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Pendulum - Stability and fixed points

  1. Jan 13, 2015 #1
    1. The problem statement, all variables and given/known data

    (a): Show the lagrangian derivative in phase space
    (b)i: Show how the phase space evolves over time and how they converge
    (b)ii: Find the fixed points and stability and sketch phase diagram
    (c)i: Find fixed points and stability
    (c)ii: Show stable limit cycles exist for T>ga and why they cannot emerge by hopf bifurcations


    2vtqfcz.png
    2. Relevant equations


    3. The attempt at a solution

    Part(a)

    Shown using conservation of rate of flow of mass.

    Part(b)i

    [tex]\Delta \dot V = \Delta V \left( \frac{\partial \dot \theta}{\partial \theta} + \frac{\partial \dot \omega}{\partial \omega} \right) [/tex]
    [tex] \Delta \dot V = -r \Delta V [/tex]

    [tex] \Delta V = \Delta V_0 e^{-rt} [/tex]

    Thus for ##r>0## system converges to an attractor.

    Part (b)ii
    For ## \dot \omega = \dot \theta = 0##, fixed points are given by ##\theta_0 = n\pi##.

    To find their stability, we find ##|J-\lambda I| = 0##
    [tex]\lambda^2 + \lambda r + ga cos\theta = 0[/tex]
    Thus for even ##n##, fixed points are stable attractors and for odd ##n##, fixed points are unstable saddle points.

    1213bjo.png


    Part (c)i

    Fixed points are given by:

    [tex] cos \theta = \sqrt{1- \left( \frac{T}{ga} \right)^2} [/tex]

    For their stability, again we find the eigenvalues:

    [tex] \lambda^2 + \lambda r + ga cos\theta = 0 [/tex]
    [tex]\lambda = \frac{-r \pm \sqrt{ r^2 - 4ga cos\theta } }{2} [/tex]

    Thus for ## r^2 > 4ga \sqrt{1- \left( \frac{T}{ga} \right)^2}##, they are fixed stable attractors. For ## r^2 < \sqrt{1- \left( \frac{T}{ga} \right)^2} ##, they are unstable saddle points.

    Part(c)ii

    I'm not too sure about this part. I know that for ##T>ga##, ##cos \theta## becomes imaginary. How do I relate that to how the fixed point changes to a limit cycle?
     
  2. jcsd
  3. Jan 15, 2015 #2
  4. Jan 16, 2015 #3
    bumppp
     
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