447 research outputs found
Rigorous numerics for nonlinear operators with tridiagonal dominant linear part
We present a method designed for computing solutions of infinite dimensional
non linear operators with a tridiagonal dominant linear part. We
recast the operator equation into an equivalent Newton-like equation , where is an approximate inverse of the derivative
at an approximate solution . We present rigorous
computer-assisted calculations showing that is a contraction near
, thus yielding the existence of a solution. Since does not have an asymptotically diagonal dominant structure, the
computation of is not straightforward. This paper provides ideas for
computing , and proposes a new rigorous method for proving existence of
solutions of nonlinear operators with tridiagonal dominant linear part.Comment: 27 pages, 3 figures, to be published in DCDS-A (Vol. 35, No. 10)
October 2015 issu
Continuation of homoclinic orbits in the suspension bridge equation: a computer-assisted proof
In this paper, we prove existence of symmetric homoclinic orbits for the
suspension bridge equation for all parameter values
. For each , a parameterization of the stable
manifold is computed and the symmetric homoclinic orbits are obtained by
solving a projected boundary value problem using Chebyshev series. The proof is
computer-assisted and combines the uniform contraction theorem and the radii
polynomial approach, which provides an efficient means of determining a set,
centered at a numerical approximation of a solution, on which a Newton-like
operator is a contraction.Comment: 37 pages, 6 figure
Rigorous numerics for piecewise-smooth systems : a functional analytic approach based on Chebyshev series
In this paper, a rigorous computational method to compute solutions of piecewise-smooth systems using a functional analytic approach based on Chebyshev series is introduced. A general theory, based on the radii polynomial approach, is proposed to compute crossing periodic orbits for continuous and discontinuous (Filippov) piecewise-smooth systems. Explicit analytic estimates to carry the computer-assisted proofs are presented. The method is applied to prove existence of crossing periodic orbits in a model nonlinear Filippov system and in the Chua’s circuit system. A general formulation to compute rigorously crossing connecting orbits for piecewise-smooth systems is also introduced
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