Uppgift 6.1: 25
Vektorfältet
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(1.1) |
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(1.2) |
ger med substitutionen
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(1.3) |
systemet
![`:=`(syst, [(D(x))(t) = eval(f, fcn), (D(y))(t) = eval(g, fcn)])](images/datorle6_8.gif){kind=small}
![[(D(x))(t) = `+`(`*`(4, `*`(y(t), `*`(`+`(1, `*`(`^`(x(t), 2)), `*`(`^`(y(t), 2))))))), (D(y))(t) = `+`(`-`(`*`(x(t), `*`(`+`(1, `*`(`^`(x(t), 2)), `*`(`^`(y(t), 2)))))))]](images/datorle6_9.gif){kind=small} |
(1.4) |
Fasporträttet ser ut så här
![DEplot(syst, [x(t), y(t)], t = 0 .. 3, [[x(0) = 0, y(0) = `/`(1, 4)], [x(0) = 0, y(0) = `/`(1, 2)], [x(0) = 0, y(0) = `/`(3, 4)]], x = -1 .. 1, y = -1 .. 1, arrows = MEDIUM, stepsize = 0.2e-1, linecol...](images/datorle6_10.gif){kind=small}
![DEplot(syst, [x(t), y(t)], t = 0 .. 3, [[x(0) = 0, y(0) = `/`(1, 4)], [x(0) = 0, y(0) = `/`(1, 2)], [x(0) = 0, y(0) = `/`(3, 4)]], x = -1 .. 1, y = -1 .. 1, arrows = MEDIUM, stepsize = 0.2e-1, linecol...](images/datorle6_11.gif){kind=small}
![DEplot(syst, [x(t), y(t)], t = 0 .. 3, [[x(0) = 0, y(0) = `/`(1, 4)], [x(0) = 0, y(0) = `/`(1, 2)], [x(0) = 0, y(0) = `/`(3, 4)]], x = -1 .. 1, y = -1 .. 1, arrows = MEDIUM, stepsize = 0.2e-1, linecol...](images/datorle6_12.gif){kind=small}
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Elimineras t får vi om y = y(x) ekvationen
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(1.5) |
Lösningen i implicit form blir
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(1.6) |
Med liapunovfunktionen
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(1.7) |
blir derivatan längs fältet (f,g)
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(1.8) |
vilket trivialt visar att (0,0) är en stabil men inte asymptotiskt stabil jämviktspunkt.