Beyond DSA: Conjugacy-based Comparison of Dynamical Systems

Comparing whether two dynamical systems implement the same computation despite differences in coordinates or measurements is a central problem in neuroscience and machine learning. Dynamical Similarity Analysis [DSA; Ostrow et al., 2023] addresses this problem by aligning finite-dimensional Koopman approximations of the two systems through an orthogonal similarity transformation. Here we show that orthogonal alignment is neither necessary nor sufficient for topological conjugacy: genuinely conjugate systems may be related by a non-orthogonal basis-transfer matrix that DSA cannot capture, while non-conjugate systems may have orthogonally equivalent Koopman operators that DSA will fail to distinguish. We then use this observation to formulate \emph{Conjugacy-based Similarity Analysis} (CSA), which restricts alignments to those induced by candidate state-space bijections rather than arbitrary orthogonal matrices. We prove that CSA's fitted alignment is the finite-data projection of the composition operator associated with the candidate bijection, and use controlled examples to show why this distinction matters when observable dictionaries are chosen explicitly or implicitly from data. Together, these results clarify what Koopman-based similarity measures must ensure to support claims of identifying conjugacies between computational systems.