11 research outputs found
Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model
We present a computer-assisted proof of heteroclinic connections in the
one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a
fourth-order parabolic partial differential equation subject to homogeneous
Neumann boundary conditions, which contains as a special case the celebrated
Cahn-Hilliard equation. While the attractor structure of the latter model is
completely understood for one-dimensional domains, the diblock copolymer
extension exhibits considerably richer long-term dynamical behavior, which
includes a high level of multistability. In this paper, we establish the
existence of certain heteroclinic connections between the homogeneous
equilibrium state, which represents a perfect copolymer mixture, and all local
and global energy minimizers. In this way, we show that not every solution
originating near the homogeneous state will converge to the global energy
minimizer, but rather is trapped by a stable state with higher energy. This
phenomenon can not be observed in the one-dimensional Cahn-Hillard equation,
where generic solutions are attracted by a global minimizer
Towards computational Morse-Floer homology: forcing results for connecting orbits by computing relative indices of critical points
To make progress towards better computability of Morse-Floer homology, and
thus enhance the applicability of Floer theory, it is essential to have tools
to determine the relative index of equilibria. Since even the existence of
nontrivial stationary points is often difficult to accomplish, extracting their
index information is usually out of reach. In this paper we establish a
computer-assisted proof approach to determining relative indices of stationary
states. We introduce the general framework and then focus on three example
problems described by partial differential equations to show how these ideas
work in practice. Based on a rigorous implementation, with accompanying code
made available, we determine the relative indices of many stationary points.
Moreover, we show how forcing results can be then used to prove theorems about
connecting orbits and traveling waves in partial differential equations.Comment: 30 pages, 4 figures. Revised accepted versio
CAPD::DynSys: a flexible C++ toolbox for rigorous numerical analysis of dynamical systems
We present the CAPD::DynSys library for rigorous numerical analysis of
dynamical systems. The basic interface is described together with several
interesting case studies illustrating how it can be used for computer-assisted
proofs in dynamics of ODEs.Comment: 25 pages, 4 figures, 11 full C++ example
Validated Numerical Approximation of Stable Manifolds for Parabolic Partial Differential Equations
This paper develops validated computational methods for studying infinite dimensional stable manifolds at equilibrium solutions of parabolic PDEs, synthesizing disparate errors resulting from numerical approximation. To construct our approximation, we decompose the stable manifold into three components: a finite dimensional slow component, a fast-but-finite dimensional component, and a strongly contracting infinite dimensional “tail”. We employ the parameterization method in a finite dimensional projection to approximate the slow-stable manifold, as well as the attached finite dimensional invariant vector bundles. This approximation provides a change of coordinates which largely removes the nonlinear terms in the slow stable directions. In this adapted coordinate system we apply the Lyapunov-Perron method, resulting in mathematically rigorous bounds on the approximation errors. As a result, we obtain significantly sharper bounds than would be obtained using only the linear approximation given by the eigendirections. As a concrete example we illustrate the technique for a 1D Swift-Hohenberg equation.</p
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described