Tuesday, August 25, 2009

frequency amplitude dependence

So, consider a dynamical system that has an elliptical fixed point. The eigenvalues of the map linearized about the fixed point gives the frequency associated with rotation about that fixed point. Move away from the fixed point and the frequency changes. This is one definition of a non-linear system. A pendulum, for example, has this property; and this changing frequency explains why it doesn't make a perfect clock.

This simple property is what I am trying to get to the bottom of. It becomes less simple when one wants to think of this continuum of frequencies as the spectrum of some operator. So the operator has become rather more complicated from the simple matrix expanded about the fixed point. Suddenly it has taken on an infinity of eigenvalues! Once we say this, however, we are left with the question of what the eigenvectors mean. Are they functions that are basically delta functions except where the orbit is? This is a rather messy singular quantity. How about the eigenvectors of the adjoint operator? These are the quantities we've been looking at, but they seem to be rather strange.
This is the line thinking I've been following lately. I'm hoping I can find a simple reference that explains all this. But I can't seem to.

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