Invertibility conditions for J_φ ensuring existence of the surrogate system

Investigate conditions under which the Jacobian matrix J_φ(x) = [∂(P_N φ_{λ_i})/∂x_j](x) is invertible on a domain D, thereby ensuring that the surrogate vector field \widetilde{F}(x) = J_φ(x)^{-1} [λ_1 P_N φ_{λ_1}(x), …, λ_n P_N φ_{λ_n}(x)]^T is well-defined and can be used to provide stability guarantees based on the truncated Koopman eigenfunctions.

Background

To strengthen stability guarantees, the paper constructs a surrogate system with vector field \widetilde{F}(x) defined via the truncated principal eigenfunctions P_N φ{λ_i}. This construction requires the invertibility of the Jacobian Jφ(x) of the mapping formed by these eigenfunctions.

The authors explicitly caution that the existence of the surrogate system can be hindered by invertibility issues of J_φ and state that this aspect requires further investigation, which they leave for future research.