// Numbas version: exam_results_page_options
{"name": "Dynamical system 6:Centre.", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["xr", "yr", "v", "f"], "name": "Dynamical system 6:Centre.", "tags": ["centres", "differential equations", "Differential equations", "dynamical system", "feedback", "fixed points", "interactive", "Jsxgraph", "jsxgraph", "navigation change", "phase space", "stable", "systems of differential equations", "unstable"], "preamble": {"css": "", "js": "var pc;\nvar brd;\nvar P1;\nvar brd1;\nvar g1;\nvar g2;\nvar fin;\nvar m;\nvar adv;\nfunction updateBoards(a,b,c,d) {\n  function f(x,yy) {\n    var y1 = yy[0];\n    var y2 = yy[1];\n    var z1 = a*y1+b*y2;\n    var z2 = c*y1+d*y2;\n    return [z1,z2];\n  }\n  \n  function ode() {\n    //solution curve data array produced in the interval between 0 and 10, 200 steps\n    return JXG.Math.Numerics.rungeKutta('heun', [P1.Y(),P1.X()], [0, 10], 500, f);\n  }\n  function ode1() {\n    //solution curve data array produced in the interval between 0 and -10, 200 steps\n    return JXG.Math.Numerics.rungeKutta('heun', [P1.Y(),P1.X()], [0, -10], 500, f);\n  }\n  \n  pc.updateDataArray = function() {\n    var data = ode();\n    var l=data.length;\n    var data1=ode1();\n    var l1=data1.length;\n    this.dataX = [];\n    this.dataY = [];\n    for(var i=0; i<l1; i++) {\n      this.dataX[i] = data1[l1-i-1][1];\n      this.dataY[i] = data1[l1-i-1][0];\n    }\n    for(var i=l1; i<l+l1; i++) {\n      this.dataX[i] = data[i-l1][1];\n      this.dataY[i] = data[i-l1][0];\n    }\n  };\n  brd.update();\n  \n  g1.updateDataArray = function() {\n    var h = 10/500;\n    var data = ode();\n    var l=data.length;\n    var data1=ode1();\n    var l1=data1.length;\n    this.dataX = [];\n    this.dataY = [];\n    for(var i=0; i<l1; i++) {\n      this.dataX[i] = -10+i*h;\n      this.dataY[i] = data1[l1-i-1][0];\n    }\n    for(var i=l1; i<l+l1; i++) {\n      this.dataX[i] = (i-l1)*h;\n      this.dataY[i] = data[i-l1][0];\n    }\n  };\n  g2.updateDataArray = function() {\n    var h = 10/500;\n    var data = ode();\n    var l=data.length;\n    var data1=ode1();\n    var l1=data1.length;\n    this.dataX = [];\n    this.dataY = [];\n    for(var i=0; i<l1; i++) {\n      this.dataX[i] = -10+i*h;\n      this.dataY[i] = data1[l1-i-1][1];\n    }\n    for(var i=l1; i<l+l1; i++) {\n      this.dataX[i] = (i-l1)*h;\n      this.dataY[i] = data[i-l1][1];\n    }\n  };\n  brd1.update();\n}\n\nquestion.signals.on('HTMLAttached',function() {\n  \n  var div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',{showNavigation:true,axis:true, boundingbox:[-50,50,50,-50],showNavigation:true});\n  brd=div.board;\n  //initial values at t=0 for x and y\n  P1 = brd.create('point',[-5,5], {name:''});\n  //dragging P1 around changes the initial values and the solution curve is redrawn.\n  function showCoords(){\n    var xval=Numbas.math.niceNumber(P1.Y());\n    var yval=Numbas.math.niceNumber(P1.X());\n    var html = $(question.display.html);\n    html.find('#xvalue').text(yval);\n    html.find('#yvalue').text(xval);\n  }\n  showCoords();\n  \n  P1.on('drag', showCoords);\n  \n  \n  pc = brd.create('curve', [[0],[0]], {strokeColor:'green', strokeWidth:2, name:'y_3', withLabel:false});\n  //These are the eigendirections giving asymptotic behaviour.\n  //var e1= brd.create('line',[[0,0],[al,1]],{fixed:'true'});\n  //var e2= brd.create('line',[[0,0],[be,1]],{fixed:'true'});\n  //Lines at which the solution curves are stationary or have infinite gradient\n  //brd.create('line',[[0,0],[-a/b,1]],{color:'black',dash:1,fixed:'true'});\n  //brd.create('line',[[0,0],[-c/d,1]],{color:'red',dash:1,fixed:'true'});\n  $(question.display.html).find('#board').append(div);\n  \n  var div1 = Numbas.extensions.jsxgraph.makeBoard('400px','400px',{axis:true,showNavigation:true, boundingbox:[-5,10,5,-10]});\n  brd1=div1.board;\n  g1 = brd1.create('curve', [[0],[0]], {strokeColor:'blue', strokeWidth:2, name:'y_1', withLabel:false});\n  g2 = brd1.create('curve', [[0],[0]], {strokeColor:'black', strokeWidth:2, name:'y_2', withLabel:false});\n  \n  $(question.display.html).find('#board1').append(div1);\n  \n  \n  \n  //the variable values a, v are captured on running the question.\n  var scope = question.scope;\n  var v = Numbas.jme.unwrapValue(scope.variables.v);\n  var a = Numbas.jme.unwrapValue(scope.variables.f);\n  ko.computed(function() {\n    var b = parseFloat(question.parts[0].gaps[0].display.studentAnswer());\n    var c = parseFloat(question.parts[0].gaps[1].display.studentAnswer());\n    var d = parseFloat(question.parts[0].gaps[2].display.studentAnswer());\n    var ans1=a+d;\n    var det= a*d-b*c;\n    var ans2 =v*v-det; \n    var ty;\n    var t1= ans1*ans1-4*det;\n    //Feedback to user if the diagonal element is not correct once all numbers have been input.\n    fin=isNaN(b)||isNaN(c)||isNaN(d);\n    if(!fin){\n      if(t1<0) {\n        if(ans1>0){\n          ty='Unstable spiral.';\n        } \n        else if(ans1<0){\n          ty='Stable spiral.';\n        }\n        else {\n          ty='Centre.';\n        }\n      }\n      else {\n        if(det<0){\n          ty='Saddle.';\n        }\n        else if ((det>0)&&(ans1>0)){\n          ty='Unstable node.';\n        } \n        else {\n          ty='Stable node.';\n        }\n      }\n    }\n    else {ty=''};\n    //if either the input has not been completed or the correct answer has been input - no message.\n    if(fin){adv='You have not entered all the numerical values into the matrix';}\n    if(fin||((ans1==0)&&(ans2==0))){ m='';}\n    //all entries have been input and an incorrect diagonal value has been input or gives the wrong eigenvalue.\n    else if(ans1!=0){m='You did not enter the correct value for the diagonal.';}\n    else {m='You have the wrong eigenvalue.';}\n    \n    var html = $(question.display.html);\n    html.find('#type').text(ty);\n    //This is displayed in the question as an aid to the user.\n    html.find('#mess').text(m);\n    \n    /*values for hidden gapfills which are marked*/\n    question.parts[0].gaps[3].display.studentAnswer(ans1);  \n    question.parts[0].gaps[4].display.studentAnswer(ans2);\n    \n    updateBoards(a,b,c,d);\n  });\n});\n"}, "advice": "<p>Given the system of differential equations:</p>\n<p>\\[\\begin{align}\\dot{x}&amp;=\\simplify[std]{a*x+b*y},\\\\\\dot{y}&amp;=\\simplify[std]{c*x+d*y}.\\end{align}\\]</p>\n<p>It can be written in the form $\\dot{\\boldsymbol{x}}=\\mathsf{A}\\boldsymbol{x}$, where $\\boldsymbol{x}=\\pmatrix{x,y}^\\mathsf{T}$ and&nbsp;</p>\n<p>\\[\\mathsf{A}=\\pmatrix{a&amp; b\\\\c &amp; d}\\]</p>\n<p>In this case, with $a=\\var{f}$ &nbsp;we want to enter values for $b,\\;c,\\;d$ such that the system gives a centre.</p>\n<p>For this to happen we need the eigenvalues of $A$ to be purely imaginary and you are given that they are $\\simplify{{v}*i}$ and $\\simplify{{-v}*i}$.</p>\n<p>The characteristic polynomial for $A$ is given by&nbsp;</p>\n<p>\\[\\det\\left(\\mathsf{A}-\\lambda\\mathsf{I}\\right)=0,\\]</p>\n<p>i.e. &nbsp;$(a-\\lambda)(d-\\lambda)-bc=0$. &nbsp;This leads to $\\lambda^2-(a+d)\\lambda+(ad-bc)=0$.</p>\n<p>So in order to get purely imaginary eigenvalues $\\pm \\var{abs(v)*i}$ we need :</p>\n<p>1. $a+d=0 \\Rightarrow d=\\var{-f}$ as $a=\\var{f}$.</p>\n<p>2. $ad-bc=\\var{v^2} \\Rightarrow bc=-\\var{v^2}-\\var{f^2}$.</p>\n<p>Hence if you set $d=\\var{-f}$ and you &nbsp;choose values of &nbsp;$b,\\;c$ such that $bc=-\\var{f^2+v^2}$, this will give the required system of differential equations with phase space a centre and the required eigenvalues.</p>", "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 0, "prompt": "<p>The system can be written in the form $\\dot{\\boldsymbol{x}}=\\mathsf{A}\\boldsymbol{x}$, where $\\boldsymbol{x}=\\pmatrix{x,y}^\\mathsf{T}$.</p>\n<p>Input the components of the matrix $\\mathsf{A}$ in order to obtain a centre where the eigenvalues of $A$ are $\\simplify{{v}*i}$ and&nbsp;$\\simplify{{-v}*i}$.</p>\n<p>You are given that $a=\\var{f}$.</p>\n<table>\n<tbody>\n<tr>\n<td rowspan=\"2\">$\\mathsf{A}=\\Bigg($</td>\n<td>$\\var{f}$</td>\n<td>[[0]]</td>\n<td rowspan=\"2\">$\\Bigg)$</td>\n</tr>\n<tr>\n<td>[[1]]</td>\n<td>[[2]]</td>\n</tr>\n</tbody>\n</table>\n<p style=\"color: red;\" id=\"mess\"></p>\n<p style=\"color: red;\" id=\"type\"></p>\n<p>Once you have input appropriate values into the matrix, the diagram below shows the plot of $(x(t),y(t))$.</p>\n<p>At $t=0$ we have initally $x=-5,\\;\\;y=5$. Moving the point gives phase diagrams for the following initial values at $t=0$:</p>\n<p>$x=\\;$<span style=\"color: red;\" id=\"xvalue\"></span> &nbsp; &nbsp; &nbsp; $y=\\;$<span style=\"color: red;\" id=\"yvalue\"></span></p>\n<p></p>\n<p></p>\n<div id=\"board\"></div>\n<p></p>\n<!--<p>If needed, you can use the navigation bar at the bottom right of the graph &nbsp;to zoom in and out of the graph to get a better picture .</p>-->\n<p></p>\n<p>You can click on Steps to see the solutions for $x(t),\\;y(t)$ after you have input values into the matrix.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<h4>Graph of $x(t),\\;y(t)$</h4>\n<div id=\"board1\"></div>\n<p></p>\n<p>$x(t)$ is in black, $y(t)$ in blue.</p>\n<p>You can use the navigation bar to zoom in and out of the graph.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "1", "minValue": "1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {"mark": {"order": "instead", "script": "!doesMarking;\nthis.answered=true;"}}, "marks": 0, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "1", "minValue": "1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {"mark": {"order": "instead", "script": "!doesMarking;\nthis.answered=true;"}}, "marks": 0, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "1", "minValue": "1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {"mark": {"order": "instead", "script": "!doesMarking;\nthis.answered=true;"}}, "marks": 0, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "0", "minValue": "0", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "0", "minValue": "0", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 2, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Consider the following two-dimensional dynamical system . You have to find values for $a,\\;b\\;c,\\;d$ such that the system's phase space is a centre.</p>\n<p>\\[\\begin{align}\\dot{x}&amp;=\\simplify[std]{a*x+b*y},\\\\\\dot{y}&amp;=\\simplify[std]{c*x+d*y}.\\end{align}\\]</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"xr": {"definition": "50", "templateType": "anything", "group": "Ungrouped variables", "name": "xr", "description": ""}, "yr": {"definition": "50", "templateType": "anything", "group": "Ungrouped variables", "name": "yr", "description": ""}, "f": {"definition": "random(-9..9 except[0,1,-1])", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "description": ""}, "v": {"definition": "random(1..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "v", "description": ""}}, "metadata": {"notes": "<p></p>\n<p><strong>01/04/2014:</strong></p>\n<p>Created.</p>\n<p><strong>02/02/2014:</strong></p>\n<p>Graphs of &nbsp;x(t), y(t) included.</p>\n<p><strong>04/08/2014</strong></p>\n<p>Changed showNavigation to false and commented out line re using Jsxgraph navigation in the Part as this does not work at present - need to upload new version of jsxgraph?</p>", "description": "<p>Asking users to input coefficients of a system of diff equations so that the phase space is a centre. All systems input by the user are graphed together with immediate feedback. Also included in the Steps are the graphs of the solutions for $x(t),\\; y(t);\\; x(0)=-5,\\;y(0)=5.$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}