When does bifurcation occur and what type(s) of bifurcation are they?

[totoroooo]

A light wheel of radius a has a uniform semicircular rim of mass M, and my rotate freely in a vertical plane about a horizontal axis through its center. A light string passes around the wheel and suspends a mass m. You may assume that this mechanical system is governed by the equation:
(M+m)a^2 (d^x/dt^2)= ag(m -2Msin(x)/(pi))
where x is the angle between the downward vertical and the diameter through the center of mass of the heavy rim.

where does bifurcation occur and what type(s) of bifurcation are they? how does the wheel behave when k is large?

 

[Mentz114]

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[totoroooo]

oh i am sorry about that.

note that k in the question refers to m/M.

so i think when k is large, d^2(x)/dt^2 tends to g/a, which means the wheel behaves like a simple pendulum??

and one of the equilibrium points is, of course, (k,x) = (0,0). then (0, pi) and (0,2*pi) are equilibrium points as well.

another observation is when k = 2/pi, sin (x) = 1. i think bifurcation occurs here, but i am not sure what would happen when k>(2/pi), because it seems like that means sin(x) >1...