Duck traps: two-dimensional critical manifolds in planar systems. Issue 4 (2nd October 2019)

• Record Type: Journal Article
• Authors: Kuehn, Christian; Münch, Christian
• Is Part Of: Dynamical systems. Volume 34:Issue 4(2019)
• Page Start: 584
• Page End: 612
• Publication Date: 2019-10-02
• Subjects: Multiple time scale dynamics -- fast-slow systems -- fold point -- canard -- piecewise smooth system -- blow-up method; 34D15 -- 34E15; Differentiable dynamical systems -- Periodicals; 515.35205
• DOI: 10.1080/14689367.2019.1575337
• Languages: English
• ISSNs: 1468-9367

Abstract

ABSTRACT: In this work we consider two-dimensional critical manifolds in planar fast-slow systems near fold and so-called canard ( = 'duck') points. These higher-dimension, and lower-codimension, situation is directly motivated by the case of hysteresis operators limiting onto fast-slow systems as well as by systems with constraints. We use geometric desingularization via blow-up to investigate two situations for the slow flow: generic fold (or jump) points, and canards in one-parameter families. We directly prove that the fold case is analogous to the classical fold involving a one-dimensional critical manifold. However, for the canard case, considerable differences and difficulties appear. Orbits can get trapped in the two-dimensional manifold after a canard-like passage thereby preventing small-amplitude oscillations generated by the singular Hopf bifurcation occurring in the classical canard case, as well as certain jump escapes.