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For small systems of differential equations, it is easy to solve for the eigenvalues of the Jacobian. From there, it is then easy to interpret what those eigenvalues mean, and to see how changing certain parameters might push an eigenvalue across a threshold to change the stability of a fixed point, or produce oscillations, or whatever.

But generally, for anything but very small systems of differential equations, the eigenvalues can only be estimated numerically. So you don't have formula for the eigenvalues that can be intuitively interpreted.

So imagine you have a system of differential equations and you can't explicitly solve for the eigenvalues, you can only obtain numerical estimates. If, under certain conditions, a particular eigenvalue is doing something of interest (such as switching from negative to positive), is there any way to gain some insight into what is causing the movement in that eigenvalue? How would one approach a problem like that? Or is this even a question worth asking?

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I'm not sure i understood your question,but i'll give it a try. You could easily use a numerical method (like power method or inverse power etc) to find all eigenvalues of interest. I believe the best way to interpret some interesting movement is to perform an iterative method that holds (all-1) parameters constant and change with some predefined step the value of the non-constant parameter until something interesting happens.In every iteration you will use the numerical method of your choice for computing the eigenvalues. By summing up the results from all iterations you could easily gain some insight about which parameters affects which eigenvalue.

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    This$ $is basically what I ended up doing. It's like sensitivity analysis of eigenvalues, which some people have done before to approach this type of problem.2012-02-28