Computer-Assisted Proofs in Delayed and Infinite-Dimensional Dynamical Systems
Computer-assisted proofs (CAPs), also known as validated or rigorous numerics, have become a powerful methodology for turning numerical evidence into mathematical theorems. Recent advances in combining tools from nonlinear analysis, spectral methods, interval arithmetic, and high-performance computing now make it possible to rigorously prove existence, uniqueness and qualitative properties of solutions in ordinary, delay and partial differential equations, including equilibria, periodic orbits, invariant manifolds, connecting orbits and chaotic dynamics.
This CISM advanced course will provide a coherent introduction to modern CAP techniques for delayed and infinite-dimensional dynamical systems. It will guide participants from the basic principles of a posteriori validation to state-of-the-art methods for delay differential equations and evolution PDEs. A central theme will be the rigorous validation paradigm: compute a high-quality numerical approximation, embed it in a suitable Banach-space formulation, and prove, by explicit estimates and Newton-Kantorovich or contraction arguments, that a true solution exists nearby.
General outline
The course will be organized in six complementary lecture series combining theory, proof strategies and computational demonstrations. Topics will include: foundations of validated nonlinear analysis; semigroup and functional-analytic approaches to delay equations; Fourier and Chebyshev techniques and coefficient-space methods; rigorous eigenvalue and stability computations; validated computation of invariant manifolds and connecting orbits; and rigorous time-stepping for PDE-grounded infinite-dimensional dynamics. Particular attention will be paid to closing estimates: choosing effective norms, controlling finite-dimensional truncations and infinite-dimensional tails, bounding nonlinear terms, and managing rounding and discretization errors.
The course will also introduce modern software workflows for rigorous computation, including interval arithmetic tools and Julia-based implementations such as RadiiPolynomial.jl. Laboratory activities and demonstrations will show how theoretical ingredients - radii polynomials, validated Newton steps, operator bounds and tail estimates - translate into implementable routines. Concrete applications will include, e.g., celestial mechanics (3- and N-body problems), fluid dynamics (Navier-Stokes), geodesic flows on manifolds and chaotic dynamics in general. Participants will be asked to bring their own laptop.
Target audience
The course is mainly addressed to graduate students, postdoctoral researchers and early-career scientists in Mathematics, Engineering, Physics and related fields concerned with dynamical systems, computational methods, delay equations and PDE models. It may also interest researchers working in applications such as biology, population dynamics and economics. Starting from core principles, the lectures will be accessible to participants with some background in analysis, dynamical systems and numerical methods, while also providing a roadmap to current research frontiers and open problems.