Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs

Abstract

In this work, we introduce a method based on piecewise polynomial interpolation to enclose rigorously solutions of nonlinear ODEs. Using a technique which we call a priori bootstrap, we transform the problem of solving the ODE into one of looking for a fixed point of a high order smoothing Picard-like operator. We then develop a rigorous computational method based on a Newton-Kantorovich type argument (the radii polynomial approach) to prove existence of a fixed point of the Picard-like operator. We present all necessary estimates in full generality and for any nonlinearities. With our approach, we study two systems of nonlinear equations: the Lorenz system and the ABC flow. For the Lorenz system, we solve Cauchy problems and prove existence of periodic and connecting orbits at the classical parameters, and for ABC flows, we prove existence of ballistic spiral orbits.

Rigorous numerics for ODEs using Chebyshev series and domain decomposition

“…Let us, however, mention a few key papers to briefly sketch what kind of methods have been developed by the rigorous numerics community. In [11,21,9,45,2,1] functional analytic methods, similar in spirit to ours, are used to solve BVPs: the differential equation is reformulated into an equivalent fixed-point problem and is solved by verifying the conditions of the Contraction Mapping Principle with the aid of a computer. Fundamentally different approaches based on topological rather than functional analytic methods, such as the Conleyindex and covering relations, have been proven to be very effective as well (see e.g.…”