{A} propos de canards (Apropos canards)

• Published in: Transactions of the American Mathematical Society Vol. 364; no. 6; pp. 3289 - 3309
• Main Author: WECHSELBERGER, MARTIN
• Format: Journal Article
• Language: English
• Published: American Mathematical Society 01-06-2012
• Subjects: Canonical forms; Chaos theory; Eigenvalues; Eigenvectors; Jacobians; Mathematical manifolds; Perturbation theory; Tangents; Trajectories; Singularly perturbed systems; folded singularities; canards; differential-algebraic equations
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-2012-05575-9

We extend canard theory of singularly perturbed systems to the general case of k fast dimensions, with k\ge 2 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) slow variables and the critical eigendirection of the fast variables. If one further assumes that the m-1 Jacobian matrix of the fast equation have all negative real part, then the (k+m)-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. Consequently, results on generic canards from the well-known case k=2.