How to write a system of equations as a dynamical system?

Question

I am having a lot of trouble understanding how to move from a system of ODE's to a dynamical systems point of view (that will allow me to make a phase-plane analysis).

Assume I want to write the following system (I invented it just as an example): $$ \begin{split} y''+2x+3y'+4x'=0\\ 2y''+2x-4y'+2x'=0 \end{split} $$ in a matrix form: $$ \frac{d}{dt} \begin{bmatrix} x\\x'\\y\\y' \end{bmatrix} = A\begin{bmatrix} x\\x'\\y\\y' \end{bmatrix} $$ where $A$ is a $4\times 4$ matrix.

I know I can write this system as: $$ \begin{bmatrix} 0&0&0&1\\0&0&0&2\end{bmatrix} \frac{d}{dt} \begin{bmatrix} x\\x'\\y\\y' \end{bmatrix} = \begin{bmatrix} -2&4&0&-3\\-2&-2&0&4\end{bmatrix} \begin{bmatrix} x\\x'\\y\\y' \end{bmatrix} $$ but it does not help me much (the inverse of the matrix in the LHS is not a well defined notion...).

In addition, I know that if I would only had the first equation, I would be able to write it as: $$ \frac{d}{dt} \begin{bmatrix} x\\x'\\y\\y' \end{bmatrix} = \begin{bmatrix} 0&1&0&0\\0&0&0&0\\0&0&0&1\\-2&-4&0&-3 \end{bmatrix}\begin{bmatrix} x\\x'\\y\\y' \end{bmatrix} $$ but how can I combine the two equations into such a form (in order to make a phase-plane analysis) ??

Just to clarify- I know that one possible solution is to isolate $y''$ from the first equation, substitute in the second one, and then move to a matrix form, but I don't want to do it (too messy and not very helpful in case of a lot of variables and a lot of equations).

Will you please help me?

Thanks

Answer 1

Note that $\det \mathbf{A}=0$ and both $x'$ and $y'$ appear on both sides. How about in this way?

$$\left( \begin{array}{c} x' \\ y'' \end{array} \right)= \left( \begin{array}{cc} -\frac{1}{3} & -\frac{5}{3} \\ -\frac{2}{3} & \frac{11}{3} \end{array} \right) \left( \begin{array}{c} x \\ y' \end{array} \right)$$

Answer 2

Strictly speaking, if you decide just to write down two linear equations involving $x$, $y$ and their (first and second) time derivatives, there is no a priori reason to assume those two equations are consistent with each other. Therefore, the best thing to do is to write each equation separately as a four-dimensional linear system of the form $\mathbf{x}' = A \mathbf{x}$, and solve those two. If a solution obeys both original equations, it has to solve both dynamical systems simultaneously. By comparing their general solutions, you can then see if these overlap in some way.

Note that I didn't go into details regarding the example you gave, since I've got the impression your question is not about this example in particular, but about the general approach.