Discretized fast-slow systems near transcritical singularities. (30th May 2019)

• Record Type: Journal Article
• Authors: Engel, Maximilian; Kuehn, Christian
• Is Part Of: Nonlinearity. Volume 32:Number 7(2019)
• Page Start: 2365
• Page End: 2391
• Publication Date: 2019-05-30
• Subjects: transcritical bifurcation -- slow manifolds -- invariant manifolds -- loss of normal hyperbolicity -- discretization -- blow-up method; 34E15 -- 34E20 -- 37M99 -- 37G10 -- 34C45 -- 39A99; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ab15c1
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We study 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.