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What are the local and global behavior of solutions of

$r'=r-r^3$

$\theta'=(\sin\theta)^2+a$

at the bifurcation value $a=-1$?

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    What have you done? Are you asking or telling?2012-11-30
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    I don't know how to describe the local and global behaviors.2012-11-30
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    Start by looking at the critical points and limit cycles, and see if they are attractors or not.2012-11-30
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    These are the equilibrium points I found: when a>0, no equilibria. when a<0, no equilibria, but when a=0, I have the origin(0,0), (1,0),(1,pi).2012-11-30
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    Still looking for a solution2012-11-30
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    I'm pretty busy right now, but tomorrow I might have some free time.2012-11-30
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    Could you please just direct me and I will do everything else2012-11-30
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    The first thing to do is see that $r= 0$ and $r=1$ are the in the limit set of the ode. You should check what happens when solutions are in the region inside the unit disk and what happens when they are outside. Do they go to $r = 0$ or $r = 1$. If so, how they do it? Does the value of $a$ has anything to do with the behavior of the solution (righthandedness, boundness, etc)? Do solutions outside the unit circle go away from it or towards it. What does $a$ has to do with the sign of $\theta'$? Finally, is there a Lyapunov Function, what about stability of the limit cycle? There is a start :)2012-11-30
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    Thanks a lot, that definitely helps.2012-11-30

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