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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.
Motivated by the normal form of a fast-slow ordinary differential equation exhibiting a pitchfork singularity we consider the discrete-time dynamical system that is obtained by an application of the explicit Euler method. Tracking trajectories in the
We study the problem of preservation of canard connections 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 requir
Fast-slow dynamical systems have subsystems that evolve on vastly different timescales, and bifurcations in such systems can arise due to changes in any or all subsystems. We classify bifurcations of the critical set (the equilibria of the fast subsy
Although the spatially continuous version of the reaction-diffusion equation has been well studied, in some instances a spatially-discretized representation provides a more realistic approximation of biological processes. Indeed, mathematically the d
We show that every (invertible, or noninvertible) minimal Cantor system embeds in $mathbb{R}$ with vanishing derivative everywhere. We also study relations between local shrinking and periodic points.