单值切奇点附近由迟滞和时滞诱导的类霍普夫分岔
Hopf-like bifurcations induced by hysteresis and time-delay near monodromic tangential singularities
AI总结：
该研究针对平面分段解析向量场，在切换条件引入迟滞或时滞时，于单值切奇点附近推导极限环渐近表达式，发现迟滞与时滞诱导的极限环增长率及与伪霍普夫分岔的增长率均存在差异。
AI中文摘要：
我们研究平面分段解析向量场，重点关注迟滞或时滞引入切换条件时的合并焦点及其他单值切奇点。若该奇点是系统在瞬时切换下的渐近稳定解，则迟滞或时滞的引入会在局部形成吸引极限环。我们推导了极限环大小和周期的渐近表达式，适用于任意切触阶数及最小非零李雅普诺夫系数的阶数。研究发现，迟滞和时滞诱导的极限环增长率存在差异，且与相关伪霍普夫分岔的增长率也不同。
英文摘要：
We investigate planar piecewise-analytic vector fields, focusing on the merged focus and other monodromic tangential singularities when hysteresis or time-delay is incorporated into the switching condition. If the singularity is an asymptotically stable solution of the system with an instantaneous switch, then the introduction of hysteresis or time-delay causes an attracting limit cycle to be formed locally. We derive asymptotic expressions for the size and period of the limit cycle, allowing any degrees of tangency and any order for the smallest non-zero Lyapunov coefficient. We find that the growth rate of the limit cycle differs for hysteresis and time-delay, and differs to that of the related pseudo-Hopf bifurcation.
