Comments on: Entrainment by oscillations http://alanrendall.wordpress.com/2012/02/25/entrainment-by-oscillations/ A mathematician thinks aloud Thu, 09 May 2013 07:10:22 +0000 hourly 1 http://wordpress.com/ By: hydrobates http://alanrendall.wordpress.com/2012/02/25/entrainment-by-oscillations/#comment-803 Sun, 26 Feb 2012 09:54:58 +0000 http://alanrendall.wordpress.com/?p=1316#comment-803 Hi Juliette,

Thanks for your comment. The class demonstration you mention sounds very interesting. I should talk to Bernold about this subject when I see him again.

]]> By: Juliette Hell http://alanrendall.wordpress.com/2012/02/25/entrainment-by-oscillations/#comment-799 Sat, 25 Feb 2012 20:20:30 +0000 http://alanrendall.wordpress.com/?p=1316#comment-799 Hi!
I did not look up in the paper you quote, but with the few details you wrote here, I have the following intuition…
This matrix measure is the derivative of the function which maps a matrix on its matrix norm, taken at the identity in direction of A. The condition says the slope in this direction is negative, meaning going this way, you’ll find contractions. Furthermore, the flow near an equilibrium for small t has leading order id+tA, i.e. it runs in the direction mentioned above.
My intuition wisely ignored the non-autonomous thing, and why you require the uniform negativity …
Another thing is: if you think of A diagonal, µ(A) negative just means that it has only stable eigenvalues. (at least with the usual matrix norm)

And last but not least, I had to think about the experience Bernold showed in class. The one with the metronomes sitting on a board, the board on two rolling cans, and (anti)synchronization taking place or not, depending on the original tuning of the metronomes. I love it!

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