Duck traps: two-dimensional critical manifolds in planar systems. (English) Zbl 1440.34060
Authors’ abstract: We consider two-dimensional critical manifolds in planar fast-slow systems near fold and so-called canard (=‘duck’) points. This 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.
Reviewer: Vladimir Sobolev (Samara)
MSC:
| 34E15 | Singular perturbations for ordinary differential equations |
| 34C45 | Invariant manifolds for ordinary differential equations |
| 34E17 | Canard solutions to ordinary differential equations |
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
Keywords:
multiple time scale dynamics; fast-slow systems; fold point; canard; piecewise smooth system; blow-up methodReferences:
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