Pendulum - Stability and fixed points

In summary, the conversation discusses finding the lagrangian derivative in phase space and how the phase space evolves over time and converges. It also covers finding fixed points and their stability, as well as sketching a phase diagram. Finally, it mentions the existence of stable limit cycles for certain conditions and why they cannot emerge through hopf bifurcations.
  • #1
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Homework Statement



(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[/B]

2vtqfcz.png

Homework Equations

The Attempt at a Solution



Part(a)
[/B]
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?
 
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  • #3
bumppp
 

1. What is a pendulum?

A pendulum is a weight suspended from a pivot point that is free to swing back and forth. It is commonly used in timekeeping devices such as clocks and metronomes.

2. What is meant by stability in a pendulum?

Stability in a pendulum refers to how well it maintains its regular swinging motion. A stable pendulum will swing back and forth evenly without losing energy, while an unstable pendulum will lose energy and eventually come to a stop.

3. What is a fixed point in a pendulum?

A fixed point, also known as a point of equilibrium, is where the pendulum comes to rest when it is not in motion. This point is determined by the length of the pendulum and the force of gravity acting on the weight.

4. How does the length of a pendulum affect its stability?

The longer the length of a pendulum, the more stable it will be. This is because a longer pendulum has a longer period of swing, allowing it to maintain its motion for a longer period of time before coming to a stop.

5. Can the weight of a pendulum affect its stability?

Yes, the weight of a pendulum can affect its stability. A heavier weight will cause the pendulum to swing slower and have a longer period of swing, making it more stable. However, if the weight is too heavy, it may cause the pendulum to lose its regular motion and become unstable.

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