2024/11/07 by Engel, Maximilian, Kühn, Christian, Petrera, Matteo +1
#500 Naturwissenschaften und Mathematik::510 Mathematik::510 Mathematik #blow-up method #canards #discretization #invariant manifolds #loss of normal hyperbolicity #maps #slow manifolds
paper · doi:10.14279/depositonce-21696
We study the problem of preservation of maximal canards for time discretized fast–slow systems with canard fold points. In order to ensure such preservation, certain favorable structure-preserving properties of the discretization scheme are required. Conventional schemes do not possess such properties. We perform a detailed analysis for an unconventional discretization scheme due to Kahan. The analysis uses the blow-up method to deal with the loss of normal hyperbolicity at the canard point. We show that the structure-preserving properties of the Kahan discretization for quadratic vector fields imply a similar result as in continuous time, guaranteeing the occurrence of maximal canards between attracting and repelling slow manifolds upon variation of a bifurcation parameter. The proof is based on a Melnikov computation along an invariant separating curve, which organizes the dynamics of the map similarly to the ODE problem.