Question

Show that if n is an odd integer, then the critical point (npi,0) is a saddle point for the damped pendulum system

Answer 1

\[x'=y, y'=-\omega^{2}sinx-cy\]

Answer 2

Please help

Answer 3

@cruffo

Answer 4

@wio

Answer 5

So, it's true you have critical points at every \((k\pi, 0)\). But this problem is not separable, so it seems a direct approach by looking for a solution may not be optimal. Epecially since your not asked to solve, just to show the behavior. So....

Answer 6

I would think that just looking at the signs of the slopes at the critical points should work...
Remind me a bit
-1 is saddle?
1 is source
0 is sink....

Looking though my notes now, but if you recall, please respond.

Answer 7

I think youre right

Answer 8

But im not sure how to show

Answer 9

i'm still looking at it...

Answer 10

So, looking at the graph of (x, x') - fixing x, and solving y' for critical values, you can crease a phase diagram, to determine stable/unstable equilibrium. see pic

Answer 11

the key is to linearilize
see p. 18 http://people.uncw.edu/hermanr/mat463/ODEBook/Book/Nonlinear.pdf

Answer 12

thank you