Limiting Cycles and Equilibrium Points

[mcafej]

Homework Statement

Determine Equilibrium points, limiting cycles, and their stabilities for the following equations

r'=r(r-1)(r-3)
θ'=1

The Attempt at a Solution

So I know one equilibrium point is going to be (0,0) because r=0 is a limiting cycle (I believe), and that is simply a point. I also know that r=1 and r=3 are going to be some part of the solution, but I'm not sure how to use that knowledge to compute the equilibrium points. My problem is the θ'. I know that x=rcosθ and y=rsinθ, but I don't really know what to do with the θ'=1 part of the system.

 

[epenguin]

Homework Statement

Determine Equilibrium points, limiting cycles, and their stabilities for the following equations

r'=r(r-1)(r-3)
θ'=1

The Attempt at a Solution

So I know one equilibrium point is going to be (0,0) because r=0 is a limiting cycle (I believe), and that is simply a point. I also know that r=1 and r=3 are going to be some part of the solution, but I'm not sure how to use that knowledge to compute the equilibrium points. My problem is the θ'. I know that x=rcosθ and y=rsinθ, but I don't really know what to do with the θ'=1 part of the system.

Polar co-ordinates - when r = 0 it really doesn't make too much sense to talk about θ or θ' does it? Nor to call it a cycle. So is r = 0 an equilibrium point or not? If so that's all the description you need of this case.

Don't worry about those trig formulae. What does θ' mean about what any solution point is doing? What is it doing when r= 1 or 3?

You are given a d.e. in 2 time-dependent variables where they are nicely separated into two equations. Lucky. In most cases you either struggle to express the equations in such a way or it cannot be done. You can solve for r as a function of t, and if you really want to show off i think you can manage to solve dθ/dt = 1 too , but you are not asked to do anything so hard, you are just asked to say what the stability of the limit cycles you find is.

Again to find those stabilities is nothing difficult, you just need to sketch r' against r and consider what it means. This whole exercise is no calculation at all, it is just meant to check you know what stuff means.