// Numbas version: finer_feedback_settings
{"name": "Classify fixed points of a 2D dynamical system", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Classify fixed points of a 2D dynamical system", "tags": ["checked2015"], "metadata": {"description": "<p>Nature of fixed points of a 2D dynamical system.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Consider the two-dimensional dynamical system</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>", "advice": "<p>a)</p>\n<p>The elements of the matrix $\\mathsf{A}$ are the coefficients of $x$ and $y$ in the system of equations, hence</p>\n<p>\\[\\mathsf{A}=\\pmatrix{\\var{a} &amp; \\var{b}\\\\\\var{c} &amp; \\var{d}}.\\]</p>\n<p>&nbsp;</p>\n<p>b)</p>\n<p>The trace $\\tau$ and determinant $\\delta$ of a matrix</p>\n<p>\\[\\mathsf{A}=\\pmatrix{a &amp; b\\\\c &amp; d}\\]</p>\n<p>are given by $\\tau=a+d$ and $\\delta=ad-bc$.</p>\n<p>A straight forward substitution for $a=\\var{a}$, $b=\\var{b}$, $c=\\var{c}$, and $d=\\var{d}$ reveals that $\\tau=\\var{tau}$ and $\\delta=\\var{delta}$.</p>\n<p>&nbsp;</p>\n<p>c)</p>\n<p>The eigenvalues $\\lambda$ of a matrix $\\mathsf{A}$ can be calculated from the expression</p>\n<p>\\[\\mathsf{A}\\boldsymbol{x}=\\lambda\\boldsymbol{x},\\]</p>\n<p>which is equivalent to</p>\n<p>\\[\\left(\\mathsf{A}-\\lambda\\mathsf{I}\\right)\\boldsymbol{x}=\\boldsymbol{0},\\]</p>\n<p>where $\\mathsf{I}$ is the identity matrix.</p>\n<p>This linear system has a solution only when</p>\n<p>\\[\\det\\left(\\mathsf{A}-\\lambda\\mathsf{I}\\right)=0,\\]</p>\n<p>i.e. when $(a-\\lambda)(d-\\lambda)-bc=0$. &nbsp;This leads to $\\lambda^2-(a+d)\\lambda+(ad-bc)=0$, which is equivalent to</p>\n<p>\\[\\lambda^2-\\tau\\lambda+\\delta=0.\\]</p>\n<p>The eigenvalues $\\lambda_1$ and $\\lambda_2$ of the matrix $\\mathsf{A}$ are therefore given by</p>\n<p>\\[\\lambda_{1,2}=\\frac{\\tau\\pm\\sqrt{\\tau^2-4\\delta}}{2}.\\]</p>\n<p>Following through this calculation for the specific values of $a$, $b$, $c$, and $d$ in this question leads to $\\lambda_1=\\var{lam1}$ and $\\lambda_2=\\var{lam2}$.</p>\n<p>&nbsp;</p>\n<p>d)</p>\n<p>The eigenvectors corresponding to the eigenvalues $\\lambda_1$ and $\\lambda_2$ can be found by solving the linear system of equations given by</p>\n<p>\\[\\mathsf{A}\\boldsymbol{x}=\\lambda\\boldsymbol{x},\\]</p>\n<p>or</p>\n<p>\\[\\begin{align}ax+by&amp;=\\lambda x,\\\\cx+dy&amp;=\\lambda y,\\end{align}\\]</p>\n<p>where $\\lambda$ is either $\\lambda_1$ or $\\lambda_2$.</p>\n<p>Because this is an eigenvalue problem, we can arbitrarily choose one of the eigenvector components to be $1$, the $x$-component in this case. &nbsp;The $y$-component can then be found from</p>\n<p>\\[y=\\frac{x(\\lambda-a)}{b}\\quad\\text{or}\\quad y=\\frac{cx}{\\lambda-d},\\]</p>\n<p>with $x=1$. &nbsp;Both expressions are equivalent, and will lead to the same value for $\\lambda$.</p>\n<p>Making the necessary substitutions reveals that the $y$-component of the eigenvector corresponding to $\\lambda_1$ is $\\var{vec1}$, and the $y$-component of the eigenvector corresponding to $\\lambda_2$ is $\\var{vec2}$.</p>\n<p>&nbsp;</p>\n<p>e)</p>\n<p>The nature of the fixed point at the origin can be determined by examining the eigenvalues. &nbsp;If the eigenvalues are real, then the sign of each eigenvalue determines the nature of the fixed point. &nbsp;If the eigenvalues are complex, then the sign of the real part of each eigenvalue determines the nature of the fixed point.</p>\n<p>In this case, {naturetext}.</p>", "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "extensions": [], "variables": {"natures": {"name": "natures", "group": "Ungrouped variables", "definition": "shuffle(['unstable node','stable node','saddle point','unstable spiral','stable spiral','centre'] except nature)", "description": "", "templateType": "anything"}, "vec2": {"name": "vec2", "group": "Ungrouped variables", "definition": "(lam2-a)/b", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "sign(b)*((tau-d)*d-delta)", "description": "", "templateType": "anything"}, "r": {"name": "r", "group": "Ungrouped variables", "definition": "random(2..18#2 except abs(tau))", "description": "", "templateType": "anything"}, "delta": {"name": "delta", "group": "Ungrouped variables", "definition": "0.25*(tau^2+sign(random(-1,1))*r^2)", "description": "", "templateType": "anything"}, "tau": {"name": "tau", "group": "Ungrouped variables", "definition": "random([0]+repeat(2*random(0..9)*sign(random(-1,1)),5))", "description": "", "templateType": "anything"}, "lam2": {"name": "lam2", "group": "Ungrouped variables", "definition": "(tau-sqrt(disc))/2", "description": "", "templateType": "anything"}, "nature": {"name": "nature", "group": "Ungrouped variables", "definition": "if (im(lam1)=0 and im(lam2)=0,\n      switch (\n    lam1>0 and lam2>0, 'unstable node',\n    lam1<0 and lam2<0, 'stable node',\n    lam1>0 and lam2<0, 'saddle point'\n      ),\n      switch (\n    re(lam1)>0, 'unstable spiral',\n    re(lam1)<0, 'stable spiral',\n    re(lam1)=0, 'centre'\n      )\n    )", "description": "", "templateType": "anything"}, "lam1": {"name": "lam1", "group": "Ungrouped variables", "definition": "(tau+sqrt(disc))/2", "description": "", "templateType": "anything"}, "disc": {"name": "disc", "group": "Ungrouped variables", "definition": "tau^2-4*delta", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "tau-d", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..15 except abs(tau))*sign(random(-1,1))", "description": "", "templateType": "anything"}, "vec1": {"name": "vec1", "group": "Ungrouped variables", "definition": "(lam1-a)/b", "description": "", "templateType": "anything"}, "naturetext": {"name": "naturetext", "group": "Ungrouped variables", "definition": "switch (\n      nature='unstable node', \"because $\\\\lambda_1>0$ and $\\\\lambda_2>0$, the fixed point is an unstable node\",\n      nature='stable node', \"because $\\\\lambda_1<0$ and $\\\\lambda_2<0$, the fixed point is a stable node\",\n      nature='saddle point', \"because $\\\\lambda_1>0$ and $\\\\lambda_2<0$, the fixed point is a saddle point\",\n      nature='unstable spiral', \"because $\\\\mathrm{Im}(\\\\lambda_1)\\\\ne 0$ and $\\\\mathrm{Im}(\\\\lambda_2)\\\\ne 0$, and $\\\\mathrm{Re}(\\\\lambda_1)=\\\\mathrm{Re}(\\\\lambda_2)>0$, the fixed point is an unstable spiral\",\n      nature='stable spiral', \"because $\\\\mathrm{Im}(\\\\lambda_1)\\\\ne 0$ and $\\\\mathrm{Im}(\\\\lambda_2)\\\\ne 0$, and $\\\\mathrm{Re}(\\\\lambda_1)=\\\\mathrm{Re}(\\\\lambda_2)<0$, the fixed point is a stable spiral\",\n      nature='centre', \"because $\\\\mathrm{Im}(\\\\lambda_1)\\\\ne 0$ and $\\\\mathrm{Im}(\\\\lambda_2)\\\\ne 0$, and $\\\\mathrm{Re}(\\\\lambda_1)=\\\\mathrm{Re}(\\\\lambda_2)=0$, the fixed point is a centre\",\n      false\n    )", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": "100"}, "ungrouped_variables": ["a", "tau", "c", "b", "d", "nature", "vec1", "lam2", "lam1", "vec2", "naturetext", "delta", "r", "disc", "natures"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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>Identify the components of the matrix $\\mathsf{A}$.</p>\n<p>$\\mathsf{A}=$ [[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([[a,b],[c,d]])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the trace $\\tau$, and the determinant $\\delta$ of the matrix $\\mathsf{A}$.</p>\n        <p>Trace $\\tau=$ [[0]].</p>\n        <p>Determinant $\\delta=$ [[1]].</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Trace $\\tau$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "tau", "maxValue": "tau", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Determinant $\\delta$", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "delta", "maxValue": "delta", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": "0", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the eigenvalues $\\lambda_1$ and $\\lambda_2$ of the matrix $\\mathsf{A}$.</p>\n<p>If the eigenvalues have <strong>zero imaginary part</strong>, enter the eigenvalue with the <strong>largest real part</strong> first. &nbsp;If the eigenvalues have <strong>non-zero imaginary part</strong>, enter the eigenvalue with the <strong>largest imaginary part</strong> first.</p>\n<p>$\\lambda_1=$ [[0]]</p>\n<p>$\\lambda_2=$ [[1]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "$\\lambda_1$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{lam1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": true, "customName": "$\\lambda_2$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{lam2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Assuming that the $x$-component of each eigenvector is $1$, find the $y$-components of the eigenvectors corresponding to the eigenvalues $\\lambda_1$ and $\\lambda_2$.</p>\n        <p>$y$-component of eigenvector corresponding to $\\lambda_1$: [[0]].</p>\n        <p>$y$-component of eigenvector corresponding to $\\lambda_2$: [[1]].</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Corresponding to $\\lambda_1$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{vec1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}, {"type": "jme", "useCustomName": true, "customName": "Corresponding to $\\lambda_2$", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{vec2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": []}], "sortAnswers": false}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Determine the nature of the fixed point at the origin.</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 1, "showCellAnswerState": true, "choices": ["<p>{nature}</p>", "<p>{natures[0]}</p>", "<p>{natures[1]}</p>", "<p>{natures[2]}</p>", "<p>{natures[3]}</p>", "<p>{natures[4]}</p>"], "matrix": [1, 0, 0, 0, 0, 0], "distractors": ["", "", "", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", 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