Determining the normal form of a map $f=\lambda z+\cdots$, where $\lambda>0$ and $\lambda\neq 1$, is a classical problem in dynamical systems. The goal is to “simplify” $f$ by finding a parabolic change of coordinates $\varphi=z+\cdots$ such that $\varphi^{-1}\circ f\circ\varphi=f_0$, where $f_0$ is a chosen normal form. This problem has been successfully solved in several settings, including analytic diffeomorphisms, various classes of real maps, Dulac maps, and logarithmic transseries. In this work, we investigate the normal form problem in the broader framework of surreal numbers. We review the main techniques used in existing normal form constructions and discuss how these methods can be extended to the surreal number setting. The study of normal forms in this context is connected with the dynamics of analytic planar vector fields because such objects arise as asymptotic expansions of Poincar'e maps associated with hyperbolic and semi-hyperbolic polycycles of such fields. This is connected to the classical Dulac problem concerning the non-accumulation of limit cycles near polycycles of analytic planar vector fields. A deeper understanding of the formal dynamics of these asymptotic expansions may provide valuable insight into the dynamics of the corresponding Poincar'e maps and could contribute to further progress on understanding of the Dulac problem. This is joint work in progress with V. Mantova, J.-P. Rolin, and T. Servi.

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D. Peran, M. Resman, J.-P. Rolin, and T. Servi, \emph{Normal forms of hyperbolic logarithmic transseries}, J. Differential Equations 348 (2023), 154–190