Discretized fast–slow systems near pitchfork singularities

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.

KEYWORDS:

• Pitchfork bifurcation
• slow manifolds
• invariant manifolds
• loss of normal hyperbolicity
• blow-up method
• discretization

MATHEMATICS SUBJECT CLASSIFICATION (2010):

• 34E15
• 37M99
• 37G10
• 34C45
• 39A99

Acknowledgments

Luca Arcidiacono acknowledges support from the graduate program TopMath of the Elite Network of Bavaria and the TopMath Graduate Center of TUM Graduate School.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Luca Arcidiacono http://orcid.org/0000-0002-4610-8805

Maximilian Engel http://orcid.org/0000-0002-1406-8052

Christian Kuehn http://orcid.org/0000-0002-7063-6173

Additional information

Funding

This work was supported by the German Research Foundation (DFG) via the SFB-TRR 109.