33 research outputs found
Existence and instability of steady states for a triangular cross-diffusion system: a computer-assisted proof
In this paper, we present and apply a computer-assisted method to study
steady states of a triangular cross-diffusion system. Our approach consist in
an a posteriori validation procedure, that is based on using a fxed point
argument around a numerically computed solution, in the spirit of the
Newton-Kantorovich theorem. It allows us to prove the existence of various non
homogeneous steady states for different parameter values. In some situations,
we get as many as 13 coexisting steady states. We also apply the a posteriori
validation procedure to study the linear stability of the obtained steady
states, proving that many of them are in fact unstable
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
Computation of maximal local (un)stable manifold patches by the parameterization method
In this work we develop some automatic procedures for computing high order
polynomial expansions of local (un)stable manifolds for equilibria of
differential equations. Our method incorporates validated truncation error
bounds, and maximizes the size of the image of the polynomial approximation
relative to some specified constraints. More precisely we use that the manifold
computations depend heavily on the scalings of the eigenvectors: indeed we
study the precise effects of these scalings on the estimates which determine
the validated error bounds. This relationship between the eigenvector scalings
and the error estimates plays a central role in our automatic procedures. In
order to illustrate the utility of these methods we present several
applications, including visualization of invariant manifolds in the Lorenz and
FitzHugh-Nagumo systems and an automatic continuation scheme for (un)stable
manifolds in a suspension bridge problem. In the present work we treat
explicitly the case where the eigenvalues satisfy a certain non-resonance
condition.Comment: Revised version, typos corrected, references adde
Rigorous numerics for nonlinear operators with tridiagonal dominant linear parts
We present a method designed for computing solutions of infinite
dimensional nonlinear operators f(x) = 0 with a tridiagonal dominant linear
part. We recast the operator equation into an equivalent Newton-like equation
x = T(x) = x - Af(x), where A is an approximate inverse of the derivative
Df(¯x) at an approximate solution ¯x. We present rigorous computer-assisted
calculations showing that T is a contraction near ¯x, thus yielding the existence
of a solution. Since Df(¯x) does not have an asymptotically diagonal dominant
structure, the computation of A is not straightforward. This paper provides
ideas for computing A, and proposes a new rigorous method for proving existence
of solutions of nonlinear operators with tridiagonal dominant linear
part