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{"name": "Determine long-term behaviour of 1D dynamical systems", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"r1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "r1"}, "incans21": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if (ans2=0,a2,\n      if (ans2=a2,b2,\n    if (ans2=b2,infinity,0)\n      )\n    )", "description": "", "name": "incans21"}, "x01": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..9)", "description": "", "name": "x01"}, "x02": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..15)", "description": "", "name": "x02"}, "a1": {"templateType": 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variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "r2"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "a2"}}, "ungrouped_variables": ["r1", "r2", "incans13", "ans1", "incans11", "incans22", "incans23", "incans21", "a1", "a2", "b2", "incans12", "x02", "x01", "ans2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Determine long-term behaviour of 1D dynamical systems", "functions": {}, "showQuestionGroupNames": false, "parts": [{"displayType": "radiogroup", "choices": ["<p>As $t\\rightarrow\\infty$ $x\\rightarrow\\var{ans1}$.</p>", "<p>As $t\\rightarrow\\infty$ $x\\rightarrow\\var{incans11}$.</p>", "<p>As $t\\rightarrow\\infty$ $x\\rightarrow\\var{incans12}$.</p>", "<p>As $t\\rightarrow\\infty$ $x\\rightarrow\\var{incans13}$.</p>"], "displayColumns": 1, "prompt": "<p>\\[\\dot{x}=f(x)=\\simplify{{r1}*x*(1-x/{a1})}\\]</p>\n        <p>with initial condition $x(0)=\\var{x01}$ and $x\\geqslant 0$.</p>", "distractors": ["", "", "", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0, 0, 0], "marks": 0}, {"displayType": "radiogroup", "choices": ["<p>As $t\\rightarrow\\infty$ $x\\rightarrow\\var{ans2}$.</p>", "<p>As $t\\rightarrow\\infty$ $x\\rightarrow\\var{incans21}$.</p>", "<p>As $t\\rightarrow\\infty$ $x\\rightarrow\\var{incans22}$.</p>", "<p>As $t\\rightarrow\\infty$ $x\\rightarrow\\var{incans23}$.</p>"], "displayColumns": 1, "prompt": "<p>\\[\\dot{x}=g(x)=\\simplify{{r2}*x*({a2}-x)*(x-{b2})}\\]</p>\n        <p>with initial condition $x(0)=\\var{x02}$ and $x\\geqslant 0$.</p>", "distractors": ["", "", "", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0, 0, 0], "marks": 0}], "statement": "<p>For the given one-dimensional dynamical systems and initial conditions below, determine the long-term behaviour of the solution $x(t)$ as $t\\rightarrow\\infty$.</p>", "tags": ["checked2015", "MAS2106"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Determine the long-term behaviour of 1D dynamical systems.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>It is helpful to draw the one-dimensional phase flow for systems of this type.</p>\n    <p>a)</p>\n    <p>The ODE in this part can be written in the general form</p>\n    <p>\\[\\dot{x}=f(x)=rx\\left(1-\\frac{x}{a}\\right),\\]</p>\n    <p>where $r=\\var{r1}$ and $a=\\var{a1}$.</p>\n    <p>The system has fixed points at $x=0$ and $x=a$, and the function $f(x)$ is a parabola.</p>\n    <p>If $r&gt;0$, then the function (with phase flow as arrows) is as shown below.</p>\n    <p><img src=\"resources/images/parabola1.png\"/></p>\n    <p>If $r&lt;0$, then the function (with phase flow as arrows) is as shown below.</p>\n    <p><img src=\"resources/images/parabola2.png\"/></p>\n    <p>Hence:</p>\n    <ul>\n    <li>if $r&gt;0$, then $x\\rightarrow a$ as $t\\rightarrow\\infty$, irrespective of whether $x(0)&lt; a$ or $x(0)&gt;a$;</li>\n    <li>if $r&lt;0$, then $x\\rightarrow 0$ as $t\\rightarrow\\infty$ if $0\\leqslant x(0)&lt; a$, and $x\\rightarrow\\infty$ as $t\\rightarrow\\infty$ if $x(0)&gt;a$;</li>\n    <li>if $x(0)=0$ or $x(0)=a$, then $x(t)$ will remain at these points for all time, since they are the fixed points $x_0$ of the system.</li>\n    </ul>\n    <p></p>\n    <p>If $r&gt;0$, then $x_0=0$ is unstable and $x_0=a$ is stable.</p>\n    <p>If $r&lt;0$, then $x_0=0$ is stable and $x_0=a$ is unstable.</p>\n    <p>For the particular values of $r$ and $a$ in this part $x\\rightarrow\\var{ans1}$ as $t\\rightarrow\\infty$.</p>\n    <p></p>\n    <p>b)</p>\n    <p>The ODE in this part can be written in the general form</p>\n    <p>\\[\\dot{x}=g(x)=rx(a-x)(x-b),\\]</p>\n    <p>where $r=\\var{r2}$, $a=\\var{a2}$, and $b=\\var{b2}$.</p>\n    <p>The system has fixed points at $x=0$, $x=a$, and $x=b$, and the function $g(x)$ is a cubic.</p>\n    <p>If $r&gt;0$, then the function (with phase flow as arrows) is as shown below.</p>\n    <p><img src=\"resources/images/cubic1.png\"/></p>\n    <p>If $r&lt;0$, then the function (with phase flow as arrows) is as shown below.</p>\n    <p><img src=\"resources/images/cubic2.png\"/></p>\n    <p>Hence:</p>\n    <ul>\n    <li>if $r&gt;0$, then $x\\rightarrow 0$ as $t\\rightarrow\\infty$ if $0\\leqslant x(0)&lt; a$, and $x\\rightarrow b$ as $t\\rightarrow\\infty$ if $x(0)&gt;a$;</li>\n    <li>if $r&lt;0$, then $x\\rightarrow a$ as $t\\rightarrow\\infty$ if&nbsp;$0\\leqslant x(0)&lt; b$, and $x\\rightarrow\\infty$ as $t\\rightarrow\\infty$ if $x(0)&gt;b$;</li>\n    <li>if $x(0)=0$, $x(0)=a$, or $x(0)=b$, then $x(t)$ will remain at these points for all time, since they are the fixed points $x_0$ of the system.</li>\n    </ul>\n    <p></p>\n    <p>If $r&gt;0$, then $x_0=0$ and $x_0=b$ are stable, and $x_0=a$ is unstable.</p>\n    <p>If $r&lt;0$, then $x_0=0$ and $x_0=b$ are unstable, and $x_0=a$ is stable.</p>\n    <p>For the particular values of $r$, $a$, and $b$ in this part $x\\rightarrow\\var{ans2}$ as $t\\rightarrow\\infty$.</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}