Discretized fast–slow systems near pitchfork singularities. Issue 7 (3rd July 2019)

• Record Type: Journal Article
• Authors: Arcidiacono, Luca; Engel, Maximilian; Kuehn, Christian
• Is Part Of: Journal of difference equations and applications. Volume 25:Issue 7(2019)
• Page Start: 1024
• Page End: 1051
• Publication Date: 2019-07-03
• Subjects: Pitchfork bifurcation -- slow manifolds -- invariant manifolds -- loss of normal hyperbolicity -- blow-up method -- discretization; 34E15 -- 37M99 -- 37G10 -- 34C45 -- 39A99; Difference equations -- Periodicals; 515.625
• DOI: 10.1080/10236198.2019.1647185
• Languages: English
• ISSNs: 1023-6198

Abstract

ABSTRACT: 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 vicinity of the singularity we show, how the slow manifold extends beyond the singular point and give an estimate on the contraction rate of a transition mapping. The proof relies on the blow-up method suitably adapted to the discrete setting where precise estimates for a cubic map in the central rescaling chart make a key technical contribution.