I suggest Shilnikov's book.

Thanks Justus, seems like a lot of what I was looking for is covered in this book. Could it be that Shilnikov uses partially other terminology than other authors, like Wiggins or Arrowsmith?

One more question: if I reduced the dynamical system at some non-hyperbolic fixed point to its normal form, is it sufficient to investigate the equations with zero linear part in order to analyze the bifurcation behavior? As I understood, this is basically equivalent to the reduction to the center manifold, is it?

It rather depends on what you mean by bifurcation of vector fields. Just local bifurcations of equilibria, bifurcations of periodic orbits, or global bifurcations, routes to chaos ...

Two books are Guckenheimer and Holmes (which does treat codimension 2 bifurcations of equilibria) and Golubitsky and Schaeffer (and Stewart for vol II, which includes symmetric bifurcations). The first book (G&H) has a chapter on codim 2 local bifurcations. The other book (G&S) has a clear statement listing all bifurcations of equilibria of codimension up to 3, but in one state variable.

Edit: Reading your question again, I guess you do mean local bifurcations of fixed points. So the books I mentioned should cover everything.

Although it might be overkill, containing much more than you ask for, I suggest

"Elements of Applied Bifurcation Theory" by Yuri Kuznetsov. There is an edition from 2004, and as such, is the most recent and comprehensive textbook.

If you look at it, you can easily restrict to the relevant sections in chapters 3,6 and 7 for codim 1 and chapter 8 for codim 2 bifurcations. His text is also a good merge/mixture of Russian and Western school bifurcation theory.

" if I reduced the dynamical system at some non-hyperbolic fixed point to its normal form, is it sufficient to investigate the equations with zero linear part in order to analyze the bifurcation behavior? " -- sounds about right. If you are not specifically looking for an analytic result, it is best to use specialized software of automatic continuation for such analyses. Specifically, the following packages can help:

- matcont is a matlab interface to the newest of the new in bifurcation theory. Very nice, but I don't work with matlab.

- AUTO2000/2007 (basically by the same authors as matcont) allows for auto-continuation. 2000 interfaces in C, and 2007 in F. There's also python frontends, as well as B. Ermentrout's XPPAUT.

Whether you can reduce to the zero linear part (the kernel) depends on what you're looking for. If it's only equilibria you want then yes you can (and this would be the approach in Golubitsky-Scheaffer). On the other hand, if you want more about the dynamics then in general you can't reduce to zero linear part - you need to include the nilpotent linear part. And specifically the set of 2x2 nilpotent matrices has codimension 2, so if you have 2-parameter families, you might expect to see them. This is the Bogdanov-Takens bifurcation (or vice versa?)

You can find a detailed description of this in the book of Guckenheimer and Holmes and that of Kuznetsov that Meijer recommended.

Kuznetsov Yu. Elements of applied Bifurcation Theory.

In general we can classify local codimension 1 bifurcations: in two way : saddle node bifurcation and Hopf bifurcation. In case of codimension 2 bifurcation it will be of 5 types in nature. degenerated saddle node bifurcation (violation of nondegeneracy condition i.e. the coefficient of quadratic term is zero, known as Cusp bifurcation), Bogdanov Takens bifurcation ( it is a codimension 2 bifurcation since the parameters can be changed in such a way such that both the eigenvalues are zero, this bifurcation consists of a series of bifurcations depending on the change of two parameters.), Degenerated Hopf or Bautin bifurcation ( this can be derived whenever the nondegenaracy condition is violated in Hopf bifurcation), Hopf-Hpf bifurcation ( In this case we need at least four dimensional phase space, varying two parameters the linearization matrix has two pair of purely imaginary eigenvalues), fold-Hopf bifurcation ( In this case we need at least three dimensional phase space, varying two parameters the linearization matrix has one zero eigenvalue and a pair of purely imaginary eigenvalues). So, given a 2D vector field we have only three possibilities for codimension 2 local bifurcation. The book "Elements of Applied Bifurcation Theory"of Yuri Kuznetsov has considered normal forms and unfolding of all such class of bifurcations, I hope you will get your answer in a proper way if you follow this book.

I guess that the second question might be concerning Bogdanov-Takens bifurcation or degenerated Bogdanov-Takens bifurcation.

To Jochen Damerau :

The method of analysis of structure and asymptotics of non-hyperbolic equilibrium of 2D-ODE system using Newton diagram, was developed in 1980 and applied it a lot: for example,

Berezovskaya F., Karev G., Arditi R. (2001). Parametric analysis of the ratio-dependent predator-prey models. J. Math. Biol., 43, Berezovskaya, Novozhilov, Karev,(2007) Population models with singular equilibrium, Mathbiosi, 208,

Generalization of the method is in

F. Berezovskaya Asymptotics of orbits of a Kolmogorov type planar vector field with a fixed Newton polygon , PROCEEDINGS of the AMS 2014,2671-2681