# System of Nonlinear ODEs.

What is the procedure to solving the following system of ODEs?

$$\frac{d}{dt}\begin{pmatrix} x\\ y\\ z \end{pmatrix}=\begin{pmatrix} -\sigma & \sigma & 0\\ \rho & -1 & -x\\ y & 0 & -\beta \end{pmatrix}\begin{pmatrix} x\\ y\\ z \end{pmatrix},$$

where $\sigma$, $\rho$, $\beta\in\mathbb{R^+}$.

Is this a nonlinear system? Apparently, it has something to do with the Lorenz attractor.

I picked up my old differential equations book, and the second to last chapter was on systems of linear ODEs. It only covers systems where the biggest matrix up there is made up of constants alone; I could not find something that resembled anything like this.

I am now very curious as to how one would go about solving this problem. Any hint would be greatly appreciated.

Today, I discovered that these are indeed the Lorenz equations:

\begin{align} \dot x&=\sigma(y-x)\\ \dot y&=x(\tau-z)-y\\ \dot z&=xy-\beta z \end{align}

$\sigma=10$, $\tau=28$ and $\beta=8/3$

Moreover, every single approach to them that I have seen is numerical. Can they be solved analytically?

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Yes, it is nonlinear. – Siminore Sep 8 '12 at 8:06
Of course it is nonlinear! We have products like $x y$ and $x z$, which are monomials of degree $2$. Moreover, I do not think that the Lorenz system is integrable, so an analytical solution is probably impossible. – Rod Carvalho Sep 11 '12 at 14:18