Discretized Fast-Slow Systems near Transcritical Singularities

TL;DR

This paper analyzes how explicit Euler discretization affects the behavior of fast-slow systems near transcritical singularities, showing that qualitative dynamics are preserved with sufficiently small step sizes.

Contribution

It extends the analysis of slow manifolds to discretized systems near transcritical points, providing explicit step size bounds and detailed proofs using blow-up methods.

Findings

Qualitative behavior is preserved under discretization with small step sizes.

Step size bounds are explicitly quantified relative to time scale separation.

The continuous-time results are recovered as a special case.

Abstract

We extend slow manifolds near a transcritical singularity in a fast-slow system given by the explicit Euler discretization of the corresponding continuous-time normal form. The analysis uses the blow-up method and direct trajectory-based estimates. We prove that the qualitative behaviour is preserved by a time-discretization with sufficiently small step size. This step size is fully quantified relative to the time scale separation. Our proof also yields the continuous-time results as a special case and provides more detailed calculations in the classical (or scaling) chart.