Dynamical system and representation theory

In this talk, a practical method is described for computing the classical normal form of vector fields near the bifurcation point. Some necessary formulas are derived and applied to the anharmonic oscillator, the Bogdanov-Takens bifurcation, the 3D nilpotent problem, and elastic pipe conveying fluid,  to demonstrate the applicability of the theoretical results.
 Then, a review will be given of the developments in the last decade concerning the classification of unique normal forms in 3D nilpotent problems.
  This work generalizes the work on the Bogdanov-Takens bifurcation and its unique normal form, which took off with the papers of Baider and Sanders (1991-92).  Here the application of the Jacobson-Morozov theorem led to a systematic approach to computing the unique normal form in a number of cases. Some of the subcases of the 2D double-zero bifurcation analysis are still open.
One can imagine that the complications of analyzing the 3D triple-zero bifurcation are rather challenging. Nevertheless, progress has been made in the last decade and it is time to list what has been done and what still needs to be done. We apply the Jacobson–Morozov theorem to embed this class of three-dimensional vector fields into a sl_2-triple. Three irreducible families are produced this way.  The first task is to find the structure constants of these families. In this talk, we also show how the Clebsch-Gordan formula is employed to find explicit formulas for the structure constants. We demonstrate that these families can generate some Lie sub-algebras with respect to the triple-zero bifurcation point, thereby creating smaller subproblems that can be studied independently in their own right (like the Hamiltonian case in the 2D analysis).
Further, we discuss possible generalizations toward a general n-dimensional theory.