21 research outputs found
Stationary coexistence of hexagons and rolls via rigorous computations
In this work we introduce a rigorous computational method for finding heteroclinic solutions of a
system of two second order differential equations. These solutions correspond to standing waves
between rolls and hexagonal patterns of a two-dimensional pattern formation PDE model. After
reformulating the problem as a projected boundary value problem (BVP) with boundaries in the
stable/unstable manifolds, we compute the local manifolds using the parameterization method and
solve the BVP using Chebyshev series and the radii polynomial approach. Our results settle a
conjecture by Doelman et al. [European J. Appl. Math., 14 (2003), pp. 85–110] about the coexistence
of hexagons and rolls
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
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