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

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 nonlinearity. Using our approach, we study two systems of nonlinear equations, namely the Lorenz system and the ABC flow. In the Lorenz system, we solve Cauchy problems and prove existence of periodic and connecting orbits for the classical parameters, and for ABC flows, we prove existence of ballistic spiral orbits.