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Building upon the pioneering work [Merle, Rapha\" el, Rodnianski, and Szeftel, Ann. of
Math., 196 (2): 567-778, 2022, Ann. of Math., 196 (2): 779-889, 2022, Invent. Math., 227 (1) …

Judicious use of interval arithmetic, combined with careful pen and paper estimates, leads to
effective strategies for computer assisted analysis of nonlinear operator equations. The …

In this paper, a method for computing periodic orbits of the Kuramoto--Sivashinsky PDE via
rigorous numerics is presented. This is an application and an implementation of the …

This work is concerned with efficient numerical methods for computing high order Taylor and
Fourier-Taylor approximations of unstable manifolds attached to equilibrium and periodic …

We study polynomial expansions of local unstable manifolds attached to equilibrium
solutions of parabolic partial differential equations. Due to the smoothing properties of …

We present numerical results and computer assisted proofs of the existence of periodic
orbits for the Kuramoto--Sivashinky equation. These two results are based on writing down …

This lecture describes validated numerical tools which are used for global analysis of
nonlinear systems. The main focus is dynamics near and between equilibrium solutions of …

We propose a general framework for proving that a compact, infinite-dimensional map has
an invariant set on which the dynamics is semiconjugated to a subshift of finite type. The …

In this paper, we develop computer-assisted techniques for the analysis of periodic orbits of
ill-posed partial differential equations. As a case study, our proposed method is applied to …