# Computing the fundamental matrix and its monodromy matrix

Assume $\dot x(t)=v(x,t)$ is a $T$-periodic, with respect to $t$ dynamical system. That is: $x\in\mathbb R^n,\quad t\in\mathbb R,\qquad v(x,t+T)=v(x,t)$.

Let $x_0$ be a smooth periodic solution in elementary functions. How can one use Mathematica to compute symbolically a fundamental matrix for the linearized system around $x_0$ and its monodromy matrix?

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You'll get better and faster answers if you show what you've tried and reference any documentation you haven't understood. –  Jagra Jul 16 '13 at 14:31
Please note that the community considers it good practice for a questioner to accept answers that address the question. Best to wait a full day to see what else comes, you may get better ones, but the courtesy of acceptance acknowledges the effort and interest of those that provide answers and makes them feel appreciated ;-) –  Jagra Jul 16 '13 at 15:50
The question should be expanded to give more details, and to include some Mathematica code you have already tried. Otherwise it's more a math question, the answer to which you can look up in textbooks. –  Jens Jul 16 '13 at 17:36
I know how to answer the math question behind this post. It is the Mathematica implementation I am strugling with. As you can see from this question, @Jens, I can't even find a fixed point of a composition of elementary functions. –  Student Jul 19 '13 at 6:22

Just a start and really just an extended comment to help you figure this out for yourself.

Take a more detailed look at Fundamental Matrix Solutions. A bit more detail may give you a better idea of how to approach this in Mathematica.

As,

...a monodromy matrix is the inverse of the fundamental matrix of a system of ODEs evaluated at zero times the fundamental matrix evaluated at the period of the coefficients of the system.

you need to know how to do basic matrix operations. This old (v3) overview should give you the general idea.

Ok, can't help myself as the question caught my interest. See this Differential Equations with Linear Algebra: Mathematica Help page 16:

All I really know how to do is look stuff up. Anyone could have done this in 5-10 minutes.

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+1 for the link. I was not familiar with fundamental matrix solutions. –  rcollyer Jul 16 '13 at 15:01
@rcollyer - Me neither, but its kind of cool;-) –  Jagra Jul 16 '13 at 15:11
I don't know if the correspondence is accurate, but it reminds me of path integrals and the lead into Feynman diagrams. There is probably something there, but I think it may take some time to tease out. Or, more correctly, get my brain in gear to do it! :) –  rcollyer Jul 16 '13 at 15:41
@rcollyer - Wow, what a great connection! That makes me even more interested in these things. Thx. –  Jagra Jul 16 '13 at 15:51
@rcollyer Some people see Feynman path integrals everywhere. They may need professional help. But seriously, (a) the answer doesn't really deal with the monodromy matrix, and (b) the answer has no mathematical relation to path integrals. –  Jens Jul 16 '13 at 17:22