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width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/10268/#Item_6" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#literature'>Literature</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#categorytheoretic_formulations'>Category-theoretic formulations</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>A <strong>dynamical system</strong> is a <a class="existingWikiWord" href="/nlab/show/space">space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (often a bare <a class="existingWikiWord" href="/nlab/show/set">set</a> or a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>) together with a “law of motion” expressed by the <a class="existingWikiWord" href="/nlab/show/action">action</a> of a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>×</mo><mi>X</mi><mo>⟶</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A \times X \longrightarrow X \,. </annotation></semantics></math></div> <p>To model continuous <a class="existingWikiWord" href="/nlab/show/time">time</a>-evolution one may take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">(</mo><mi>ℝ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = (\mathbb{R}, +)</annotation></semantics></math> to be the additive <a class="existingWikiWord" href="/nlab/show/group">group</a> of <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>.</p> <p>For discrete time evolution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = (\mathbb{Z}, +)</annotation></semantics></math> is the additive group of <a class="existingWikiWord" href="/nlab/show/integers">integers</a> (or just the monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℕ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{N}, +)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>). In this case the law of motion is often given by an <a class="existingWikiWord" href="/nlab/show/initial+value+problem">initial value problem</a> for a <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a> “of evolution type”.</p> <p>In this case, the dynamical system is equivalently a space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> equipped with an <a class="existingWikiWord" href="/nlab/show/automorphism">automorphism</a> (this being the action of the <a class="existingWikiWord" href="/nlab/show/unit+element">unit element</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>, sometimes called the “shift”).</p> <p>Sometimes the evolution is only partially defined; this is most often in dynamical systems induced by evolution differential equations which do not necessarily have existence of solutions for arbitrary large time, or the dynamical system is defined only for nonnegative time.</p> <p>The definition evidently makes sense quite generally <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> various ambient <a class="existingWikiWord" href="/nlab/show/categories">categories</a>: For instance one considers complex dynamics, <a class="existingWikiWord" href="/nlab/show/algebraic+dynamics">algebraic dynamics</a>, arithmetic dynamics and so on.</p> <p>Dynamical systems are used to describe not only physical motions but also the behaviour of parameters of various systems, e.g. in sociological, financial, weather and other models.</p> <h2 id="literature">Literature</h2> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+systems+theory">categorical systems theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/zeta+function+of+a+dynamical+system">zeta function of a dynamical system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+topology">arithmetic topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+dynamical+systems">cohomology of dynamical systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chaos">chaos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Furstenberg-Zimmer+structure+theorem">Furstenberg-Zimmer structure theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/shifts">shifts</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <ul> <li>Boris Hasselblatt, Anatole Katok, <em>Handbook of dynamical systems</em>, <a href="http://www.math.psu.edu/katok_a/handbook.html">contents</a></li> </ul> <p>See also</p> <ul> <li>Wikipedia <a href="http://en.wikipedia.org/wiki/Dynamical_system">dynamical system</a>, <a href="http://en.wikipedia.org/wiki/Autonomous_system_%28mathematics%29">autonomous system</a></li> </ul> <h3 id="categorytheoretic_formulations">Category-theoretic formulations</h3> <p>Discussion of dynamical systems in terms of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> (see also at <em><a class="existingWikiWord" href="/nlab/show/categorical+systems+theory">categorical systems theory</a></em>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/William+Lawvere">William Lawvere</a>, <a href="http://www.ima.umn.edu/preprints/Functional-Remarks-General-Concept-Chaos"><em>Functorial Remarks on the General Concept of Chaos</em></a>, IMA reprint 87, 1984 (<a href="http://www.ima.umn.edu/sites/default/files/87s.pdf">pdf</a>)</p> </li> <li> <p>George Dimitrov, Fabian Haiden, <a class="existingWikiWord" href="/nlab/show/Ludmil+Katzarkov">Ludmil Katzarkov</a>, <a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, <em>Dynamical systems and categories</em>, [<a href="http://arxiv.org/abs/1307.8418">arxiv/1307.8418</a>]</p> </li> <li> <p>Mike Behrisch, Sebastian Kerkhoff, Reinhard Pöschel, Friedrich Martin Schneider, Stefan Siegmund, <em>Dynamical systems in categories</em>. Applied Categorical Structures 25, 2017. (<a href="https://tud.qucosa.de/api/qucosa%3A27340/attachment/ATT-0/">link</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Barr">Michael Barr</a>, <a class="existingWikiWord" href="/nlab/show/John+Kennison">John Kennison</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Raphael">Robert Raphael</a>, <em>Flows: cocyclic and almost cocyclic</em>, Theory Appl. Categories <strong>25</strong> 18 (2010) 490–507 [<a href="http://www.tac.mta.ca/tac/volumes/25/18/25-18abs.html">tac:25-18</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Patrick+Schultz">Patrick Schultz</a>, <a class="existingWikiWord" href="/nlab/show/David+I.+Spivak">David I. Spivak</a>, <a class="existingWikiWord" href="/nlab/show/Christina+Vasilakopoulou">Christina Vasilakopoulou</a>, <em>Dynamical Systems and Sheaves</em>, Appl Categor Struct <strong>28</strong> (2020) 1–57 [<a href="https://arxiv.org/abs/1609.08086">arXiv:1609.08086</a>, <a href="https://doi.org/10.1007/s10485-019-09565-x">doi:10.1007/s10485-019-09565-x</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/David+Jaz+Myers">David Jaz Myers</a>, <em>Double Categories of Open Dynamical Systems</em>, EPTCS <strong>333</strong> (2021) 154-167 [<a href="https://arxiv.org/abs/2005.05956">arXiv:2005.05956</a>, <a href="https://doi.org/10.4204/EPTCS.333.11">doi:10.4204/EPTCS.333.11</a>]</p> </li> </ul> <p>Monograph:</p> <ul> <li id="Myers"><a class="existingWikiWord" href="/nlab/show/David+Jaz+Myers">David Jaz Myers</a>, <em>Categorical systems theory</em>, book project [<a href="https://github.com/DavidJaz/DynamicalSystemsBook/tree/master/book">github</a>, <a href="http://davidjaz.com/Papers/DynamicalBook.pdf">pdf</a>]</li> </ul> <p>Exposition:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/David+Jaz+Myers">David Jaz Myers</a>, <em><a href="https://topos.site/blog/2021/11/categorical-systems-theory/">Categorical systems theory</a></em>, <a href="https://topos.site/blog/">Topos Institute Blog</a> (Nov 2021)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/David+Jaz+Myers">David Jaz Myers</a>, <em>Double Categories of Dynamical Systems</em> (2020) [<a href="http://davidjaz.com/Talks/DJM_Dyn2020.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/David+Jaz+Myers">David Jaz Myers</a>, <em>A general definition of open dynamical systems</em>, talk at <em>MIT Category Theory Seminar</em> (2020) [video:<a href="https://youtu.be/8T-Km3taNko">YT</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 23, 2024 at 11:13:00. 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