论文标题
Lyapunov稳定性的新示例
A new example on Lyapunov stability
论文作者
论文摘要
The purpose of this paper is to present an example of an Ordinary Differential Equation $x'=F(x)$ in the infinite-dimensional Hilbert space $\ell^2$ with $F$ being of class $\mathcal{C}^1$ in the Fréchet sense, such that the origin is an asymptotically stable equilibrium point but the spectrum of the linearized operator $DF(0)$ intersects the半平面$ \ re（z）> 0 $。据我们所知，直到现在，直到现在，这种可能的存在还是不存在的例子一直是一个悬而未决的问题。一个类似的示例，但是不可变形的图，而不是由ODE定义的流程，最近是由作者在最近的一篇论文中构建的。这两个示例使用了不同的技术，但两者都基于因链球菌而在操作者理论中的经典示例。
The purpose of this paper is to present an example of an Ordinary Differential Equation $x'=F(x)$ in the infinite-dimensional Hilbert space $\ell^2$ with $F$ being of class $\mathcal{C}^1$ in the Fréchet sense, such that the origin is an asymptotically stable equilibrium point but the spectrum of the linearized operator $DF(0)$ intersects the half-plane $\Re(z)>0$. The possible existence or not of an example of this kind has been an open question until now, to our knowledge. An analogous example, but of a non-invertible map instead of a flow defined by an ODE was recently constructed by the authors in a recent paper. The two examples use different techniques, but both are based on a classical example in Operator Theory due to S. Kakutani.
