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Change ODE system to Polar to apply PoincareBendixson 
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#1
Dec1512, 01:16 PM

P: 2

Question:
Show that the system x'= xyx[x^2 + (3/2)y^2] y'= x+y y[x^2 + (1/2)y^2] has at least one periodic orbit. I know that I need to apply PoincareBendixson Theorem. I can prove the first three points of it easily, but to create a trapping region, I believe that I need to switch this system to polar. I know that I need to make the substitutions y=r*sin(θ), dy= sin(θ)*dr + r*cos(θ) θ', x= r*cos(θ), dx= cos(θ)*dr  r*sin(θ)θ'. But when I do make the substitution, it makes the equations worse than before. Is there some other way to come up with a r min and r max for the trapping region? I just assumed it was a polar type question because its phase plane has a limit cycle on it. 


#2
Dec1612, 04:14 PM

HW Helper
Thanks
P: 1,001

If [itex]V(x,y) = \frac12(x^2 + y^2) = \frac12 r^2[/itex] then
[tex]\dot V = x\dot x + y \dot y = x^2 + y^2  x^4  \frac{5}{2} x^2y^2  \frac12 y^4[/tex] which after some rearrangement yields [tex]\dot V = x^2 + y^2  \left(x^2 + \frac54y^2\right)^2 + \frac{17}{16}y^4[/tex] Looking at that, you can see that there exist [itex]0 < m < M[/itex] such that if [itex]0 < r < m[/itex] then [itex]\dot V > 0[/itex] (because the origin is unstable, so trajectories are locally away from it) and if [itex]r > M[/itex] then [itex]\dot V < 0[/itex] (because if [itex]y[/itex] is fixed then [itex]\dot V < 0[/itex] for [itex]x[/itex] sufficiently large), so that the trapping region is [itex]m \leq r \leq M[/itex]. I don't think it's necessary to calculate [itex]m[/itex] and [itex]M[/itex]; it is enough to show that they exist. 


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