Linear approximation to a system in the neighbourhood of the origin?

Question

What would a linear approximation to the following system near the origin be?

$${dx \over dt}=-y-x(x^2+y^2), {dy \over dt}=x-y(x²+y²)$$

I have no idea how to find this... I'm looking at this as an example, can you help me?

Thank you very much!

Answer 1

If we are near the origin then $\sqrt{x^2 + y^2}$ is small. In fact all terms $x^py^q$ such that $p + q > 1$ are of smaller magnitude than terms in $x$ and $y$. So the approximation is relatively simple:

$$x' = -y, \ \ \ y' = x$$

This is a linear approximation in as much as $x'$ and $y'$ are equal to separate functions $f(x,y)$ which only have linear terms. That is, $f(x,y) = ax + by + c \ $ for constants $a, b, c$. If we had a linear approximation about another point $(p,q)$, then the linear approximation looks like

$$f(x,y) = a(x-p) + b(y-q) + c$$