Duck-shaped solutions and winding. (Q1173675)

This paper studies, using non-standard analysis, phase portraits of duck- shaped solutions of a singular perturbation problem in \(\mathbb{R}^ 3\) of the form: \(x'=f(x,y,z,\epsilon)\), \(y'=g(x,y,z,\epsilon)\), \(\epsilon z'=h(x,y,z,\epsilon)\). The study is done in the case when the slow surface \((h=0)\) has a generic fold in the \(z\)-direction. Within this general framework, the study is restricted to a hopefully prototype equation: \(x'=ay+bz\), \(y'=1\), \(\epsilon z'=-(z^ 2+x)\), where \(b>0\). Amongst other results, the author proves a formula relating the number of jumps of a smooth duck-shaped solution to a sort of winding number of that solution about a certain global solution of the constrained equation \((\epsilon=0)\). Most theoretical results are illustrated by numerical results. The paper concludes with a discussion and a comparison with a similar work by G. Wallet where, in contrast to here, the slow manifold has dimension one.