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<!DOCTYPE html> <html lang="en"> <head> <title>oscJacobian -- create the Jacobian for the oscillator system associated to a graph</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="generation and analysis of oscillator steady states for small graphs" href="index.html">Oscillators</a> :: <a title="create the Jacobian for the oscillator system associated to a graph" href="_osc__Jacobian.html">oscJacobian</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="_osc__Quadrics.html">next</a> | <a href="_is__Stable__Solution.html">previous</a> | <a href="_osc__Quadrics.html">forward</a> | <a href="_is__Stable__Solution.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>oscJacobian -- create the Jacobian for the oscillator system associated to a graph</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">oscJacobian(G,R)</code></dd> <dd><code class="language-macaulay2">oscJacobian(I)</code></dd> <dd><code class="language-macaulay2">oscJacobian(G)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">G</span>, <span>a <a href="../../Graphs/html/___Graph.html">graph</a></span>, an undirected, connected graph $G$</span></li> <li><span><span class="tt">S</span>, <span>a <a title="the class of all rings" href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, created with <a title="create a polynomial ring for a given graph or number of oscillators" href="_osc__Ring.html">oscRing</a></span></li> <li><span><span class="tt">I</span>, <span>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, an ideal, intended to be ideal created by oscSystem</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><span class="tt">Reduced</span><span class="tt"> => </span><span>a <a title="the class of boolean values" href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <span>default value false</span>, if true, then the angles are reduced to the first angle being 0. That is, the number of oscillators will be one less than the number of vertices of $G$</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, the $n \times n$ Jacobian matrix of the system, as a matrix of polynomials involving the cosines and sines of angles $\theta_1, \ldots, \theta_{n-1}$, where we set $\theta_0 = 0$ if $\texttt{Reduced => true}$</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>The matrix is a symmetric $n \times n$ matrix (with determinant zero).</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}}) o1 = Graph{0 => {1, 3}} 1 => {0, 2} 2 => {1, 3} 3 => {0, 2} o1 : Graph</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : oscRing(G, CoefficientRing => CC) o2 = CC [x ..y ] 53 0 3 o2 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : S = oo o3 = S o3 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : I = oscSystem(G,S) o4 = ideal (x y + x y - x y - x y , - x y + x y + x y - x y , - x y + 1 0 3 0 0 1 0 3 1 0 0 1 2 1 1 2 2 1 ------------------------------------------------------------------------ 2 2 2 2 x y + x y - x y , - x y - x y + x y + x y , x + y - 1, x + y - 1 2 3 2 2 3 3 0 3 2 0 3 2 3 0 0 1 1 ------------------------------------------------------------------------ 2 2 2 2 1, x + y - 1, x + y - 1) 2 2 3 3 o4 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : Jac = oscJacobian(G,S) o5 = | -x_0x_1-x_0x_3-y_0y_1-y_0y_3 x_0x_1+y_0y_1 | x_0x_1+y_0y_1 -x_0x_1-x_1x_2-y_0y_1-y_1y_2 | 0 x_1x_2+y_1y_2 | x_0x_3+y_0y_3 0 ------------------------------------------------------------------------ 0 x_0x_3+y_0y_3 | x_1x_2+y_1y_2 0 | -x_1x_2-x_2x_3-y_1y_2-y_2y_3 x_2x_3+y_2y_3 | x_2x_3+y_2y_3 -x_0x_3-x_2x_3-y_0y_3-y_2y_3 | 4 4 o5 : Matrix S <-- S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : assert(det Jac == 0)</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : assert(Jac - transpose Jac == 0)</code></pre> </td> </tr> </table> <div> <p>We can find the eigenvalues of the Jacobian at approximate points, and see if they are stable (all eigenvalues negative, except for the one required 0), unstable (a positive eigenvalue), or semistable (no positive eigenvalues, up to a certain tolerance).</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i8 : realsols = findRealSolutions I warning: some solutions are not regular: {8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 37, 38, 39, 40, 41, 48, 49, 50, 51, 52, 53, 54} o8 = {{-1, 1, 1, 1, 0, 0, 0, 0}, {1, 1, -1, 1, 0, 0, 0, 0}, {1, -1, 1, 1, 0, ------------------------------------------------------------------------ 0, 0, 0}, {1, 1, 1, -1, 0, 0, 0, 0}, {-1, -1, -1, 1, 0, 0, 0, 0}, {1, ------------------------------------------------------------------------ -1, 1, 1, 0, 0, 0, 0}, {1, 1, 1, -1, 0, 0, 0, 0}, {-1, 1, -1, -1, 0, 0, ------------------------------------------------------------------------ 0, 0}, {1, -1, -1, -1, 0, 0, 0, 0}, {-1, -1, -1, 1, 0, 0, 0, 0}, {-1, ------------------------------------------------------------------------ -1, 1, -1, 0, 0, 0, 0}, {-1, 1, -1, -1, 0, 0, 0, 0}, {-1, -1, -1, 1, 0, ------------------------------------------------------------------------ 0, 0, 0}, {1, 1, -1, 1, 0, 0, 0, 0}, {1, 1, 1, -1, 0, 0, 0, 0}, {-1, -1, ------------------------------------------------------------------------ 1, 1, 0, 0, 0, 0}, {-1, -1, 1, 1, 0, 0, 0, 0}, {1, -1, -1, 1, 0, 0, 0, ------------------------------------------------------------------------ 0}, {-1, 1, -1, -1, 0, 0, 0, 0}, {1, -1, -1, -1, 0, 0, 0, 0}, {1, -1, 1, ------------------------------------------------------------------------ 1, 0, 0, 0, 0}, {-1, 1, 1, -1, 0, 0, 0, 0}, {-1, -1, 1, 1, 0, 0, 0, 0}, ------------------------------------------------------------------------ {-1, 1, 1, -1, 0, 0, 0, 0}, {1, -1, -1, 1, 0, 0, 0, 0}, {1, 1, -1, -1, ------------------------------------------------------------------------ 0, 0, 0, 0}, {-1, 1, -1, -1, 0, 0, 0, 0}, {-1, 1, 1, 1, 0, 0, 0, 0}, {1, ------------------------------------------------------------------------ -1, 1, 1, 0, 0, 0, 0}, {1, -1, -1, 1, 0, 0, 0, 0}, {1, 1, -1, -1, 0, 0, ------------------------------------------------------------------------ 0, 0}, {-1, -1, 1, -1, 0, 0, 0, 0}, {-1, -1, -1, 1, 0, 0, 0, 0}} o8 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : jacs = for pt in realsols list sub(Jac, matrix{pt}) o9 = {| 2 -1 0 -1 |, | -2 1 0 1 |, | 0 -1 0 1 |, | 0 1 0 -1 |, | | -1 0 1 0 | | 1 0 -1 0 | | -1 2 -1 0 | | 1 -2 1 0 | | | 0 1 -2 1 | | 0 -1 2 -1 | | 0 -1 0 1 | | 0 1 0 -1 | | | -1 0 1 0 | | 1 0 -1 0 | | 1 0 1 -2 | | -1 0 -1 2 | | ------------------------------------------------------------------------ 0 1 0 -1 |, | 0 -1 0 1 |, | 0 1 0 -1 |, | 0 -1 0 1 |, | 2 1 -2 1 0 | | -1 2 -1 0 | | 1 -2 1 0 | | -1 2 -1 0 | | -1 0 1 0 -1 | | 0 -1 0 1 | | 0 1 0 -1 | | 0 -1 0 1 | | 0 -1 0 -1 2 | | 1 0 1 -2 | | -1 0 -1 2 | | 1 0 1 -2 | | -1 ------------------------------------------------------------------------ -1 0 -1 |, | 0 1 0 -1 |, | -2 1 0 1 |, | 0 -1 0 1 |, | 0 1 0 1 0 | | 1 -2 1 0 | | 1 0 -1 0 | | -1 2 -1 0 | | 1 -2 1 -2 1 | | 0 1 0 -1 | | 0 -1 2 -1 | | 0 -1 0 1 | | 0 1 0 1 0 | | -1 0 -1 2 | | 1 0 -1 0 | | 1 0 1 -2 | | -1 0 ------------------------------------------------------------------------ 0 -1 |, | -2 1 0 1 |, | 0 1 0 -1 |, | 0 1 0 -1 |, | 0 1 0 1 0 | | 1 0 -1 0 | | 1 -2 1 0 | | 1 0 -1 0 | | 1 0 -1 0 -1 | | 0 -1 2 -1 | | 0 1 0 -1 | | 0 -1 0 1 | | 0 -1 0 -1 2 | | 1 0 -1 0 | | -1 0 -1 2 | | -1 0 1 0 | | -1 0 1 ------------------------------------------------------------------------ -1 |, | 0 -1 0 1 |, | 0 -1 0 1 |, | 2 -1 0 -1 |, | 0 -1 0 1 0 | | -1 0 1 0 | | -1 2 -1 0 | | -1 0 1 0 | | -1 2 -1 0 1 | | 0 1 0 -1 | | 0 -1 0 1 | | 0 1 -2 1 | | 0 -1 0 1 0 | | 1 0 -1 0 | | 1 0 1 -2 | | -1 0 1 0 | | 1 0 1 -2 ------------------------------------------------------------------------ |, | 0 -1 0 1 |, | 0 1 0 -1 |, | 0 -1 0 1 |, | 0 -1 0 1 |, | | | -1 0 1 0 | | 1 0 -1 0 | | -1 0 1 0 | | -1 0 1 0 | | | | 0 1 0 -1 | | 0 -1 0 1 | | 0 1 0 -1 | | 0 1 0 -1 | | | | 1 0 -1 0 | | -1 0 1 0 | | 1 0 -1 0 | | 1 0 -1 0 | | ------------------------------------------------------------------------ 0 1 0 -1 |, | 0 -1 0 1 |, | 2 -1 0 -1 |, | 0 -1 0 1 |, | 0 1 0 -1 0 | | -1 2 -1 0 | | -1 0 1 0 | | -1 2 -1 0 | | -1 0 -1 0 1 | | 0 -1 0 1 | | 0 1 -2 1 | | 0 -1 0 1 | | 0 -1 0 1 0 | | 1 0 1 -2 | | -1 0 1 0 | | 1 0 1 -2 | | 1 ------------------------------------------------------------------------ -1 0 1 |, | 0 1 0 -1 |, | -2 1 0 1 |, | 0 1 0 -1 |} 0 1 0 | | 1 0 -1 0 | | 1 0 -1 0 | | 1 -2 1 0 | 1 0 -1 | | 0 -1 0 1 | | 0 -1 2 -1 | | 0 1 0 -1 | 0 -1 0 | | -1 0 1 0 | | 1 0 -1 0 | | -1 0 -1 2 | o9 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : jacs/eigenvalues o10 = {{2.82843 }, {-2.82843 }, {-2.82843 }, {2.82843 }, {-1.19387e-16} {1.19387e-16 } {2.82843 } {-2.82843 } {-2.82843 } {2.82843 } {3.92523e-17} {-3.92523e-17} {2.46519e-32 } {-2.46519e-32} {7.22038e-33} {-7.22038e-33} ----------------------------------------------------------------------- {2.82843 }, {-2.82843 }, {2.82843 }, {-2.82843 }, {-3.93452e-16} {3.93452e-16} {-3.93452e-16} {3.93452e-16} {-2.82843 } {2.82843 } {-2.82843 } {2.82843 } {0 } {0 } {0 } {0 } ----------------------------------------------------------------------- {2.82843 }, {2.82843 }, {-2.82843 }, {-2.82843 }, {-1.19387e-16} {-2.82843 } {1.19387e-16 } {2.82843 } {-2.82843 } {-3.92523e-17} {2.82843 } {3.92523e-17} {2.46519e-32 } {-7.22038e-33} {-2.46519e-32} {7.22038e-33} ----------------------------------------------------------------------- {2.82843 }, {-2.82843 }, {2.82843 }, {2 }, {-3.93452e-16} {3.14062e-17} {-2.82843 } {-6.4281e-17} {-2.82843 } {2.82843 } {1.46388e-16 } {-2 } {0 } {0 } {-4.21003e-17} {0 } ----------------------------------------------------------------------- {2 }, {2 }, {-2.82843 }, {2.82843 }, {-6.4281e-17} {-6.4281e-17} {2.82843 } {-3.14062e-17} {-2 } {-2 } {-1.46388e-16} {-2.82843 } {0 } {0 } {4.21003e-17 } {0 } ----------------------------------------------------------------------- {-2.82843 }, {2 }, {2 }, {2 }, {3.93452e-16} {-6.4281e-17} {-6.4281e-17} {-6.4281e-17} {2.82843 } {-2 } {-2 } {-2 } {0 } {0 } {0 } {0 } ----------------------------------------------------------------------- {2 }, {2 }, {-2.82843 }, {2.82843 }, {-6.4281e-17} {-6.4281e-17} {3.93452e-16} {-3.14062e-17} {-2 } {-2 } {2.82843 } {-2.82843 } {0 } {0 } {0 } {0 } ----------------------------------------------------------------------- {-2.82843 }, {2 }, {2 }, {-2.82843 }, {2.82843 } {-2 } {2.84023e-16} {3.14062e-17} {-1.46388e-16} {1.74515e-24 } {-2 } {2.82843 } {4.21003e-17 } {-1.74515e-24} {0 } {0 } ----------------------------------------------------------------------- {2.82843 }} {-2.82843 } {1.46388e-16 } {-4.21003e-17} o10 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : jacs/eigenvalues/identifyStability o11 = {Unstable, Unstable, Unstable, Unstable, Unstable, Unstable, Unstable, ----------------------------------------------------------------------- Unstable, Unstable, Unstable, Unstable, Unstable, Unstable, Unstable, ----------------------------------------------------------------------- Unstable, Unstable, Unstable, Unstable, Unstable, Unstable, Unstable, ----------------------------------------------------------------------- Unstable, Unstable, Unstable, Unstable, Unstable, Unstable, Unstable, ----------------------------------------------------------------------- Unstable, Unstable, Unstable, Unstable, Unstable} o11 : List</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="create a polynomial ring for a given graph or number of oscillators" href="_osc__Ring.html">oscRing</a> -- create a polynomial ring for a given graph or number of oscillators</span></li> <li><span><a title="the ideal of the reduced equilibrium points of a dynamical system of oscillators" href="_osc__System.html">oscSystem</a> -- the ideal of the reduced equilibrium points of a dynamical system of oscillators</span></li> <li><span><a title="find real solutions, at least one per component for well-conditioned systems" href="_find__Real__Solutions.html">findRealSolutions</a> -- find real solutions, at least one per component for well-conditioned systems</span></li> <li><span><a title="Identify the stability of a list of eigenvalues, or of potential solutions to the oscillator system" href="_identify__Stability.html">identifyStability</a> -- Identify the stability of a list of eigenvalues, or of potential solutions to the oscillator system</span></li> <li><span><a title="find eigenvalues of a matrix" href="../../Macaulay2Doc/html/_eigenvalues.html">eigenvalues</a> -- find eigenvalues of a matrix</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">oscJacobian</span>:</h2> <ul> <li><kbd>oscJacobian(Graph)</kbd></li> <li><kbd>oscJacobian(Graph,Ring)</kbd></li> <li><kbd>oscJacobian(Ideal)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="create the Jacobian for the oscillator system associated to a graph" href="_osc__Jacobian.html">oscJacobian</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Oscillators/Documentation.m2:318:0</span>.</p> </div> </div> </div> </body> </html>
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