How Does the Bifurcation Diagram in AUTO Represent Mathematical Concepts?

[member 428835]

Hi PF

I am given the ODE $\ddot{\theta} +q\sin\theta+\mu\cos\theta=0$, where I introduce $x_1 = \theta$ and $x_2=\dot{\theta}$ to yield$$
\dot{x_2}=-q\sin x_1-\mu\cos x_1\\
x_2=\dot{x_1}
$$
subject to $x_2(0)=x_2(1)=0$ and $\int_0^1\sin x_1\, d t = 0$. I then plug this system into the program AUTO, a bifurcation detection software. The bifurcation plot is attached. My question is, what does this graph represent mathematically? What would the value of one point, say 17, represent?

Thanks a ton for your help!

 

[jedishrfu]

Can you post the graph directly as an image instead of a pdf? Not everyones browser opens pdf files especially with the issues Adobe has with malware.

 

[member 428835]

I'm happy to, but I don't know how to do this?

 

[Greg Bernhardt]

Here