I have to find takens normal form for system , where . Can anybody help me please? Thanks

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- November 2nd 2010, 05:49 AMsiditakens normal form
I have to find takens normal form for system , where . Can anybody help me please? Thanks

- November 2nd 2010, 05:59 AMAckbeet
I'm finding it extremely difficult to find information on the "Takens normal form". Most of what I see is in research papers. Is this the Bogdanov-Takens normal form? Or the Poincare-Takens normal form? Could you please provide a definition of the Takens normal form?

- November 2nd 2010, 06:04 AMsidi
It's Bogdanov-Takens normal form.

- November 2nd 2010, 06:32 AMAckbeet
Hmm. Still very hard to find info on this. It looks like the Bogdanov-Takens normal form is something seen in bifurcation theory. I found one reference that defined the Bogdanov normal form (which can also be derived using the Takens Theorem, so I'm assuming it's what you mean here), that says you need to write the following system:

where as

where and this system is versal.

This definition is on page 263 of Medved's*Fundamentals of Dynamical Systems and Bifurcation Theory*. Medved defines*versal*in this fashion:

The germ is called the versal deformation or the versal unfolding of the germ if for an arbitrary natural number an arbitrary -parametric deformation of the germ is orbitally topologically equivalent to a germ induced from the germ We also say that the germ is versal.

I have absolutely no idea what all this means. However, I will say that your system is linear, whereas the Bogdanov-Takens normal form appears to be defined only for linear systems. How does this definition carry over for linear systems?