One Hat Cyber Team
Your IP :
216.73.216.115
Server IP :
194.44.31.54
Server :
Linux zen.imath.kiev.ua 4.18.0-553.77.1.el8_10.x86_64 #1 SMP Fri Oct 3 14:30:23 UTC 2025 x86_64
Server Software :
Apache/2.4.37 (Rocky Linux) OpenSSL/1.1.1k
PHP Version :
5.6.40
Buat File
|
Buat Folder
Eksekusi
Dir :
~
/
usr
/
share
/
doc
/
Macaulay2
/
Oscillators
/
html
/
Edit File:
_osc__System.html
<!DOCTYPE html> <html lang="en"> <head> <title>oscSystem -- the ideal of the reduced equilibrium points of a dynamical system of oscillators</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="the ideal of the reduced equilibrium points of a dynamical system of oscillators" href="_osc__System.html">oscSystem</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="___S__C__T_spgraphs_spwith_spexotic_spsolutions.html">next</a> | <a href="_osc__Ring.html">previous</a> | <a href="___S__C__T_spgraphs_spwith_spexotic_spsolutions.html">forward</a> | <a href="_osc__Ring.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>oscSystem -- the ideal of the reduced equilibrium points of a dynamical system of oscillators</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">oscSystem(G,R)</code></dd> <dd><code class="language-macaulay2">oscSystem(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">R</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> </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>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, in the ring R</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>$R$ should be a ring created with <a title="create a polynomial ring for a given graph or number of oscillators" href="_osc__Ring.html">oscRing</a>. The dynamical system involved is the oscillator system associated to $G$: one angle per vertex. If $a_{ij} = 1$ if $(i,j)$ is an edge of the undirected graph $G$, and is zero otherwise, then the system is $d\theta_i/dt = \sum_j a_{ij} \sin(\theta_j - \theta_i)$ where we consider only reduced equilibrium solutions $\theta_0 = 0$.</p> <p>This function returns the ideal of equilibrium points, where angles $(0, \theta_1, ..., \theta_{n-1})$ are represented via cosines and sines of the angles.</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 : R = oo o3 = R o3 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : I = oscSystem(G,R) 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 R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : netList I_* +---------------------------+ o5 = |x y + x y - x y - x y | | 1 0 3 0 0 1 0 3 | +---------------------------+ |- x y + x y + x y - x y | | 1 0 0 1 2 1 1 2| +---------------------------+ |- x y + x y + x y - x y | | 2 1 1 2 3 2 2 3| +---------------------------+ |- x y - x y + x y + x y | | 3 0 3 2 0 3 2 3| +---------------------------+ | 2 2 | |x + y - 1 | | 0 0 | +---------------------------+ | 2 2 | |x + y - 1 | | 1 1 | +---------------------------+ | 2 2 | |x + y - 1 | | 2 2 | +---------------------------+ | 2 2 | |x + y - 1 | | 3 3 | +---------------------------+</code></pre> </td> </tr> </table> <div> <p>We can find approximations to the 26 complex solutions to this system. If the system has positive dimension (not the case here), the idea is that this set of points should contain at least one on each component.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i6 : solveSystem I_* o6 = {{-.992345+.03175*ii, .992345-.03175*ii, -.992345+.03175*ii, ------------------------------------------------------------------------ .992345-.03175*ii, -.201665-.156233*ii, .201665+.156233*ii, ------------------------------------------------------------------------ -.201665-.156233*ii, .201665+.156233*ii}, {.996984-.001895*ii, ------------------------------------------------------------------------ .996984-.001895*ii, .996984-.001895*ii, .996984-.001895*ii, ------------------------------------------------------------------------ -.0810557-.0233023*ii, -.0810557-.0233023*ii, -.0810557-.0233023*ii, ------------------------------------------------------------------------ -.0810557-.0233023*ii}, {-.996984+.001895*ii, .996984-.001895*ii, ------------------------------------------------------------------------ -.996984+.001895*ii, .996984-.001895*ii, -.0810557-.0233023*ii, ------------------------------------------------------------------------ .0810557+.0233023*ii, -.0810557-.0233023*ii, .0810557+.0233023*ii}, ------------------------------------------------------------------------ {.992345-.03175*ii, .992345-.03175*ii, .992345-.03175*ii, ------------------------------------------------------------------------ .992345-.03175*ii, -.201665-.156233*ii, -.201665-.156233*ii, ------------------------------------------------------------------------ -.201665-.156233*ii, -.201665-.156233*ii}, {.996984-.001895*ii, ------------------------------------------------------------------------ -.996984+.001895*ii, .996984-.001895*ii, -.996984+.001895*ii, ------------------------------------------------------------------------ .0810557+.0233023*ii, -.0810557-.0233023*ii, .0810557+.0233023*ii, ------------------------------------------------------------------------ -.0810557-.0233023*ii}, {-.992345+.03175*ii, -.992345+.03175*ii, ------------------------------------------------------------------------ -.992345+.03175*ii, -.992345+.03175*ii, .201665+.156233*ii, ------------------------------------------------------------------------ .201665+.156233*ii, .201665+.156233*ii, .201665+.156233*ii}, ------------------------------------------------------------------------ {-.996984+.001895*ii, -.996984+.001895*ii, -.996984+.001895*ii, ------------------------------------------------------------------------ -.996984+.001895*ii, .0810557+.0233023*ii, .0810557+.0233023*ii, ------------------------------------------------------------------------ .0810557+.0233023*ii, .0810557+.0233023*ii}, {.992345-.03175*ii, ------------------------------------------------------------------------ -.992345+.03175*ii, .992345-.03175*ii, -.992345+.03175*ii, ------------------------------------------------------------------------ .201665+.156233*ii, -.201665-.156233*ii, .201665+.156233*ii, ------------------------------------------------------------------------ -.201665-.156233*ii}, (-1, 1, 1, 1, 1.18306e-12-1.29026e-12*ii, ------------------------------------------------------------------------ -3.29597e-17-7.45931e-17*ii, 1.18307e-12-1.29022e-12*ii, ------------------------------------------------------------------------ 4.68375e-17+1.29237e-16*ii), (1, 1, -1, 1, 1.18306e-12-1.29026e-12*ii, ------------------------------------------------------------------------ 3.29597e-17+7.45931e-17*ii, 1.18307e-12-1.29022e-12*ii, ------------------------------------------------------------------------ -4.68375e-17-1.29237e-16*ii), (1, -1, 1, 1, 1.31991e-12+4.4264e-13*ii, ------------------------------------------------------------------------ 4.75125e-14+2.70308e-14*ii, -1.41494e-12-4.96702e-13*ii, ------------------------------------------------------------------------ -4.75125e-14-2.70309e-14*ii), (1, 1, 1, -1, -1.31991e-12-4.4264e-13*ii, ------------------------------------------------------------------------ 4.75125e-14+2.70308e-14*ii, 1.41494e-12+4.96702e-13*ii, ------------------------------------------------------------------------ -4.75125e-14-2.70309e-14*ii), (-1, -1, -1, 1, ------------------------------------------------------------------------ 6.92389e-14+4.46618e-14*ii, 6.9238e-14+4.46689e-14*ii, ------------------------------------------------------------------------ 6.9238e-14+4.46557e-14*ii, -6.9238e-14-4.46689e-14*ii), (1, -1, 1, 1, ------------------------------------------------------------------------ -3.53415e-14-2.04511e-14*ii, 3.53433e-14+2.04539e-14*ii, ------------------------------------------------------------------------ -3.53454e-14-2.04576e-14*ii, -3.5345e-14-2.04575e-14*ii), (1, 1, 1, -1, ------------------------------------------------------------------------ -6.92389e-14-4.46618e-14*ii, -6.9238e-14-4.46689e-14*ii, ------------------------------------------------------------------------ -6.9238e-14-4.46557e-14*ii, 6.9238e-14+4.46689e-14*ii), (-1, 1, -1, -1, ------------------------------------------------------------------------ 3.50926e-14+2.09739e-14*ii, -3.50943e-14-2.09776e-14*ii, ------------------------------------------------------------------------ 3.50948e-14+2.09807e-14*ii, 3.50943e-14+2.0981e-14*ii), (1, -1, -1, -1, ------------------------------------------------------------------------ -1.18306e-12+1.29026e-12*ii, 3.29597e-17+7.45931e-17*ii, ------------------------------------------------------------------------ -1.18307e-12+1.29022e-12*ii, -4.68375e-17-1.29237e-16*ii), (-1, -1, -1, ------------------------------------------------------------------------ 1, 1.31991e-12+4.4264e-13*ii, -4.75125e-14-2.70308e-14*ii, ------------------------------------------------------------------------ -1.41494e-12-4.96702e-13*ii, 4.75125e-14+2.70309e-14*ii), (-1, -1, 1, ------------------------------------------------------------------------ -1, -1.18306e-12+1.29026e-12*ii, -3.29597e-17-7.45931e-17*ii, ------------------------------------------------------------------------ -1.18307e-12+1.29022e-12*ii, 4.68375e-17+1.29237e-16*ii), (-1, 1, -1, ------------------------------------------------------------------------ -1, -1.31991e-12-4.4264e-13*ii, -4.75125e-14-2.70308e-14*ii, ------------------------------------------------------------------------ 1.41494e-12+4.96702e-13*ii, 4.75125e-14+2.70309e-14*ii), ------------------------------------------------------------------------ {-2.04165e-12+4.20039e-12*ii, -1.41421, 2.04165e-12-4.20037e-12*ii, ------------------------------------------------------------------------ 1.41421, -1, ii, 1, ii}, {-1.41421, 1.9403e-13+1.98468e-13*ii, 1.41421, ------------------------------------------------------------------------ -1.94022e-13-1.98477e-13*ii, -ii, -1, -ii, 1}, (-1, -1, -1, 1, ------------------------------------------------------------------------ -1.86483e-17*ii, -1.86483e-17*ii, -1.82146e-17*ii, 1.86483e-17*ii), (1, ------------------------------------------------------------------------ 1, -1, 1, -4.6557e-15+3.22738e-14*ii, -1.09181e-14-9.90581e-15*ii, ------------------------------------------------------------------------ 4.65592e-15-3.2274e-14*ii, 1.5979e-15+7.44515e-14*ii), (1, 1, 1, -1, ------------------------------------------------------------------------ -3.26995e-13-8.62092e-13*ii, -3.40408e-13-4.93195e-13*ii, ------------------------------------------------------------------------ -3.5382e-13-1.24295e-13*ii, 3.40408e-13+4.93195e-13*ii), (-1, -1, 1, 1, ------------------------------------------------------------------------ -6.91634e-14-4.4691e-14*ii, 6.92762e-14+4.4755e-14*ii, ------------------------------------------------------------------------ 6.91643e-14+4.46904e-14*ii, -6.92762e-14-4.4755e-14*ii), (-1, -1, 1, 1, ------------------------------------------------------------------------ -4.33681e-17+3.72966e-17*ii, 1.21431e-17-1.05384e-16*ii, ------------------------------------------------------------------------ 4.33681e-17-3.59955e-17*ii, -1.21431e-17+1.02782e-16*ii), (1, -1, -1, 1, ------------------------------------------------------------------------ -6.91374e-14-4.50054e-14*ii, -6.93213e-14-4.47682e-14*ii, ------------------------------------------------------------------------ 6.91409e-14+4.50083e-14*ii, 6.93291e-14+4.4768e-14*ii), (-1, 1, -1, -1, ------------------------------------------------------------------------ -3.26995e-13-8.62092e-13*ii, 3.40408e-13+4.93195e-13*ii, ------------------------------------------------------------------------ -3.5382e-13-1.24295e-13*ii, -3.40408e-13-4.93195e-13*ii), (1, -1, -1, ------------------------------------------------------------------------ -1, 4.6557e-15-3.22738e-14*ii, -1.09181e-14-9.90581e-15*ii, ------------------------------------------------------------------------ -4.65592e-15+3.2274e-14*ii, 1.5979e-15+7.44515e-14*ii), (1, -1, 1, 1, ------------------------------------------------------------------------ 6.93178e-14+4.46583e-14*ii, -6.93178e-14-4.4665e-14*ii, ------------------------------------------------------------------------ 6.93178e-14+4.46511e-14*ii, 6.93178e-14+4.4665e-14*ii), ------------------------------------------------------------------------ {5.1318e-12+3.74496e-12*ii, -1.41421, -5.13181e-12-3.74495e-12*ii, ------------------------------------------------------------------------ 1.41421, 1, ii, -1, ii}, {1.41421, -5.38406e-12+5.14141e-12*ii, ------------------------------------------------------------------------ -1.41421, 5.38407e-12-5.1414e-12*ii, -ii, -1, -ii, 1}, ------------------------------------------------------------------------ {2.04165e-12-4.20039e-12*ii, -1.41421, -2.04165e-12+4.20037e-12*ii, ------------------------------------------------------------------------ 1.41421, -1, -ii, 1, -ii}, {-1.41421, -5.38406e-12+5.14141e-12*ii, ------------------------------------------------------------------------ 1.41421, 5.38407e-12-5.1414e-12*ii, -ii, 1, -ii, -1}, {-1.41421, ------------------------------------------------------------------------ 5.9116e-12-3.61391e-12*ii, 1.41421, -5.91161e-12+3.61393e-12*ii, ii, -1, ------------------------------------------------------------------------ ii, 1}, {-6.09525e-12-1.32367e-12*ii, 1.41421, ------------------------------------------------------------------------ 6.09525e-12+1.3237e-12*ii, -1.41421, -1, ii, 1, ii}, (-1, 1, 1, -1, ------------------------------------------------------------------------ -2.16634e-13-7.6029e-14*ii, 2.16695e-13+7.61439e-14*ii, ------------------------------------------------------------------------ 2.16634e-13+7.6029e-14*ii, -2.16695e-13-7.61439e-14*ii), (-1, -1, 1, 1, ------------------------------------------------------------------------ 4.33238e-13+1.52107e-13*ii, 4.33263e-13+1.52094e-13*ii, ------------------------------------------------------------------------ -4.33238e-13-1.52107e-13*ii, -4.33263e-13-1.52094e-13*ii), (-1, 1, 1, ------------------------------------------------------------------------ -1, -1.38778e-17-3.46945e-18*ii, 1.38778e-17+5.20417e-18*ii, ------------------------------------------------------------------------ 1.38778e-17+3.03577e-18*ii, -1.38778e-17-4.77049e-18*ii), (1, -1, -1, 1, ------------------------------------------------------------------------ 4.44517e-15+2.60321e-15*ii, -4.44512e-15-2.60359e-15*ii, ------------------------------------------------------------------------ -4.44517e-15-2.60321e-15*ii, 4.44512e-15+2.60359e-15*ii), (1, 1, -1, -1, ------------------------------------------------------------------------ -4.44517e-15-2.60322e-15*ii, -4.44512e-15-2.60359e-15*ii, ------------------------------------------------------------------------ 4.44517e-15+2.60322e-15*ii, 4.44512e-15+2.60359e-15*ii), ------------------------------------------------------------------------ {-1.31172e-12+2.43682e-12*ii, 1.41421, 1.31171e-12-2.43681e-12*ii, ------------------------------------------------------------------------ -1.41421, 1, ii, -1, ii}, {1.41421, 5.38406e-12-5.14141e-12*ii, ------------------------------------------------------------------------ -1.41421, -5.38407e-12+5.1414e-12*ii, ii, -1, ii, 1}, {1.41421, ------------------------------------------------------------------------ 1.9403e-13+1.98468e-13*ii, -1.41421, -1.94022e-13-1.98477e-13*ii, -ii, ------------------------------------------------------------------------ 1, -ii, -1}, {6.09525e-12+1.32367e-12*ii, -1.41421, ------------------------------------------------------------------------ -6.09525e-12-1.3237e-12*ii, 1.41421, 1, -ii, -1, -ii}, ------------------------------------------------------------------------ {6.09525e-12+1.32367e-12*ii, 1.41421, -6.09525e-12-1.3237e-12*ii, ------------------------------------------------------------------------ -1.41421, -1, -ii, 1, -ii}, {-1.41421, 5.38406e-12-5.14141e-12*ii, ------------------------------------------------------------------------ 1.41421, -5.38407e-12+5.1414e-12*ii, ii, 1, ii, -1}, (-1, 1, -1, -1, ------------------------------------------------------------------------ -6.93178e-14-4.46583e-14*ii, 6.93178e-14+4.4665e-14*ii, ------------------------------------------------------------------------ -6.93178e-14-4.46511e-14*ii, -6.93178e-14-4.4665e-14*ii), (-1, 1, 1, 1, ------------------------------------------------------------------------ -4.6557e-15+3.22738e-14*ii, 1.09181e-14+9.90581e-15*ii, ------------------------------------------------------------------------ 4.65592e-15-3.2274e-14*ii, -1.5979e-15-7.44515e-14*ii), (1, -1, 1, 1, ------------------------------------------------------------------------ 3.26995e-13+8.62092e-13*ii, -3.40408e-13-4.93195e-13*ii, ------------------------------------------------------------------------ 3.5382e-13+1.24295e-13*ii, 3.40408e-13+4.93195e-13*ii), (1, -1, -1, 1, ------------------------------------------------------------------------ 5.5258e-13-9.09342e-12*ii, -4.7102e-13-7.4734e-12*ii, ------------------------------------------------------------------------ -5.5258e-13+9.09342e-12*ii, 4.7102e-13+7.4734e-12*ii), (1, 1, -1, -1, ------------------------------------------------------------------------ -3.16495e-13+7.53139e-13*ii, -2.32767e-12-2.29808e-13*ii, ------------------------------------------------------------------------ 3.16495e-13-7.53139e-13*ii, 2.32767e-12+2.29807e-13*ii), (-1, -1, 1, -1, ------------------------------------------------------------------------ 4.6557e-15-3.22738e-14*ii, 1.09181e-14+9.90581e-15*ii, ------------------------------------------------------------------------ -4.65592e-15+3.2274e-14*ii, -1.5979e-15-7.44515e-14*ii), (-1, -1, -1, 1, ------------------------------------------------------------------------ 3.26995e-13+8.62092e-13*ii, 3.40408e-13+4.93195e-13*ii, ------------------------------------------------------------------------ 3.5382e-13+1.24295e-13*ii, -3.40408e-13-4.93195e-13*ii), {1.41421, ------------------------------------------------------------------------ -1.9403e-13-1.98468e-13*ii, -1.41421, 1.94022e-13+1.98477e-13*ii, ii, 1, ------------------------------------------------------------------------ ii, -1}, {2.04165e-12-4.20039e-12*ii, 1.41421, ------------------------------------------------------------------------ -2.04165e-12+4.20037e-12*ii, -1.41421, 1, -ii, -1, -ii}} o6 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : #oo o7 = 57</code></pre> </td> </tr> </table> <div> <p>We can find approximations to the 6 real solutions to this system.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i8 : findRealSolutions I warning: some solutions are not regular: {6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 37, 38, 39, 40, 41, 42, 43, 44, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60} 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}, {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 : #oo o9 = 41</code></pre> </td> </tr> </table> <div> <p>The angles of these solutions (in degrees, not radians, and the 3 refers to the numbner of oscillators).</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i10 : netList getAngles(3, findRealSolutions I, Radians=>false) warning: some solutions are not regular: {8, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 27, 28, 29, 32, 33, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 61, 62, 63, 64, 67, 68, 69} +---+---+---+ o10 = |315|180|180| +---+---+---+ |45 |0 |180| +---+---+---+ |135|180|180| +---+---+---+ |135|0 |0 | +---+---+---+ |45 |180|0 | +---+---+---+ |315|0 |180| +---+---+---+ |315|0 |0 | +---+---+---+ |225|180|0 | +---+---+---+ |225|0 |180| +---+---+---+ |315|180|180| +---+---+---+ |135|180|0 | +---+---+---+ |225|0 |180| +---+---+---+ |45 |0 |180| +---+---+---+ |135|180|180| +---+---+---+ |135|180|0 | +---+---+---+ |135|180|0 | +---+---+---+ |225|0 |0 | +---+---+---+ |45 |180|180| +---+---+---+ |45 |180|0 | +---+---+---+ |225|0 |180| +---+---+---+ |135|0 |0 | +---+---+---+ |315|0 |0 | +---+---+---+ |45 |0 |180| +---+---+---+ |315|180|180| +---+---+---+ |135|180|180| +---+---+---+ |225|180|0 | +---+---+---+ |45 |180|180| +---+---+---+ |225|0 |0 | +---+---+---+ |135|180|0 | +---+---+---+ |45 |180|180| +---+---+---+ |315|0 |180| +---+---+---+ |45 |0 |180| +---+---+---+ |135|180|180| +---+---+---+ |225|180|0 | +---+---+---+ |135|0 |0 | +---+---+---+ |315|0 |0 | +---+---+---+ |45 |180|0 | +---+---+---+ |315|180|180| +---+---+---+ |225|0 |180| +---+---+---+ |225|0 |0 | +---+---+---+ |45 |180|180| +---+---+---+ |315|0 |180| +---+---+---+ |315|0 |180| +---+---+---+ |315|0 |0 | +---+---+---+ |225|180|0 | +---+---+---+ |45 |180|0 | +---+---+---+</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="create the Jacobian for the oscillator system associated to a graph" href="_osc__Jacobian.html">oscJacobian</a> -- create the Jacobian for the oscillator system associated to a graph</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="Compute angles from a list of solutions" href="_get__Angles.html">getAngles</a> -- Compute angles from a list of solutions</span></li> <li><span><a title="solve a system of polynomial equations" href="../../NumericalAlgebraicGeometry/html/_solve__System_lp__List_rp.html">solveSystem</a> -- solve a system of polynomial equations</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">oscSystem</span>:</h2> <ul> <li><kbd>oscSystem(Graph)</kbd></li> <li><kbd>oscSystem(Graph,Ring)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="the ideal of the reduced equilibrium points of a dynamical system of oscillators" href="_osc__System.html">oscSystem</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.m2:263:0</span>.</p> </div> </div> </div> </body> </html>
Simpan