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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.) –  Hans Lundmark Mar 28 '11 at 15:03
    
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?) –  Sam Lisi Mar 29 '11 at 0:13
    
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. –  Alex Becker Mar 29 '11 at 0:55

1 Answer 1

If $M$ is invertible, proceed to multiply both sides by $M^{-1},$ you will have a standard $3 \times 3$ system: \[ \frac{\mathrm{d}}{\mathrm{d}t} \left(\matrix{x \cr y \cr z\cr}\right) = \left( \matrix{ F(x,y,z,t) \cr G(x,y,z,t)\cr H(x,y,z,t)\cr}\right). \]

Of course it's impossible to solve such a system in general, but the standard numerical methods for ODE systems can be used.

If $M$ is not invertible, it may be doubtful whether solutions will even exist. What you have here is a Differential-Algebraic Equation problem (DAE).

Please See example: Differential Algebraic Equation

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The differential equations in question admit a geometric interpretation that ensures a solution will exist anywhere within the region $D$ of $R^4$ that is of concern, yet I have not been able to prove that $\operatorname{det}M \neq 0$ in $D$, which is why I am concerned with the behavior of these solutions (partly in the hope that their behavior will be errant enough to prove that the determinant cannot be zero in $D$). As for numerical methods, do you know of any references for bounds on the instability of these solutions (so that I can prove an approximate solution is "close enough")? –  Alex Becker Mar 28 '11 at 17:13
    
@Alex, if you are interested in numerical solutions, Matlab has some built-in functions to deal with this kind of problem. You can find the relevant page of the user manual here mathworks.com/help/techdoc/ref/ode23.html. –  Sam Lisi Mar 29 '11 at 0:11

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