// Numbas version: exam_results_page_options
{"name": "Dynamical system 9: Saddle.", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "parts": [{"stepsPenalty": 0, "showCorrectAnswer": true, "variableReplacements": [], "showFeedbackIcon": true, "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 saddle.</p>\n<p>In order to achieve this you have to supply the diagonal elements of the matrix, the entries for $b,\\;c$ are given. In this case the eigenvalues of $A$ are real and of opposite signs.&nbsp;</p>\n<p>You are given that $b=\\var{f},\\;c=\\var{g}$.</p>\n<table>\n<tbody>\n<tr>\n<td rowspan=\"2\">$\\mathsf{A}=\\Bigg($</td>\n<td>[[0]]</td>\n<td>$\\var{f}$</td>\n<td rowspan=\"2\">$\\Bigg)$</td>\n</tr>\n<tr>\n<td>$\\var{g}$</td>\n<td>[[1]]</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>", "type": "gapfill", "gaps": [{"showFeedbackIcon": true, "minValue": "1", "type": "numberentry", "allowFractions": false, "maxValue": "1", "showCorrectAnswer": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "variableReplacements": [], "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "scripts": {"mark": {"script": "!doesMarking;\nthis.answered=true;", "order": "instead"}}, "marks": 0}, {"showFeedbackIcon": true, "minValue": "1", "type": "numberentry", "allowFractions": false, "maxValue": "1", "showCorrectAnswer": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "variableReplacements": [], "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "scripts": {"mark": {"script": "!doesMarking;\nthis.answered=true;", "order": "instead"}}, "marks": 0}, {"showFeedbackIcon": true, "minValue": "1", "type": "numberentry", "allowFractions": false, "maxValue": "1", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "variableReplacements": [], "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 1}, {"showFeedbackIcon": true, "minValue": "1", "type": "numberentry", "allowFractions": false, "maxValue": "1", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "variableReplacements": [], "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 2}], "steps": [{"showCorrectAnswer": true, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "<h4>Graph of $x(t),\\;y(t)$</h4>\n<p></p>\n<p><span id=\"board1\"></span></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>", "type": "information", "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0}], "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0}], "variablesTest": {"maxRuns": 100, "condition": ""}, "extensions": ["jsxgraph"], "metadata": {"description": "<p>Asking users to input coefficients of a system of diff equations so that the phase space is a saddle. 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"}, "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "ungrouped_variables": ["xr", "yr", "g", "f"], "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 saddle.</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>Note that this question is purely formative and for experimenting with. No marks are given.</p>", "preamble": {"css": "", "js": "var pc;\nvar brd;\nvar P1;\nvar brd1;\nvar g1;\nvar g2;\nvar fin;\nvar m;\nvar e1;\nvar e2;\n\nfunction updateBoards(a,b,c,d,r1,r2) {\n  function f(x,yy) {\nvar y1 = yy[0];\nvar y2 = yy[1];\nvar z1 = a*y1+b*y2;\nvar z2 = c*y1+d*y2;\nreturn [z1,z2];\n  }\n  \n  function ode() {\n//solution curve data array produced in the interval between 0 and 10, 200 steps\nreturn 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\nreturn JXG.Math.Numerics.rungeKutta('heun', [P1.Y(),P1.X()], [0, -10], 500, f);\n  }\n\npc.updateDataArray = function() {\nvar data = ode();\nvar l=data.length;\nvar data1=ode1();\nvar l1=data1.length;\nthis.dataX = [];\nthis.dataY = [];\nfor(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}\nfor(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//\n//e1= brd.create('line',[[0,0],[1, -(a-r1)/b]],{fixed:'true'});\n//e2= brd.create('line',[[0,0],[1,-(a-r2)/b]],{fixed:'true'});\nbrd.update();\n\ng1.updateDataArray = function() {\n  var h = 10/500;\n  var data = ode();\nvar l=data.length;\nvar data1=ode1();\nvar l1=data1.length;\n  this.dataX = [];\n  this.dataY = [];\n  for(var i=0; i<l1; i++) {\nthis.dataX[i] = -10+i*h;\nthis.dataY[i] = data1[l1-i-1][0];\n  }\n  for(var i=l1; i<l+l1; i++) {\nthis.dataX[i] = (i-l1)*h;\nthis.dataY[i] = data[i-l1][0];\n  }\n};\ng2.updateDataArray = function() {\n  var h = 10/500;\n  var data = ode();\nvar l=data.length;\nvar data1=ode1();\nvar l1=data1.length;\n  this.dataX = [];\n  this.dataY = [];\n  for(var i=0; i<l1; i++) {\nthis.dataX[i] = -10+i*h;\nthis.dataY[i] = data1[l1-i-1][1];\n  }\n  for(var i=l1; i<l+l1; i++) {\nthis.dataX[i] = (i-l1)*h;\nthis.dataY[i] = data[i-l1][1];\n  }\n};\nbrd1.update();\n}\n\nquestion.signals.on('HTMLAttached',function() {\nvar html = $(question.display.html);\n  var div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',{showNavigation:true,axis:true, boundingbox:[-50,50,50,-50]});\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  html.find('#xvalue').text(yval);\n  html.find('#yvalue').text(xval);\n}\nshowCoords();\n\nP1.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  e1= brd.create('line',[[0,0],[0,0]],{fixed:'true'});\n  e2= brd.create('line',[[0,0],[0,0]],{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\nvar div1 = Numbas.extensions.jsxgraph.makeBoard('400px','400px',{axis:true,showNavigation:true, boundingbox:[-5,10,5,-10]});\nbrd1=div1.board;\ng1 = brd1.create('curve', [[0],[0]], {strokeColor:'blue', strokeWidth:2, name:'y_1', withLabel:false});\ng2 = brd1.create('curve', [[0],[0]], {strokeColor:'black', strokeWidth:2, name:'y_2', withLabel:false});\n\nquestion.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 b = Numbas.jme.unwrapValue(scope.variables.f);\n  var c = Numbas.jme.unwrapValue(scope.variables.g);\n  ko.computed(function() {\nfunction sign(x) {\n    return typeof x === 'number' ? x ? x < 0 ? -1 : 1 : x === x ? 0 : NaN : NaN;\n}\nvar a = parseFloat(question.parts[0].gaps[0].display.studentAnswer());\nvar d = parseFloat(question.parts[0].gaps[1].display.studentAnswer());\nvar ans1=a+d;\nvar det= a*d-b*c;\nvar ty;\nvar t1= ans1*ans1-4*det;\nvar st1=sign(t1);\nvar ans;\nvar str=sign(det);\nvar r1=(ans1+Math.pow(t1,0.5))/2;\nvar r2=(ans1-Math.pow(t1,0.5))/2;\nif((st1>0)&&(str<0)){ans=1;} else {ans=0;};\n//Feedback to user if the diagonal element is not correct once all numbers have been input.\nfin=isNaN(a)||isNaN(d);\nif(!fin){\n  if(t1<0){\n    if(ans1>0){ty='Unstable spiral.';} \n    else if(ans1<0){ty='Stable spiral.';} else {ty='Centre.';}}\n         else {if(det<0){ty='Saddle.';}\n               else if ((det>0)&&(ans1>0)){ty='Unstable node.';} else {ty='Stable node.';}}}\nelse {ty=''};\n//if either the input has not been completed or the correct answer has been input - no message.\nif(fin||(ans==1)){ m='';}\n//all entries have been input and an incorrect diagonal value has been input or gives the wrong eigenvalue.\n  else if(st1==-1){m='The eigenvalues are complex, so you do not have a saddle.';}\n  else {m='This is not a saddle.';}\n\nhtml.find('#type').text(ty);\n//This is displayed in the question as an aid to the user.\nhtml.find('#mess').text(m);\n\n/*values for hidden gapfills which are marked*/\nquestion.parts[0].gaps[2].display.studentAnswer(st1);  \nquestion.parts[0].gaps[3].display.studentAnswer(ans);\n\nupdateBoards(a,b,c,d,r1,r2);\n});\n});\n"}, "variable_groups": [], "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 $b=\\var{f},\\;c=\\var{g}$ &nbsp;we want to enter values for $a,\\;d$ such that the system gives a saddle for its phase space.</p>\n<p>For this to happen we need the eigenvalues of $A$ to be real with one positive and one negative.</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 real eigenvalues with opposite signs we need :</p>\n<p>1. Real roots. $(a+d)^2 \\gt 4(ad-bc) \\Rightarrow (a-d)^2 \\gt -4bc$.&nbsp;</p>\n<p>Hence we have one condition is that $(a-d)^2 \\gt \\simplify[all,!collectNumbers]{-4*{f}*{g}={-4*f*g}}$</p>\n<p>2. Opposite signs. The roots are given by $\\displaystyle \\frac{(a+d) \\pm \\sqrt{(a+d)^2-4(ad-bc)}}{2}$.</p>\n<p>Hence if $\\operatorname{det}(A)=ad-bc \\lt 0$ we see that the roots are opposite in sign.</p>\n<p>So this is the other condition: $ad-bc \\lt 0 \\Rightarrow ad \\lt \\var{f}\\times \\var{g}=\\var{f*g}$</p>\n<p></p>", "variables": {"yr": {"description": "", "group": "Ungrouped variables", "definition": "50", "templateType": "anything", "name": "yr"}, "g": {"description": "", "group": "Ungrouped variables", "definition": "-sign(f)*random(1..5)", "templateType": "anything", "name": "g"}, "f": {"description": "", "group": "Ungrouped variables", "definition": "random(-9..9 except[0,1,-1])", "templateType": "anything", "name": "f"}, "xr": {"description": "", "group": "Ungrouped variables", "definition": "50", "templateType": "anything", "name": "xr"}}, "tags": ["centres", "Differential equations", "differential equations", "dynamical system", "feedback", "fixed points", "interactive", "Jsxgraph", "jsxgraph", "phase space", "saddle", "stable", "systems of differential equations", "unstable"], "name": "Dynamical system 9: Saddle.", "type": "question", "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/"}]}