Department of Mathematics and Statistics, Center for BioDynamics, Boston University, Boston, Massachusetts 02215, USA.
Chaos. 2011 Jun;21(2):023131. doi: 10.1063/1.3592798.
We study the recently observed phenomena of torus canards. These are a higher-dimensional generalization of the classical canard orbits familiar from planar systems and arise in fast-slow systems of ordinary differential equations in which the fast subsystem contains a saddle-node bifurcation of limit cycles. Torus canards are trajectories that pass near the saddle-node and subsequently spend long times near a repelling branch of slowly varying limit cycles. In this article, we carry out a study of torus canards in an elementary third-order system that consists of a rotated planar system of van der Pol type in which the rotational symmetry is broken by including a phase-dependent term in the slow component of the vector field. In the regime of fast rotation, the torus canards behave much like their planar counterparts. In the regime of slow rotation, the phase dependence creates rich torus canard dynamics and dynamics of mixed mode type. The results of this elementary model provide insight into the torus canards observed in a higher-dimensional neuroscience model.
我们研究了最近观察到的环面拟鹦鹉现象。这些是经典拟鹦鹉轨道的高维推广，经典拟鹦鹉轨道在平面系统中很常见，出现在快速-缓慢常微分方程组中，其中快速子系统包含极限环的鞍结点分岔。环面拟鹦鹉是一种轨迹，它经过鞍结点附近，随后在缓慢变化的极限环的排斥分支附近花费很长时间。在本文中，我们对一个基本的三阶系统中的环面拟鹦鹉进行了研究，该系统由一个旋转的范德波尔平面系统组成，其中通过在向量场的慢分量中包含一个与相位相关的项来打破旋转对称性。在快速旋转的情况下，环面拟鹦鹉的行为与它们的平面对应物非常相似。在缓慢旋转的情况下，相位依赖性产生了丰富的环面拟鹦鹉动力学和混合模式类型的动力学。这个基本模型的结果为在更高维神经科学模型中观察到的环面拟鹦鹉提供了深入的了解。
