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arXiv · 2011.02936

A new example on Lyapunov stability

Hildebrando M. Rodrigues·J. Solà-Morales
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Abstract

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 Fr\'{e}chet 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.

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Hildebrando M. Rodrigues, J. Solà-Morales. 2020-11-05. A new example on Lyapunov stability. https://arxiv.org/abs/2011.02936

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