2020/11/05 by Hildebrando M. Rodrigues, Rodrigues, Hildebrando M., J. Solà‐Morales +1
Engineering · Physics and Astronomy · #34D20 #35B35 #37D75 #Control and Stability of Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Quantum chaos and dynamical systems #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2011.02936
openalex publication_date 2020/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The purpose of this paper is to present an example of an Ordinary Differential Equation x'=F(x) in the infinite-dimensional Hilbert space ℓ2 with F being of class C1 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.