Conditions for super-linearization of general nonlinear systems

Characterize the necessary and sufficient conditions under which a general nonlinear dynamical system admits super-linearization by augmenting variables with non-polynomial coordinate transformations so that the system can be embedded into a (finite or effectively finite) linear dynamical system with convergent and well-conditioned behavior.

Background

Super-linearization extends Carleman by introducing additional, non-polynomial variables to improve expressivity of the linear embedding. While this can increase modeling capability, stability and conditioning issues arise and there is no guarantee of convergence.

The authors explicitly state that the criteria enabling super-linearization for general nonlinear systems are unknown, highlighting a theoretical gap in understanding when such embeddings exist and are numerically reliable.

References

Moreover, the conditions under which a general dynamical system can be super-linearized remain an open question.

— Globalizing the Carleman linear embedding method for nonlinear dynamics  (2510.15715 - Novikau et al., 17 Oct 2025) in Subsection “The Carleman and Koopman embedding methods”

Both constructions are recent, and how far each extends is an open program rather than a closed result.

— Is higher-order physics different?  (2609.20759 - Villegas et al., 17 Sep 2026) in Section 4, “A matter of representation”