Topological equivalence of ODEs

Let's have ode $x' = f(x)$, $f(0) = 0$, $x\in \mathbb{R^n}$. There is clasical theorem that states that if all eigenvalues of ${\rm Df}(0)$ have nonzero real part, than $x'=f(x)$ and $x'={\rm Df}(0) x$ are topological equivalent at neighbourhood of $0$.

Is there any generalization of this theorem?

Like take truncuated(to $n$-th term) taylor expansion of $f(x)$ denote it $T_n(x)$. Than under what condition is $x'=f(x)$ topologicaly equivalent to $x'=T_n(x)$ at neighbourhood of $0$?

We know the condition for $n=0,1$.

For $n=0$ has to be $f(0)\neq 0$.

For $n=1$ has to ${\rm Df}(0)$ have eigenvalues with nonzero real part and $f(0)=0$

If conditions for $n=0,1$ fail is there any know condition for $n=2$ ?

In general, are there any known conditions for $n\geq2$?

Solution 1:

First, you use $n$ in your post twice, in different meanings. First $n$ denotes the dimension of the phase space, and second, $n$ means the order of the terms you keep. I will use $d$ instead of $n$ in the former case.

A lot is known about the case $d=2$, although not everything. Assume that we are given polynomial system on the plane $$ \dot x=P(x,y),\\ \dot y=Q(x,y), $$ where $P$ and $Q$ are polynomials, and assume that $(0,0)$ is isolated. Then it is either monodromic or not. The former means that there are no specific directions the orbits approach this point (focus or center), the latter means that there are specific directions. In this non-monodromic case we know that a neighborhood of zero is a union of sectors, which can be either parabolic (think of a node), or hyperbolic (think of a saddle), or elliptic (think of a family of homoclinics to zero). So, which terms of $P(x,y)$ and $Q(x,y)$ are responsible for the behavior, and which can be dropped? For this build a Newtons diagram, which is a convex hull of the set of points with coordinates $(\alpha,\beta)$, if there is a term in the system of the form $x^\alpha y^\beta$. Whatever you can see from the origin are the terms that are essential for the behavior. There is an algorithm how to proceed to figure out the exact structure (see, e.g., Qualitative Theory of Planar Differential Systems). So, what is not known? For example, the question to distinguish center from focus in a finite number of steps is still open.

Much much less is known for an arbitrary dimension $d\geq 3$. I can recommend try to read Power Geometry in Algebraic and Differential Equations but this is definitely not an introductory read.