Canard cycles are periodic orbits that appear as special solutions of fast-slow systems (or singularly perturbed ordinary differential equations). It is well known that canard cycles are difficult to detect, hard to reproduce numerically, and that they are sensible to exponentially small changes in parameters. In this paper, we combine techniques from geometric singular perturbation theory, the blow-up method, and control theory, to design controllers that stabilize canard cycles of planar fast-slow systems with a folded critical manifold. As an application, we propose a controller that produces stable mixed-mode oscillations in the van der Pol oscillator.

Canard 循环是周期性轨道，表现为快慢系统（或奇异摄动常微分方程）的特殊解。众所周知，鸭式循环很难检测，很难用数值重现，而且它们对参数的指数级小变化很敏感。在本文中，我们结合了几何奇异摄动理论、爆破法和控制理论的技术，设计了稳定具有折叠临界流形的平面快-慢系统的鸭式循环的控制器。作为一个应用，我们提出了一种控制器，它可以在 van der Pol 振荡器中产生稳定的混合模式振荡。