@ecterrab

Hi,

First of all, thank you very much for your intention to help!

I will start with my purpose, then I will give the definitions of normal and hypernormal forms in my understanding. Finally, I will mention works (some of them with Maple code) that deal with this issue.

1. My purpose - to investigate the behavior of the non-linear dynamic system near the fixed points (including bifurcation analysis).

Definitions:

2. Normal Form - Under this term I assume that the differential equation has two parts: (a) linear (defined in the terms of Jordan normal form) + (b) non-linear part. Usually, this form is received after the Taylor expansion,

3. Hypernormal form - sometimes I see in the literature the other name - "simplest form". In my understanding, this is a continuation of the reduction of the non-linear term in normal form as much as it is possible.

4. My expectation - (a) given any system of non-linear differential equations, I would like to receive normal and hypernormal (simplest) form. Today in the literature exists a number of algorithms for such a reduction but for very specific differential equations. In particular, for the differential equations which have zero eigenvalues. (b) since the reduction requires changing the coordinates, I would like to know, how I can get back from the reduced form to the origin system.

Literature:

a. Edneral, V. F. (2007, September). An algorithm for construction of normal forms. In *International Workshop on Computer Algebra in Scientific Computing* (pp. 134-142). Springer, Berlin, Heidelberg - **describes the algorithm for the normal form reduction.**

b. Yuan, Y., & Yu, P. (2001). Computation of simplest normal forms of differential equations associated with a double-zero eigenvalue. *International Journal of Bifurcation and Chaos*, *11*(05), 1307-1330 - *including Maple code.*

c. Murdock, J. (2003). Hypernormal form theory: foundations and algorithms. *Mathematics Preprint Archive*, *2003*(12), 225-271 - *gives a comprehensive review of the existing reduction methods to the hypernormal form*.

d. Gamero, E., Freire, E., Rodríguez-Luis, A. J., Ponce, E., & Algaba, A. (1999). Hypernormal form calculation for triple-zero degeneracies. *Bulletin of the Belgian Mathematical Society-Simon Stevin*, *6*(3), 357-368 - **shows the reduction to the hypernormal form for the system of differential equations with triple zero eigenvalues**.

Please note, I saw that there exists add-in to Maple: **BifTools**. But, it's also limited to some specific types of differential equations.

If you need any additional information, please let me know.

I did not attach my problem because we still working on it and the problem may be two or three-dimensional where the eigenvalues are not zero (at least not all of them).

Thanks in advance,

Dmitry