  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.
