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Archived Nonlinear Dynamics and Chaos, Strogatz: 2.1.5

  • Thread starter Niteo
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1. Homework Statement
a) Find a mechanical system that is approximately governed by [itex]\dot{x}=sin(x)[/itex]
b) Using your physical intuition, explain why it now becomes obvious that x*=0 is an unstable fixed point and x*=[itex]\pi[/itex] is stable.

2. Homework Equations

[itex]\dot{x}=sin(x)[/itex] (?)

3. The Attempt at a Solution
I'm thinking a pendulum can be used as a mechanical system that varies with sinθ, but I'm not sure how to solidify my answer.

Could it possibly be an inverted pendulum in a very viscous medium?
 
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haruspex

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Consider a particle sliding over terrain y=f(x). Assuming conservation of energy (and some convenient total) what is the PE at x? What does that yield for f?
 

epenguin

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Question of understanding what they want. I think all you need to do is draw a diagram of the given function (familiar!) extending a little further on both sides than the points mentioned; xdot is the ordinate, but more important, with little horizontal arrows show which way x Is moving on each side of the named points, and you will soon see what they mean about stability/instability. Just explain this in your own words.
 
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Immerse your pendulum in honey and evolve your equations. When you finish, you're gonna get a second order differencial equation. At this point, you gotta be audacious and destroy the second order term. Voilà.
 

haruspex

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Immerse your pendulum in honey and evolve your equations. When you finish, you're gonna get a second order differencial equation. At this point, you gotta be audacious and destroy the second order term. Voilà.
The OP never came back, and this is years old.
But for what it's worth, I believe my suggestion in post #2 gives a very easy model, no approximations needed.
 

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