$$\dot{x} = -\sin(x)$$
- Find the fixed points and also find out if it is attractive or repelling
- Find Lyapunov function for each of the attractive fixed points.
I thought:
Fixed points are $n\pi$ and the even integers are stable and thus attractive and the uneven integers are unstable and thus repelling. 0 is an unstable critical point.
How can I find a Lyapunov function? I couldn't find a general appliable theorem when i searched the web.
Lyapunov stability: nearby trajectories remain close for all time
Are the following steps correct ? :
a) Find fixed points
b) Linearize the system (find Jacobian) here it is simply $J = [-\cos(x)$ $ 0] $ if I'm not mistaken
c) Choose $V(x)$ to be... $\cos(x)$ maybe and find partial derivative so $\sin(x)*\dot{x}$..
Can someone please show me how to solve question 2. (that is if I did 1. correctly) Thanks