I have to find takens normal form for system $\displaystyle x'=Ax$, where $\displaystyle A=[0,1,0;0,0,1;0,0,0]$. Can anybody help me please? Thanks
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?
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:
$\displaystyle \dot{y}_{1}=y_{2}$
$\displaystyle \dot{y}_{2}=\varepsilon_{1}+\varepsilon_{2}y_{1}+y _{1}^{2}+y_{1}y_{2}Q(y_{1},\varepsilon)+y_{2}^{2}\ Phi(y,\varepsilon),$ where $\displaystyle Q,\Phi\in C^{r}, Q(0,0)=\alpha\not=0,$ as
$\displaystyle \dot{y}_{1}=y_{2},$
$\displaystyle \dot{y}_{2}=\varepsilon_{1}+\varepsilon_{2}y_{1}+y _{1}^{2}+by_{1}y_{2},$
where $\displaystyle b=\text{sgn}(\alpha),$ 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 $\displaystyle [G]_{x_{0},\epsilon_{0})}\in V^{r}_{(x_{0},\epsilon_{0})}(R^{k},r^{n})$ is called the versal deformation or the versal unfolding of the germ $\displaystyle [v]_{x_{0}}\in V^{r}_{x_{0}}(R^{n}),$ if for an arbitrary natural number $\displaystyle p$ an arbitrary $\displaystyle p$-parametric deformation of the germ $\displaystyle [v]_{x_{0}}$ is orbitally topologically equivalent to a germ induced from the germ $\displaystyle [G]_{(x_{0},\epsilon_{0})}.$ We also say that the germ $\displaystyle [G]_{(x_{0},\epsilon_{0})}$ 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?