2023/04/18 by Maximilian Engel, Engel, Maximilian, Georg A. Gottwald +1
Computer Science · Engineering · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Nonlinear Dynamics and Pattern Formation #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2304.08797
openalex publication_date 2023/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Canards are a well-studied phenomenon in fast-slow ordinary differential equations implying the delayed loss of stability after the slow passage through a singularity. Recent studies have shown that the corresponding maps stemming from explicit Runge-Kutta discretizations, in particular the forward Euler scheme, exhibit significant distinctions to the continuous-time behavior: for folds, the delay in loss of stability is typically shortened whereas, for transcritical singularities, it is arbitrarily prolonged. We employ the method of modified equations, which correspond with the fixed discretization schemes up to higher order, to understand and quantify these effects directly from a fast-slow ODE, yielding consistent results with the discrete-time behavior and opening a new perspective on the wide range of (de-)stabilization phenomena along canards.