À propos de canards (Apropos canards)

We extend canard theory of singularly perturbed systems to the general case of $k$ slow and $m$ fast dimensions, with $k\ge 2$ and $m\ge 1$ arbitrary. A folded critical manifold of a singularly perturbed system, a generic requirement for canards to exist, implies that there exists a local $(k+1)$-dimensional center manifold spanned by the $k$ slow variables and the critical eigendirection of the fast variables. If one further assumes that the $m-1$ nonzero eigenvalues of the $m\times m$ Jacobian matrix of the fast equation have all negative real part, then the $(k+m)$-dimensional singularly perturbed problem is locally governed by the flow on the $(k+1)$-dimensional center manifold. By using the blow-up technique (a desingularization procedure for folded singularities) we then show that the local flow near a folded singularity of a $k$-dimensional folded critical manifold is, to leading order, governed by a three-dimensional canonical system for any $k\ge 2$. Consequently, results on generic canards from the well-known case $k=2$ can be extended to the general case $k\ge 2$.