论文标题
可逆的非自主微分方程中的亚谐波解决方案
Subharmonic Solutions In Reversible Non-Autonomous Differential Equations
论文作者
论文摘要
我们研究了系统中的亚谐波解决方案的存在$ \ ddot {u}（t）= f（t，u（t））$，其中$ u（t）\ in \ mathbb {r}^{k} $ and $ f $是一个均匀的，$ p $ - p $ - p $ - periodic函数。在功能$ f $的一些其他对称条件下，可以在功能空间中重新审查$γ\ times \ times \ mathbb {z} _ {2} \ times d_ {m}％$ - equivariant方程$ quam $γ\ time $ gatip past，可以在功能空间中重新构建$ f $ f $的其他对称条件。 $ \ mathbb {r}^{k} $和$ d_ {m} $通过时间缩短和反射在$ u（t）$上代替。我们应用Brouwer Equivariant度量，以证明存在无限数量的函数$ f $的亚谐波解决方案，这些解决方案满足了零接近线性行为的其他假设，而无穷大的nagumo条件则满足。当系统依赖额外的参数时，我们还讨论了亚谐波解决方案的分叉。
We study the existence of subharmonic solutions in the system $\ddot {u}(t)=f(t,u(t))$, where $u(t)\in\mathbb{R}^{k}$ and $f$ is an even and $p$-periodic function in time. Under some additional symmetry conditions on the function $f$, the problem of finding $mp$-periodic solutions can be reformulated in a functional space as a $Γ\times\mathbb{Z}_{2}\times D_{m}% $-equivariant equation, where the group $Γ\times\mathbb{Z}_{2}$ acts on the space $\mathbb{R}^{k}$ and $D_{m}$ acts on $u(t)$ by time-shifts and reflection. We apply Brouwer equivariant degree to prove the existence of an infinite number of subharmonic solutions for the function $f$ that satisfies additional hypothesis on linear behavior near zero and the Nagumo condition at infinity. We also discuss the bifurcation of subharmonic solutions when the system depends on an extra parameter.
