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Homework Statement
The trajectory of an arrow in space obeys the following system of equations:
[itex]\dot{x}[/itex] = y+[itex](x^2+y^2-3)^2 (x^3-x+xy^2)[/itex]
[itex]\dot{y}[/itex] = y+[itex](x^2+y^2-3)^2 (y^3-y+x^2y)[/itex]
1. Questions
a) Derive an ODE for the radial coordiante r(t) = [itex]\sqrt[]{x^2(t)+y^2(t)}[/itex]
b) Show that the system admits two limit cycles and classify their stability
c) The target is standing at the origin (0,0) of the phase plane. Let Po=(x(0),y(0))[itex]\neq[/itex] (0,0) denote the initial position of the arrow at the time t = 0. Determine and plot the region in the (x,y) plane containing all the points Po such that the arrow will hit the target at some instant of time t > 0
The Attempt at a Solution
I derived ODE in radial coordiante and got:
[itex]\dot{r}[/itex]=[itex](r^2-3)^2(r^2-1)r[/itex]
So we have 5 points to check the stability: [itex]0, 1,-1,\sqrt{3},-\sqrt{3} [/itex]
Than I calculate the derivative of [itex]\dot{r}[/itex] and evaluate in those 5 points: I get five zeros so they are undecided.
Then I daw the plot of [itex]\dot{r}[/itex] and if the [itex]\dot{r}[/itex]>0 it does mean that arrow on the phase line is to the right and if [itex]\dot{r}[/itex]<0 the arrow is to the left.
So The phase line looks like this: <- (-\sqrt{3}) <- (-1) -> (0) <- (1) -> (\sqrt{3}) ->
Is that correct? What should I do next?