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I have been studying a particular ergodic system and it has become apparent that solutions to differential equations of the form $M(x,y,z,t)\begin{pmatrix} \frac{dx}{dt} \\ \frac{dy}{dt} \\ \frac{dz}{dt} \end{pmatrix} = \begin{pmatrix} f(x,y,z,t) \\ g(x,y,z,t) \\ h(x,y,z,t) \end{pmatrix}$ where $M$ is a $3\times 3$ matrix reveal important properties of the system. Is it possible to solve such a system explicitly in general? In most of the cases that arise, the functions (including entries in the matrix) involved are lengthy polynomials in $sin(x),cos(x),sin(y),cos(y),sin(z),cos(z),t,sin(tx),cos(tx),sin(ty),cos(ty),sin(tz),cos(tz)$and so clearly solving the equation will be challenging if it is even possible. If no explicit solution is possible, is there a standard computational technique to determine an approximate solution? Furthermore, what can be said about behavior of solutions where $\operatorname{det}M = 0$ or where the matrix fails to be diagonizable?

EDIT: After further thought I have been able to reduce this to a $2\times 2$ system in theb variables $x,y,t$. To clarify, I am onoly concerned with solutions within a region $D\subset R^3$ in which I know from the geometric interpretation of the differential equations that a solution exists, but I do not know that $\operatorname{det}M = 0$ in $D$. One reason I am interested in the behavior of solutions when the determinant is $0$ is the hope that I can use the poor behavior of these solutions to show that the determinant must be nonzero. As I expected, it seems that an exact solution is not possible for these equations in general (and I presume this is also true for the simplified version), so I would like to know where I can find references to standard numerical approximation techniques and bounds on the instability of solutions to these equations, as I wish to be able to show that an approximate solution is "close enough" to the true solution in order to use it for rigorous proofs.

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    Well, since for example the Lorentz attractor is a system of this form (with $M$ = identity), you can't hope for an explicit solution in general. (Rather the opposite: you need some kind of miracle in order for a system like that to be explicitly solvable.)2011-03-28
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    Do you have an explicit $M$ in mind? What $M$ behaves like will make a huge difference, especially since you suspect it might be singular. (I'm also not sure I understand: is it singular, or are you trying to prove it is nowhere singular?)2011-03-29
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    No, I do not have an explicit $M$ in mind, but I do know that its entries are polynomials in $t$ and the sines and cosines given above. I *suspect* that $M$ is nowhere singular in $D$ (the only region I care about), but this is in fact equivalent to an open question.2011-03-29

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