (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. </p><p>The exponent is named after <a href="/wiki/Aleksandr_Lyapunov" title="Aleksandr Lyapunov">Aleksandr Lyapunov</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_of_the_maximal_Lyapunov_exponent">Definition of the maximal Lyapunov exponent</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lyapunov_exponent&action=edit&section=1" title="Edit section: Definition of the maximal Lyapunov exponent"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The maximal Lyapunov exponent can be defined as follows: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =\lim _{t\to \infty }\lim _{|\delta \mathbf {Z} _{0}|\to 0}{\frac {1}{t}}\ln {\frac {|\delta \mathbf {Z} (t)|}{|\delta \mathbf {Z} _{0}|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>t</mi> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =\lim _{t\to \infty }\lim _{|\delta \mathbf {Z} _{0}|\to 0}{\frac {1}{t}}\ln {\frac {|\delta \mathbf {Z} (t)|}{|\delta \mathbf {Z} _{0}|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47938b708c0bc8cb7ee27b4770a5114e9b64dcdb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.39ex; height:6.509ex;" alt="{\displaystyle \lambda =\lim _{t\to \infty }\lim _{|\delta \mathbf {Z} _{0}|\to 0}{\frac {1}{t}}\ln {\frac {|\delta \mathbf {Z} (t)|}{|\delta \mathbf {Z} _{0}|}}}"></span> </p><p>The limit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\delta \mathbf {Z} _{0}|\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>δ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\delta \mathbf {Z} _{0}|\to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9de02cc1947eb72e6b76e32da768425cf72b441a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.807ex; height:2.843ex;" alt="{\displaystyle |\delta \mathbf {Z} _{0}|\to 0}"></span> ensures the validity of the linear approximation at any time.<sup id="cite_ref-cencini_1-0" class="reference"><a href="#cite_note-cencini-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>For discrete time system (maps or fixed point iterations) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n+1}=f(x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n+1}=f(x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/643700935b0ff5e6dd286c1ec5dbb1605be82171" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.383ex; height:2.843ex;" alt="{\displaystyle x_{n+1}=f(x_{n})}"></span>, for an orbit starting with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span> this translates into: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (x_{0})=\lim _{n\to \infty }{\frac {1}{n}}\sum _{i=0}^{n-1}\ln |f'(x_{i})|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda (x_{0})=\lim _{n\to \infty }{\frac {1}{n}}\sum _{i=0}^{n-1}\ln |f'(x_{i})|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cad41c8155238ec954d0fcfc61700b0bf324e3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.231ex; height:7.343ex;" alt="{\displaystyle \lambda (x_{0})=\lim _{n\to \infty }{\frac {1}{n}}\sum _{i=0}^{n-1}\ln |f'(x_{i})|}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Definition_of_the_Lyapunov_spectrum">Definition of the Lyapunov spectrum</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lyapunov_exponent&action=edit&section=2" title="Edit section: Definition of the Lyapunov spectrum"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:LyapunovDiagram.svg" class="mw-file-description"><img alt="Lyapunov exponent" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/LyapunovDiagram.svg/220px-LyapunovDiagram.svg.png" decoding="async" width="220" height="236" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/LyapunovDiagram.svg/330px-LyapunovDiagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/04/LyapunovDiagram.svg/440px-LyapunovDiagram.svg.png 2x" data-file-width="529" data-file-height="568" /></a><figcaption>The leading Lyapunov vector.</figcaption></figure> <p>For a dynamical system with evolution equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {x}}_{i}=f_{i}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {x}}_{i}=f_{i}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7f3cab0f75c97e6b4652afb5873eb6e6747ca9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.306ex; height:2.843ex;" alt="{\displaystyle {\dot {x}}_{i}=f_{i}(x)}"></span> in an <i>n</i>–dimensional phase space, the spectrum of Lyapunov exponents <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}\}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}\}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6d451c9bd8a3fa425e7a5357b23473ab6f19bfc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.964ex; height:2.843ex;" alt="{\displaystyle \{\lambda _{1},\lambda _{2},\ldots ,\lambda _{n}\}\,,}"></span> in general, depends on the starting point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span>. However, we will usually be interested in the <a href="/wiki/Attractor" title="Attractor">attractor</a> (or attractors) of a dynamical system, and there will normally be one set of exponents associated with each attractor. The choice of starting point may determine which attractor the system ends up on, if there is more than one. (For Hamiltonian systems, which do not have attractors, this is not a concern.) The Lyapunov exponents describe the behavior of vectors in the tangent space of the phase space and are defined from the <a href="/wiki/Jacobian_matrix" class="mw-redirect" title="Jacobian matrix">Jacobian matrix</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{ij}(t)=\left.{\frac {df_{i}(x)}{dx_{j}}}\right|_{x(t)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{ij}(t)=\left.{\frac {df_{i}(x)}{dx_{j}}}\right|_{x(t)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d8cdfe012771f9c6cb0e6c2f5f4e90eabfd4a5b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.337ex; height:7.009ex;" alt="{\displaystyle J_{ij}(t)=\left.{\frac {df_{i}(x)}{dx_{j}}}\right|_{x(t)}}"></span> this Jacobian defines the evolution of the tangent vectors, given by the matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>, via the equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {Y}}=JY}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>J</mi> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {Y}}=JY}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fac746e2537a450a7b2773d0453a496defe81c4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.116ex; height:2.676ex;" alt="{\displaystyle {\dot {Y}}=JY}"></span> with the initial condition <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{ij}(0)=\delta _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{ij}(0)=\delta _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72d060822d81e538c3da0fc2aaeec167cc12b311" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.407ex; height:3.009ex;" alt="{\displaystyle Y_{ij}(0)=\delta _{ij}}"></span>. The matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> describes how a small change at the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7176643e6d36fa7674dc79fdff1a4daa068f5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.301ex; height:2.843ex;" alt="{\displaystyle x(0)}"></span> propagates to the final point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54c275db3a1e620737b58e143b0818107fa5f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}"></span>. The limit <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda =\lim _{t\rightarrow \infty }{\frac {1}{2t}}\log(Y(t)Y^{T}(t))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda =\lim _{t\rightarrow \infty }{\frac {1}{2t}}\log(Y(t)Y^{T}(t))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18028f42f186b36e3185bf4f5aeae17ea3bfec84" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.346ex; height:5.343ex;" alt="{\displaystyle \Lambda =\lim _{t\rightarrow \infty }{\frac {1}{2t}}\log(Y(t)Y^{T}(t))}"></span> defines a matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span> (the conditions for the existence of the limit are given by the <a href="/wiki/Oseledets_theorem" title="Oseledets theorem">Oseledets theorem</a>). The Lyapunov exponents <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72fde940918edf84caf3d406cc7d31949166820f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}}"></span> are defined by the eigenvalues of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span>. </p><p>The set of Lyapunov exponents will be the same for almost all starting points of an <a href="/wiki/Dynamical_system#Ergodic_systems" title="Dynamical system">ergodic</a> component of the dynamical system. </p> <div class="mw-heading mw-heading2"><h2 id="Lyapunov_exponent_for_time-varying_linearization">Lyapunov exponent for time-varying linearization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lyapunov_exponent&action=edit&section=3" title="Edit section: Lyapunov exponent for time-varying linearization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To introduce Lyapunov exponent consider a fundamental matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfeb6663c0a903f587cd6d776c387370fc5c4ab7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.629ex; height:2.843ex;" alt="{\displaystyle X(t)}"></span> (e.g., for linearization along a stationary solution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span> in a continuous system), the fundamental matrix is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp \left(\left.{\frac {df^{t}(x)}{dx}}\right|_{x_{0}}t\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msub> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp \left(\left.{\frac {df^{t}(x)}{dx}}\right|_{x_{0}}t\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42fbd772d30685945d9c967b7b2678a11d3b8663" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.062ex; height:7.509ex;" alt="{\displaystyle \exp \left(\left.{\frac {df^{t}(x)}{dx}}\right|_{x_{0}}t\right)}"></span> consisting of the linearly-independent solutions of the first-order approximation of the system. The singular values <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\alpha _{j}{\big (}X(t){\big )}\}_{1}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\alpha _{j}{\big (}X(t){\big )}\}_{1}^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c70c4c8926ba455024e197837a3ea676aa7211" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.699ex; height:3.176ex;" alt="{\displaystyle \{\alpha _{j}{\big (}X(t){\big )}\}_{1}^{n}}"></span> of the matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfeb6663c0a903f587cd6d776c387370fc5c4ab7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.629ex; height:2.843ex;" alt="{\displaystyle X(t)}"></span> are the square roots of the eigenvalues of the matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(t)^{*}X(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(t)^{*}X(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da1ecb6b68c051c30fba048eb8084c457e046e8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.312ex; height:2.843ex;" alt="{\displaystyle X(t)^{*}X(t)}"></span>. The largest Lyapunov exponent <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\mathrm {max} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{\mathrm {max} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84643cf5aa56108219c7e9ba81e01aba19ca0bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.646ex; height:2.509ex;" alt="{\displaystyle \lambda _{\mathrm {max} }}"></span> is as follows<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\mathrm {max} }=\max \limits _{j}\limsup _{t\rightarrow \infty }{\frac {1}{t}}\ln \alpha _{j}{\big (}X(t){\big )}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>=</mo> <munder> <mo form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">lim sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>t</mi> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{\mathrm {max} }=\max \limits _{j}\limsup _{t\rightarrow \infty }{\frac {1}{t}}\ln \alpha _{j}{\big (}X(t){\big )}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8900d575b62717ccaba52f175fef23fe86eb400" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.479ex; height:5.676ex;" alt="{\displaystyle \lambda _{\mathrm {max} }=\max \limits _{j}\limsup _{t\rightarrow \infty }{\frac {1}{t}}\ln \alpha _{j}{\big (}X(t){\big )}.}"></span> Lyapunov proved that if the system of the first approximation is regular (e.g., all systems with constant and periodic coefficients are regular) and its largest Lyapunov exponent is negative, then the solution of the original system is <a href="/wiki/Lyapunov_stability" title="Lyapunov stability">asymptotically Lyapunov stable</a>. Later, it was stated by O. Perron that the requirement of regularity of the first approximation is substantial. </p> <div class="mw-heading mw-heading3"><h3 id="Perron_effects_of_largest_Lyapunov_exponent_sign_inversion">Perron effects of largest Lyapunov exponent sign inversion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lyapunov_exponent&action=edit&section=4" title="Edit section: Perron effects of largest Lyapunov exponent sign inversion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1930 <a href="/wiki/Oskar_Perron" title="Oskar Perron">O. Perron</a> constructed an example of a second-order system, where the first approximation has negative Lyapunov exponents along a zero solution of the original system but, at the same time, this zero solution of the original nonlinear system is Lyapunov unstable. Furthermore, in a certain neighborhood of this zero solution almost all solutions of original system have positive Lyapunov exponents. Also, it is possible to construct a reverse example in which the first approximation has positive Lyapunov exponents along a zero solution of the original system but, at the same time, this zero solution of original nonlinear system is Lyapunov stable.<sup id="cite_ref-2005-IEEE-Discrete-system-stability-Lyapunov-exponent_3-0" class="reference"><a href="#cite_note-2005-IEEE-Discrete-system-stability-Lyapunov-exponent-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2007-IJBC-Lyapunov-exponent_4-0" class="reference"><a href="#cite_note-2007-IJBC-Lyapunov-exponent-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The effect of sign inversion of Lyapunov exponents of solutions of the original system and the system of first approximation with the same initial data was subsequently called the Perron effect.<sup id="cite_ref-2005-IEEE-Discrete-system-stability-Lyapunov-exponent_3-1" class="reference"><a href="#cite_note-2005-IEEE-Discrete-system-stability-Lyapunov-exponent-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2007-IJBC-Lyapunov-exponent_4-1" class="reference"><a href="#cite_note-2007-IJBC-Lyapunov-exponent-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Perron's counterexample shows that a negative largest Lyapunov exponent does not, in general, indicate stability, and that a positive largest Lyapunov exponent does not, in general, indicate chaos. </p><p>Therefore, time-varying linearization requires additional justification.<sup id="cite_ref-2007-IJBC-Lyapunov-exponent_4-2" class="reference"><a href="#cite_note-2007-IJBC-Lyapunov-exponent-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Basic_properties">Basic properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lyapunov_exponent&action=edit&section=5" title="Edit section: Basic properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the system is conservative (i.e., there is no <a href="/wiki/Dissipation" title="Dissipation">dissipation</a>), a volume element of the phase space will stay the same along a trajectory. Thus the sum of all Lyapunov exponents must be zero. If the system is dissipative, the sum of Lyapunov exponents is negative. </p><p>If the system is a flow and the trajectory does not converge to a single point, one exponent is always zero—the Lyapunov exponent corresponding to the eigenvalue of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> with an eigenvector in the direction of the flow. </p> <div class="mw-heading mw-heading2"><h2 id="Significance_of_the_Lyapunov_spectrum">Significance of the Lyapunov spectrum</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lyapunov_exponent&action=edit&section=6" title="Edit section: Significance of the Lyapunov spectrum"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Lyapunov spectrum can be used to give an estimate of the rate of entropy production, of the <a href="/wiki/Fractal_dimension" title="Fractal dimension">fractal dimension</a>, and of the <a href="/wiki/Hausdorff_dimension" title="Hausdorff dimension">Hausdorff dimension</a> of the considered <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical system</a>.<sup id="cite_ref-2020-KuznetsovR_5-0" class="reference"><a href="#cite_note-2020-KuznetsovR-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In particular from the knowledge of the Lyapunov spectrum it is possible to obtain the so-called <a href="/wiki/Lyapunov_dimension" title="Lyapunov dimension">Lyapunov dimension</a> (or <a href="/wiki/Kaplan%E2%80%93Yorke_conjecture" title="Kaplan–Yorke conjecture">Kaplan–Yorke dimension</a>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{KY}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mi>Y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{KY}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d99ac47e414853068c2be623f4a876c27a88765d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.871ex; height:2.509ex;" alt="{\displaystyle D_{KY}}"></span>, which is defined as follows: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{KY}=k+\sum _{i=1}^{k}{\frac {\lambda _{i}}{|\lambda _{k+1}|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mi>Y</mi> </mrow> </msub> <mo>=</mo> <mi>k</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{KY}=k+\sum _{i=1}^{k}{\frac {\lambda _{i}}{|\lambda _{k+1}|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d8c27c3a76dc60b78b7978c7419a729915614a4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.438ex; height:7.343ex;" alt="{\displaystyle D_{KY}=k+\sum _{i=1}^{k}{\frac {\lambda _{i}}{|\lambda _{k+1}|}}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> is the maximum integer such that the sum of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> largest exponents is still non-negative. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{KY}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mi>Y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{KY}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d99ac47e414853068c2be623f4a876c27a88765d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.871ex; height:2.509ex;" alt="{\displaystyle D_{KY}}"></span> represents an upper bound for the <a href="/wiki/Information_dimension" title="Information dimension">information dimension</a> of the system.<sup id="cite_ref-ky_6-0" class="reference"><a href="#cite_note-ky-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Moreover, the sum of all the positive Lyapunov exponents gives an estimate of the <a href="/wiki/Kolmogorov%E2%80%93Sinai_entropy" class="mw-redirect" title="Kolmogorov–Sinai entropy">Kolmogorov–Sinai entropy</a> accordingly to Pesin's theorem.<sup id="cite_ref-pesin_7-0" class="reference"><a href="#cite_note-pesin-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Along with widely used numerical methods for estimating and computing the <a href="/wiki/Lyapunov_dimension" title="Lyapunov dimension">Lyapunov dimension</a> there is an effective analytical approach, which is based on the direct Lyapunov method with special Lyapunov-like functions.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The Lyapunov exponents of bounded trajectory and the <a href="/wiki/Lyapunov_dimension" title="Lyapunov dimension">Lyapunov dimension</a> of attractor are invariant under <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphism</a> of the phase space.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> of the largest Lyapunov exponent is sometimes referred in literature as <a href="/wiki/Lyapunov_time" title="Lyapunov time">Lyapunov time</a>, and defines the characteristic <i>e</i>-folding time. For chaotic orbits, the Lyapunov time will be finite, whereas for regular orbits it will be infinite. </p> <div class="mw-heading mw-heading2"><h2 id="Numerical_calculation">Numerical calculation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lyapunov_exponent&action=edit&section=7" title="Edit section: Numerical calculation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Lyapunov_exponents_of_the_Mandelbrot_set_(The_mini-Mandelbrot)_-_Matlab.png" class="mw-file-description"><img alt="Lyapunov exponent" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Lyapunov_exponents_of_the_Mandelbrot_set_%28The_mini-Mandelbrot%29_-_Matlab.png/220px-Lyapunov_exponents_of_the_Mandelbrot_set_%28The_mini-Mandelbrot%29_-_Matlab.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Lyapunov_exponents_of_the_Mandelbrot_set_%28The_mini-Mandelbrot%29_-_Matlab.png/330px-Lyapunov_exponents_of_the_Mandelbrot_set_%28The_mini-Mandelbrot%29_-_Matlab.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2d/Lyapunov_exponents_of_the_Mandelbrot_set_%28The_mini-Mandelbrot%29_-_Matlab.png/440px-Lyapunov_exponents_of_the_Mandelbrot_set_%28The_mini-Mandelbrot%29_-_Matlab.png 2x" data-file-width="801" data-file-height="801" /></a><figcaption>Points inside and outside <a href="/wiki/Mandelbrot_set" title="Mandelbrot set">Mandelbrot set</a> colored by Lyapunov exponent.</figcaption></figure> <p>Generally the calculation of Lyapunov exponents, as defined above, cannot be carried out analytically, and in most cases one must resort to numerical techniques. An early example, which also constituted the first demonstration of the exponential divergence of chaotic trajectories, was carried out by <a href="/w/index.php?title=Richard_H._Miller&action=edit&redlink=1" class="new" title="Richard H. Miller (page does not exist)">R. H. Miller</a> in 1964.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Currently, the most commonly used numerical procedure estimates the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> matrix based on averaging several finite time approximations of the limit defining <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span>. </p><p>One of the most used and effective numerical techniques to calculate the Lyapunov spectrum for a smooth dynamical system relies on periodic <a href="/wiki/Gram%E2%80%93Schmidt" class="mw-redirect" title="Gram–Schmidt">Gram–Schmidt</a> orthonormalization of the <a href="/wiki/Lyapunov_vector" title="Lyapunov vector">Lyapunov vectors</a> to avoid a misalignment of all the vectors along the direction of maximal expansion.<sup id="cite_ref-benettin_11-0" class="reference"><a href="#cite_note-benettin-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-shimada_13-0" class="reference"><a href="#cite_note-shimada-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> The Lyapunov spectrum of various models are described.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> Source codes for nonlinear systems such as the Hénon map, the Lorenz equations, a delay differential equation and so on are introduced.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>For the calculation of Lyapunov exponents from limited experimental data, various methods have been proposed. However, there are many difficulties with applying these methods and such problems should be approached with care. The main difficulty is that the data does not fully explore the phase space, rather it is confined to the attractor which has very limited (if any) extension along certain directions. These thinner or more singular directions within the data set are the ones associated with the more negative exponents. The use of nonlinear mappings to model the evolution of small displacements from the attractor has been shown to dramatically improve the ability to recover the Lyapunov spectrum,<sup id="cite_ref-Bryant1_19-0" class="reference"><a href="#cite_note-Bryant1-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-brown_20-0" class="reference"><a href="#cite_note-brown-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> provided the data has a very low level of noise. The singular nature of the data and its connection to the more negative exponents has also been explored.<sup id="cite_ref-bryant2_21-0" class="reference"><a href="#cite_note-bryant2-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Local_Lyapunov_exponent">Local Lyapunov exponent</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lyapunov_exponent&action=edit&section=8" title="Edit section: Local Lyapunov exponent"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Whereas the (global) Lyapunov exponent gives a measure for the total predictability of a system, it is sometimes of interest to estimate the local predictability around a point <span class="texhtml"><i>x</i><sub>0</sub></span> in phase space. This may be done through the <a href="/wiki/Eigenvalues" class="mw-redirect" title="Eigenvalues">eigenvalues</a> of the <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a> matrix <span class="texhtml"><i>J</i><sup>0</sup>(<i>x</i><sub>0</sub>)</span>. These eigenvalues are also called local Lyapunov exponents.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Local exponents are not invariant under a nonlinear change of coordinates. </p> <div class="mw-heading mw-heading2"><h2 id="Conditional_Lyapunov_exponent">Conditional Lyapunov exponent</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lyapunov_exponent&action=edit&section=9" title="Edit section: Conditional Lyapunov exponent"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This term is normally used regarding <a href="/wiki/Synchronization_of_chaos" title="Synchronization of chaos">synchronization of chaos</a>, in which there are two systems that are coupled, usually in a unidirectional manner so that there is a drive (or master) system and a response (or slave) system. The conditional exponents are those of the response system with the drive system treated as simply the source of a (chaotic) drive signal. Synchronization occurs when all of the conditional exponents are negative.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lyapunov_exponent&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Chaos_theory#Sensitivity_to_initial_conditions" title="Chaos theory">Chaos Theory</a></li> <li><a href="/wiki/Chaotic_mixing#Lyapunov_exponents" title="Chaotic mixing">Chaotic mixing</a> for an alternative derivation</li> <li><a href="/wiki/Eden%27s_conjecture" title="Eden's conjecture">Eden's conjecture</a> on the Lyapunov dimension</li> <li><a href="/wiki/Floquet_theory" title="Floquet theory">Floquet theory</a></li> <li><a href="/wiki/Liouville%27s_theorem_(Hamiltonian)" title="Liouville's theorem (Hamiltonian)">Liouville's theorem (Hamiltonian)</a></li> <li><a href="/wiki/Lyapunov_dimension" title="Lyapunov dimension">Lyapunov dimension</a></li> <li><a href="/wiki/Lyapunov_time" title="Lyapunov time">Lyapunov time</a></li> <li><a href="/wiki/Recurrence_quantification_analysis" title="Recurrence quantification analysis">Recurrence quantification analysis</a></li> <li><a href="/wiki/Oseledets_theorem" title="Oseledets theorem">Oseledets theorem</a></li> <li><a href="/wiki/Butterfly_effect" title="Butterfly effect">Butterfly effect</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lyapunov_exponent&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-cencini-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-cencini_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFCencini2010" class="citation book cs1">Cencini, M.; et al. (2010). World Scientific (ed.). <i>Chaos From Simple models to complex systems</i>. World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4277-65-5" title="Special:BookSources/978-981-4277-65-5"><bdi>978-981-4277-65-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chaos+From+Simple+models+to+complex+systems&rft.pub=World+Scientific&rft.date=2010&rft.isbn=978-981-4277-65-5&rft.aulast=Cencini&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTemam1988" class="citation book cs1"><a href="/wiki/Roger_Temam" title="Roger Temam">Temam, R.</a> (1988). <i>Infinite Dimensional Dynamical Systems in Mechanics and Physics</i>. 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Vol. Proceedings Volume 2005. pp. 596–599. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FPHYCON.2005.1514053">10.1109/PHYCON.2005.1514053</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7803-9235-9" title="Special:BookSources/978-0-7803-9235-9"><bdi>978-0-7803-9235-9</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:31746738">31746738</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+stability+by+the+first+approximation+for+discrete+systems&rft.btitle=Proceedings.+2005+International+Conference+Physics+and+Control%2C+2005&rft.pages=596-599&rft.date=2005&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A31746738%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2FPHYCON.2005.1514053&rft.isbn=978-0-7803-9235-9&rft.au=N.V.+Kuznetsov&rft.au=G.A.+Leonov&rft_id=http%3A%2F%2Fwww.math.spbu.ru%2Fuser%2Fnk%2FPDF%2F2005-IEEE-Discrete-system-stability-Lyapunov-exponent.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></span> </li> <li id="cite_note-2007-IJBC-Lyapunov-exponent-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-2007-IJBC-Lyapunov-exponent_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-2007-IJBC-Lyapunov-exponent_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-2007-IJBC-Lyapunov-exponent_4-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFG.A._LeonovN.V._Kuznetsov2007" class="citation journal cs1">G.A. 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Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0198508403" title="Special:BookSources/978-0198508403"><bdi>978-0198508403</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chaos+and+Time-Series+Analysis&rft.pub=Oxford+University+Press&rft.date=2001-09-27&rft.isbn=978-0198508403&rft.aulast=Sprott&rft.aufirst=Julien+Clinton&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSprott2005" class="citation web cs1">Sprott, Julien Clinton (May 26, 2005). <a rel="nofollow" class="external text" href="https://sprott.physics.wisc.edu/chaos/lespec.htm">"Lyapunov Exponent Spectrum Software"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Lyapunov+Exponent+Spectrum+Software&rft.date=2005-05-26&rft.aulast=Sprott&rft.aufirst=Julien+Clinton&rft_id=https%3A%2F%2Fsprott.physics.wisc.edu%2Fchaos%2Flespec.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSprott2006" class="citation web cs1">Sprott, Julien Clinton (October 4, 2006). <a rel="nofollow" class="external text" href="https://sprott.physics.wisc.edu/chaos/ddele.htm">"Lyapunov Exponents for Delay Differential Equations"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Lyapunov+Exponents+for+Delay+Differential+Equations&rft.date=2006-10-04&rft.aulast=Sprott&rft.aufirst=Julien+Clinton&rft_id=https%3A%2F%2Fsprott.physics.wisc.edu%2Fchaos%2Fddele.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTomo2022" class="citation web cs1">Tomo, Nakamura (19 October 2022). <a rel="nofollow" class="external text" href="https://sites.google.com/view/lyapunov-spectrum/home">"nonlinear systems and Lyapunov spectrum"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=nonlinear+systems+and+Lyapunov+spectrum&rft.date=2022-10-19&rft.aulast=Tomo&rft.aufirst=Nakamura&rft_id=https%3A%2F%2Fsites.google.com%2Fview%2Flyapunov-spectrum%2Fhome&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></span> </li> <li id="cite_note-Bryant1-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bryant1_19-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBryantBrownAbarbanel1990" class="citation journal cs1">Bryant, P.; Brown, R.; Abarbanel, H. (1990). "Lyapunov exponents from observed time series". <i>Physical Review Letters</i>. <b>65</b> (13): 1523–1526. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1990PhRvL..65.1523B">1990PhRvL..65.1523B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.65.1523">10.1103/PhysRevLett.65.1523</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10042292">10042292</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Lyapunov+exponents+from+observed+time+series&rft.volume=65&rft.issue=13&rft.pages=1523-1526&rft.date=1990&rft_id=info%3Apmid%2F10042292&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.65.1523&rft_id=info%3Abibcode%2F1990PhRvL..65.1523B&rft.aulast=Bryant&rft.aufirst=P.&rft.au=Brown%2C+R.&rft.au=Abarbanel%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></span> </li> <li id="cite_note-brown-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-brown_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrownBryantAbarbanel1991" class="citation journal cs1">Brown, R.; Bryant, P.; Abarbanel, H. (1991). "Computing the Lyapunov spectrum of a dynamical system from an observed time series". <i>Physical Review A</i>. <b>43</b> (6): 2787–2806. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991PhRvA..43.2787B">1991PhRvA..43.2787B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevA.43.2787">10.1103/PhysRevA.43.2787</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9905344">9905344</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+A&rft.atitle=Computing+the+Lyapunov+spectrum+of+a+dynamical+system+from+an+observed+time+series&rft.volume=43&rft.issue=6&rft.pages=2787-2806&rft.date=1991&rft_id=info%3Apmid%2F9905344&rft_id=info%3Adoi%2F10.1103%2FPhysRevA.43.2787&rft_id=info%3Abibcode%2F1991PhRvA..43.2787B&rft.aulast=Brown&rft.aufirst=R.&rft.au=Bryant%2C+P.&rft.au=Abarbanel%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></span> </li> <li id="cite_note-bryant2-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-bryant2_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBryant1993" class="citation journal cs1">Bryant, P. H. (1993). "Extensional singularity dimensions for strange attractors". <i>Physics Letters A</i>. <b>179</b> (3): 186–190. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993PhLA..179..186B">1993PhLA..179..186B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0375-9601%2893%2991136-S">10.1016/0375-9601(93)91136-S</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters+A&rft.atitle=Extensional+singularity+dimensions+for+strange+attractors&rft.volume=179&rft.issue=3&rft.pages=186-190&rft.date=1993&rft_id=info%3Adoi%2F10.1016%2F0375-9601%2893%2991136-S&rft_id=info%3Abibcode%2F1993PhLA..179..186B&rft.aulast=Bryant&rft.aufirst=P.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbarbanelBrownKennel1992" class="citation journal cs1">Abarbanel, H.D.I.; Brown, R.; Kennel, M.B. (1992). "Local Lyapunov exponents computed from observed data". <i>Journal of Nonlinear Science</i>. <b>2</b> (3): 343–365. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992JNS.....2..343A">1992JNS.....2..343A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01208929">10.1007/BF01208929</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122542761">122542761</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Nonlinear+Science&rft.atitle=Local+Lyapunov+exponents+computed+from+observed+data&rft.volume=2&rft.issue=3&rft.pages=343-365&rft.date=1992&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122542761%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01208929&rft_id=info%3Abibcode%2F1992JNS.....2..343A&rft.aulast=Abarbanel&rft.aufirst=H.D.I.&rft.au=Brown%2C+R.&rft.au=Kennel%2C+M.B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">See, e.g., <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPecoraCarrollJohnsonMar1997" class="citation journal cs1">Pecora, L. M.; Carroll, T. L.; Johnson, G. A.; Mar, D. J.; Heagy, J. F. (1997). <a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.166278">"Fundamentals of synchronization in chaotic systems, concepts, and applications"</a>. <i>Chaos: An Interdisciplinary Journal of Nonlinear Science</i>. <b>7</b> (4): 520–543. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997Chaos...7..520P">1997Chaos...7..520P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.166278">10.1063/1.166278</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/12779679">12779679</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Chaos%3A+An+Interdisciplinary+Journal+of+Nonlinear+Science&rft.atitle=Fundamentals+of+synchronization+in+chaotic+systems%2C+concepts%2C+and+applications&rft.volume=7&rft.issue=4&rft.pages=520-543&rft.date=1997&rft_id=info%3Apmid%2F12779679&rft_id=info%3Adoi%2F10.1063%2F1.166278&rft_id=info%3Abibcode%2F1997Chaos...7..520P&rft.aulast=Pecora&rft.aufirst=L.+M.&rft.au=Carroll%2C+T.+L.&rft.au=Johnson%2C+G.+A.&rft.au=Mar%2C+D.+J.&rft.au=Heagy%2C+J.+F.&rft_id=https%3A%2F%2Fdoi.org%2F10.1063%252F1.166278&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lyapunov_exponent&action=edit&section=12" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKuznetsovReitmann2020" class="citation book cs1">Kuznetsov, Nikolay; Reitmann, Volker (2020). <a rel="nofollow" class="external text" href="https://www.springer.com/gp/book/9783030509866"><i>Attractor Dimension Estimates for Dynamical Systems: Theory and Computation</i></a>. Cham: Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Attractor+Dimension+Estimates+for+Dynamical+Systems%3A+Theory+and+Computation&rft.place=Cham&rft.pub=Springer&rft.date=2020&rft.aulast=Kuznetsov&rft.aufirst=Nikolay&rft.au=Reitmann%2C+Volker&rft_id=https%3A%2F%2Fwww.springer.com%2Fgp%2Fbook%2F9783030509866&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFM.-F._DancaN.V._Kuznetsov2018" class="citation journal cs1">M.-F. Danca & N.V. Kuznetsov (2018). "Matlab Code for Lyapunov Exponents of Fractional-Order Systems". <i>International Journal of Bifurcation and Chaos</i>. <b>25</b> (5): 1850067–1851392. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1804.01143">1804.01143</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2018IJBC...2850067D">2018IJBC...2850067D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0218127418500670">10.1142/S0218127418500670</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Bifurcation+and+Chaos&rft.atitle=Matlab+Code+for+Lyapunov+Exponents+of+Fractional-Order+Systems&rft.volume=25&rft.issue=5&rft.pages=1850067-1851392&rft.date=2018&rft_id=info%3Aarxiv%2F1804.01143&rft_id=info%3Adoi%2F10.1142%2FS0218127418500670&rft_id=info%3Abibcode%2F2018IJBC...2850067D&rft.au=M.-F.+Danca&rft.au=N.V.+Kuznetsov&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></li> <li>Cvitanović P., Artuso R., Mainieri R., Tanner G. and Vattay G.<a rel="nofollow" class="external text" href="http://www.chaosbook.org/">Chaos: Classical and Quantum</a> Niels Bohr Institute, Copenhagen 2005 – <i>textbook about chaos available under <a href="/wiki/Free_Documentation_License" class="mw-redirect" title="Free Documentation License">Free Documentation License</a></i></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFreddy_ChristiansenHans_Henrik_Rugh1997" class="citation journal cs1">Freddy Christiansen & Hans Henrik Rugh (1997). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060425194442/http://www.mpipks-dresden.mpg.de/eprint/freddy/9702017/9702017.ps">"Computing Lyapunov spectra with continuous Gram–Schmidt orthonormalization"</a>. <i>Nonlinearity</i>. <b>10</b> (5): 1063–1072. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/chao-dyn/9611014">chao-dyn/9611014</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997Nonli..10.1063C">1997Nonli..10.1063C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0951-7715%2F10%2F5%2F004">10.1088/0951-7715/10/5/004</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122976405">122976405</a>. Archived from <a rel="nofollow" class="external text" href="http://www.mpipks-dresden.mpg.de/eprint/freddy/9702017/9702017.ps">the original</a> on 2006-04-25.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nonlinearity&rft.atitle=Computing+Lyapunov+spectra+with+continuous+Gram%E2%80%93Schmidt+orthonormalization&rft.volume=10&rft.issue=5&rft.pages=1063-1072&rft.date=1997&rft_id=info%3Aarxiv%2Fchao-dyn%2F9611014&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122976405%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0951-7715%2F10%2F5%2F004&rft_id=info%3Abibcode%2F1997Nonli..10.1063C&rft.au=Freddy+Christiansen&rft.au=Hans+Henrik+Rugh&rft_id=http%3A%2F%2Fwww.mpipks-dresden.mpg.de%2Feprint%2Ffreddy%2F9702017%2F9702017.ps&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSalman_HabibRobert_D._Ryne1995" class="citation journal cs1">Salman Habib & Robert D. Ryne (1995). "Symplectic Calculation of Lyapunov Exponents". <i>Physical Review Letters</i>. <b>74</b> (1): 70–73. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/chao-dyn/9406010">chao-dyn/9406010</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995PhRvL..74...70H">1995PhRvL..74...70H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.74.70">10.1103/PhysRevLett.74.70</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10057701">10057701</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:19203665">19203665</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Symplectic+Calculation+of+Lyapunov+Exponents&rft.volume=74&rft.issue=1&rft.pages=70-73&rft.date=1995&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A19203665%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1995PhRvL..74...70H&rft_id=info%3Aarxiv%2Fchao-dyn%2F9406010&rft_id=info%3Apmid%2F10057701&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.74.70&rft.au=Salman+Habib&rft.au=Robert+D.+Ryne&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGovindan_RangarajanSalman_HabibRobert_D._Ryne1998" class="citation journal cs1">Govindan Rangarajan; Salman Habib & Robert D. Ryne (1998). "Lyapunov Exponents without Rescaling and Reorthogonalization". <i>Physical Review Letters</i>. <b>80</b> (17): 3747–3750. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/chao-dyn/9803017">chao-dyn/9803017</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998PhRvL..80.3747R">1998PhRvL..80.3747R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.80.3747">10.1103/PhysRevLett.80.3747</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14483592">14483592</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Lyapunov+Exponents+without+Rescaling+and+Reorthogonalization&rft.volume=80&rft.issue=17&rft.pages=3747-3750&rft.date=1998&rft_id=info%3Aarxiv%2Fchao-dyn%2F9803017&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14483592%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.80.3747&rft_id=info%3Abibcode%2F1998PhRvL..80.3747R&rft.au=Govindan+Rangarajan&rft.au=Salman+Habib&rft.au=Robert+D.+Ryne&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFX._ZengR._EykholtR._A._Pielke1991" class="citation journal cs1">X. Zeng; R. Eykholt & R. A. Pielke (1991). "Estimating the Lyapunov-exponent spectrum from short time series of low precision". <i>Physical Review Letters</i>. <b>66</b> (25): 3229–3232. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991PhRvL..66.3229Z">1991PhRvL..66.3229Z</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.66.3229">10.1103/PhysRevLett.66.3229</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10043734">10043734</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Estimating+the+Lyapunov-exponent+spectrum+from+short+time+series+of+low+precision&rft.volume=66&rft.issue=25&rft.pages=3229-3232&rft.date=1991&rft_id=info%3Apmid%2F10043734&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.66.3229&rft_id=info%3Abibcode%2F1991PhRvL..66.3229Z&rft.au=X.+Zeng&rft.au=R.+Eykholt&rft.au=R.+A.+Pielke&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFE_AurellG_BoffettaA_CrisantiG_Paladin1997" class="citation journal cs1">E Aurell; G Boffetta; A Crisanti; G Paladin; A Vulpiani (1997). "Predictability in the large: an extension of the concept of Lyapunov exponent". <i>J. Phys. A: Math. Gen</i>. <b>30</b> (1): 1–26. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/chao-dyn/9606014">chao-dyn/9606014</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1997JPhA...30....1A">1997JPhA...30....1A</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0305-4470%2F30%2F1%2F003">10.1088/0305-4470/30/1/003</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:54697488">54697488</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Phys.+A%3A+Math.+Gen.&rft.atitle=Predictability+in+the+large%3A+an+extension+of+the+concept+of+Lyapunov+exponent&rft.volume=30&rft.issue=1&rft.pages=1-26&rft.date=1997&rft_id=info%3Aarxiv%2Fchao-dyn%2F9606014&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A54697488%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F0305-4470%2F30%2F1%2F003&rft_id=info%3Abibcode%2F1997JPhA...30....1A&rft.au=E+Aurell&rft.au=G+Boffetta&rft.au=A+Crisanti&rft.au=G+Paladin&rft.au=A+Vulpiani&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFF_GinelliP_PoggiA_TurchiH_Chaté2007" class="citation journal cs1">F Ginelli; P Poggi; A Turchi; H Chaté; R Livi; A Politi (2007). <a rel="nofollow" class="external text" href="https://wayback.archive-it.org/all/20081031185653/http://www.fi.isc.cnr.it/users/antonio.politi/Reprints/145.pdf">"Characterizing Dynamics with Covariant Lyapunov Vectors"</a> <span class="cs1-format">(PDF)</span>. <i>Physical Review Letters</i>. <b>99</b> (13): 130601. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0706.0510">0706.0510</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007PhRvL..99m0601G">2007PhRvL..99m0601G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.99.130601">10.1103/PhysRevLett.99.130601</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2158%2F253565">2158/253565</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17930570">17930570</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:21992110">21992110</a>. Archived from <a rel="nofollow" class="external text" href="http://www.fi.isc.cnr.it/users/antonio.politi/Reprints/145.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2008-10-31.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Characterizing+Dynamics+with+Covariant+Lyapunov+Vectors&rft.volume=99&rft.issue=13&rft.pages=130601&rft.date=2007&rft_id=info%3Ahdl%2F2158%2F253565&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A21992110%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2007PhRvL..99m0601G&rft_id=info%3Aarxiv%2F0706.0510&rft_id=info%3Apmid%2F17930570&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.99.130601&rft.au=F+Ginelli&rft.au=P+Poggi&rft.au=A+Turchi&rft.au=H+Chat%C3%A9&rft.au=R+Livi&rft.au=A+Politi&rft_id=http%3A%2F%2Fwww.fi.isc.cnr.it%2Fusers%2Fantonio.politi%2FReprints%2F145.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALyapunov+exponent" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Software">Software</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lyapunov_exponent&action=edit&section=13" title="Edit section: Software"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external autonumber" href="https://web.archive.org/web/20161027044059/http://www.mpipks-dresden.mpg.de/~tisean/Tisean_3.0.1/index.html">[1]</a> R. Hegger, H. Kantz, and T. Schreiber, Nonlinear Time Series Analysis, <a href="/wiki/Tisean" title="Tisean">TISEAN</a> 3.0.1 (March 2007).</li> <li><a rel="nofollow" class="external autonumber" href="https://web.archive.org/web/20130917012451/http://www.scientio.com/Products/ChaosKit">[2]</a> Scientio's ChaosKit product calculates Lyapunov exponents amongst other Chaotic measures. Access is provided online via a web service and Silverlight demo.</li> <li><a rel="nofollow" class="external autonumber" href="http://ftp2.sco.com/pub/skunkware/src/x11/misc/mathrec-1.1c.tar.gz">[3]</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220628200022/http://ftp2.sco.com/pub/skunkware/src/x11/misc/mathrec-1.1c.tar.gz">Archived</a> 2022-06-28 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> Dr. Ronald Joe Record's mathematical recreations software laboratory includes an X11 graphical client, lyap, for graphically exploring the Lyapunov exponents of a forced logistic map and other maps of the unit interval. The <a rel="nofollow" class="external text" href="http://ftp2.sco.com/pub/skunkware/src/x11/misc/mathrec-1.1c/ReadMe.html">contents and manual pages</a> of the mathrec software laboratory are also available.</li> <li><a rel="nofollow" class="external autonumber" href="http://biocircuits.ucsd.edu/pbryant/">[4]</a> Software on this page was developed specifically for the efficient and accurate calculation of the full spectrum of exponents. This includes LyapOde for cases where the equations of motion are known and also Lyap for cases involving experimental time series data. LyapOde, which includes source code written in "C", can also calculate the conditional Lyapunov exponents for coupled identical systems. It is intended to allow the user to provide their own set of model equations or to use one of the ones included. There are no inherent limitations on the number of variables, parameters etc. Lyap which includes source code written in Fortran, can also calculate the Lyapunov direction vectors and can characterize the singularity of the attractor, which is the main reason for difficulties in calculating the more negative exponents from time series data. In both cases there is extensive documentation and sample input files. The software can be compiled for running on Windows, Mac, or Linux/Unix systems. The software runs in a text window and has no graphics capabilities, but can generate output files that could easily be plotted with a program like excel.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lyapunov_exponent&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304045024/http://www.math.spbu.ru/user/nk/PDF/Lyapunov_exponent.pdf">Perron effects of Lyapunov exponent sign inversions</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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