Shadowing in the neighborhood of a hyperbolic equilibrium point for fractional equations

In this paper, we study the behavior of trajectories of abstract parabolic problems with fractional time derivative in the neighborhood of a hyperbolic equilibrium point, where the fractional derivative is understood in the Caputo – Djerbashyan sense. It is well known that for dynamical systems with integer derivative, the phase space in the neighborhood of a hyperbolic equilibrium point splits in such a way that this initial value problem reduces to initial value problems with exponentially decreasing solutions in opposite directions. In the case of a fractional derivative, the situation changes dramatically. First, there is no exponential decay. Second, the spectrum of the linearized operator admits an expansion different from the classical picture. Nevertheless, we manage to prove analogs of the results on shadowing. The main conditions of our results are satisfied, in particular, for operators with a compact resolvent and can be verified for the finite elements method and difference methods. Acknowledgements The paper was carried out at the Research Computing Center of Moscow State University named after M. V. Lomonosov as part of the research work on the topic "Research and development of methods, algorithms and software in the field of computational mathematics"and with the support of the Russian Science Foundation (grant No. 23-21-00005).