Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom.
Division of Infection, Immunity and Respiratory Medicine, University of Manchester, Southmoor Road, Manchester M23 9LT, United Kingdom.
Phys Rev E. 2019 Jan;99(1-1):012211. doi: 10.1103/PhysRevE.99.012211.
We study the dynamics of knotted vortices in a bulk excitable medium using the FitzHugh-Nagumo model. From a systematic survey of all knots of at most eight crossings we establish that the generic behavior is of unsteady, irregular dynamics, with prolonged periods of expansion of parts of the vortex. The mechanism for the length expansion is a long-range "wave-slapping" interaction, analogous to that responsible for the annihilation of small vortex rings by larger ones. We also show that there are stable vortex geometries for certain knots; in addition to the unknot, trefoil, and figure-eight knots reported previously, we have found stable examples of the Whitehead link and 6_{2} knot. We give a thorough characterization of their geometry and steady-state motion. For the unknot, trefoil, and figure-eight knots we greatly expand previous evidence that FitzHugh-Nagumo dynamics untangles initially complex geometries while preserving topology.
我们使用 FitzHugh-Nagumo 模型研究了体激发介质中扭结涡旋的动力学。通过对最多有八个交叉点的所有纽结的系统调查，我们确定了一般行为是不稳定的、不规则的动力学，涡旋的部分会经历长时间的扩张。长度扩张的机制是一种长程的“波击”相互作用，类似于较小的涡环被较大的涡环湮灭的机制。我们还表明，某些纽结存在稳定的涡旋几何形状；除了以前报道的无纽结、三叶结和 8 字结之外，我们还发现了稳定的 Whitehead 链接和 6_2 结的例子。我们对它们的几何形状和稳态运动进行了彻底的描述。对于无纽结、三叶结和 8 字结，我们极大地扩展了之前的证据，即 FitzHugh-Nagumo 动力学在保持拓扑的同时解开了初始复杂的几何形状。
