Please solve part (a) using polar coordinates. Where x = rcosθ and y = rsinθ. r^2 = x^2 + y^2 and tanθ = y/x sin^2θ + co
Posted: Thu May 12, 2022 12:10 pm
Please solve part (a) using polar coordinates.
Where x = rcosθ and y =
rsinθ.
r^2 = x^2 + y^2 and tanθ = y/x
sin^2θ + cos^2θ =
1
r dot = (x x dot + y y dot)/r
and θ dot = (y dot x - x dot y)/r^2
Sub x dot and y dot into r dot and solve for r.
Again sub x dot and y dot
into θ dot to solve for
r.
Both r dot and θ dot should give the
same solution to r.
For part (b), using r found in part (a) find out if the periodic
orbit is stable or unstable.
Thank you.
2. Consider the system = 23 + xy? - 1 - y, y = x²y + y2 + – y. (a) Transform the system into polar co-ordinates (r, 0) and show that it has exactly one periodic orbit of constant radius. (b) Classify the stability of the periodic orbit. [4 marks]
Where x = rcosθ and y =
rsinθ.
r^2 = x^2 + y^2 and tanθ = y/x
sin^2θ + cos^2θ =
1
r dot = (x x dot + y y dot)/r
and θ dot = (y dot x - x dot y)/r^2
Sub x dot and y dot into r dot and solve for r.
Again sub x dot and y dot
into θ dot to solve for
r.
Both r dot and θ dot should give the
same solution to r.
For part (b), using r found in part (a) find out if the periodic
orbit is stable or unstable.
Thank you.
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