Abstract
We show that there exist generic slow-fast systems with only one (time-scaling) parameter on the two-torus, which have canard cycles for arbitrary small values of this parameter. This is in drastic contrast with the planar case, where canards usually occur in two-parametric families. Here we treat systems with a convex slow curve. In this case there is a set of parameter values accumulating to zero for which the system has exactly one attracting and one repelling canard cycle. The basin of the attracting cycle is almost the whole torus.
Similar content being viewed by others
References
O. Anosova, On invariant manifolds in singulalry perturbed systems. J. Dynam. Control Syst. 5 (1999), No. 4, 501–507.
_____, Invariant manifolds in singularly perturbed systems. Proc. Steklov Math. Inst. 236 (2002), 19–24.
A. Denjoy, Sur les courbes définies par des équations différentielles à la surface du tore. J. Math. Pure Appl. 11 (1932), 333–375.
M. Diener, The canard unchained or how fast/slow dynamical systems bifurcate. Math. Intel. 6 (1984), 38–48.
F. Dumortier and R. Roussarie, Canard cycles and center manifolds. Mem. Amer. Math. Soc. 121 (1996).
N. Fenichel, Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equations 31 (1979), 53–98.
J. Guckenheimer and Yu. S. Ilyashenko, The duck and the devil: canards on the staircase. Moscow Math. J. 1 (2001), No. 1, 27–47.
M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points — fold and canard points in two dimensions. SIAM J. Math. Anal. 33, No. 2, 286–314.
E. F. Mishchenko and N. Kh. Rozov, Differential equations with small parameters and relaxation oscillations. Plenum Press, New York (1980).
A. Schwartz, A generalization of Poincaré–Bendixon theorem to closed two dimensional manifolds. Amer. J. Math. 85 (1963), 453–458.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schurov, I.V. Ducks on the torus: existence and uniqueness. J Dyn Control Syst 16, 267–300 (2010). https://doi.org/10.1007/s10883-010-9093-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10883-010-9093-9