Using the Hartman-Grobman theorem on a polar system I have a system of differential equations, in polar form. It is quite simple this way, but rather ugly if I transform it into Cartesian coordinates. Is there any way to apply the Hartman-Grobman theorem without having to change coordinates?
The equation is $$r' = r^3-r$$ $$\theta' = \sin(\theta)^2-\mu$$
Please DO NOT SOLVE THE PROBLEM FOR ME. I am just giving it so you know what kind of equation I am talking about. The goal is to analyze the five equilibria using Hartman-Grobman without changing coordinates.
Please just give me hint(s).
 A: As previous comments have mentioned, the application of the Hartman-Grobman theorem does not depend on the specific choice of coordinates, so you could just go ahead and determine the eigenvalues of the Jacobian at every equilibrium -- as long as the coordinates are well-defined. This is usually a non-issue, but it is good to keep in mind. Consider the following.
Polar coordinates are not defined in the origin. However, your example system has an equilibrium at the origin, basically because $r=0$ is an equilibrium of the $r'$-equation and the angular variation vanishes at the origin due to the singularity of the coordinate system. You could verify this by changing to Cartesian coordinates and determine the existence of the limit $(x,y) \to (0,0)$.
However, when you determine the Jacobian of the system at the origin, you will find that the answer does depend on the direction in which you take the limit $(x,y) \to (0,0)$. In other words, you have a two-component system $z' = f(z)$, where the function $f$ is not smooth at the origin. In particular, $f$ is continuous but not differentiable at the origin. Hence, the Hartman-Grobman theorem cannot be applied at this equilibrium -- the exact conditions of the H-G theorem depend a bit on the formulation, but the right hand side $f(z)$ needs to be at least $C^1$, in order to be able to speak about the Jacobian.
To summarise: for equilibria in the domain of definition of the coordinates, you can use the Hartman-Grobman theorem without problems. For equilibria outside the domain of definition of the coordinates, it depends on the equilibrium and on the system.
