论文标题
周期性延迟轨道和多窝隐式函数定理
Periodic delay orbits and the polyfold implicit function theorem
论文作者
论文摘要
我们考虑$ \ partial_tx（t）= x_ {t}（x（t -τ））$中的差分延迟方程，in $ \ mathbb {r}^n $，其中$（x_t）_ {x_t）_ {t \ in s^1} $是$ MATHBB的时间依赖于$ m mathbb的时间依赖于$ \ tay的$ \} $} $ and $ n $ r} $ r} $ r} $ r}。如果有一个（适当的非分类）定期解决方案$ x_0以$τ= 0 $的方式$ x_0 $，也就没有延迟了，则有充分的理由期望存在适用于所有足够小的延迟的定期解决方案家族，并通过延迟顺利参数。但是，似乎很难使用经典隐式函数定理证明这一点，因为以上方程在延迟参数中并不平滑。在本文中，我们展示了如何使用Hofer-Wysocki-Zehnder [HWZ09，HWZ17]的M-PolyFold隐式函数定理来克服自然设置中的这个问题。
We consider differential delay equations of the form $\partial_tx(t) = X_{t}(x(t - τ))$ in $\mathbb{R}^n$, where $(X_t)_{t\in S^1}$ is a time-dependent family of smooth vector fields on $\mathbb{R}^n$ and $τ$ is a delay parameter. If there is a (suitably non-degenerate) periodic solution $x_0$ of this equation for $τ=0$, that is without delay, there are good reasons to expect existence of a family of periodic solutions for all sufficiently small delays, smoothly parametrized by delay. However, it seems difficult to prove this using the classical implicit function theorem, since the equation above is not smooth in the delay parameter. In this paper, we show how to use the M-polyfold implicit function theorem by Hofer-Wysocki-Zehnder [HWZ09, HWZ17] to overcome this problem in a natural setup.
