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Polynomial interpolation
and a priori bootstrap for computer-assisted proofs in
nonlinear ODEs
Maxime Breden and
Jean-Philippe Lessard.
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.
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