arXiv subjects

Jean-Philippe Lessard

Publications and source records attributed to Jean-Philippe Lessard.

42 records · Page 3Linked to original sources

arXiv2015

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.

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arXiv2015

Global bifurcation diagram of steady states of systems of PDEs via rigorous numerics: a 3-component reaction-diffusion system

In this paper, we use rigorous numerics to compute several global smooth branches of steady states for a system of three reaction-diffusion PDEs introduced by Iida et al. [J. Math. Biol., {\bf 53}, 617--641 (2006)] to study the effect of cross-diffusion in competitive interactions. An explicit and mathematically rigorous construction of a global bifurcation diagram is done, except in small neighborhoods of the bifurcations. The proposed method, even though influenced by the work of van den Berg et al. [Math. Comp., {\bf 79}, 1565--1584 (2010)], introduces new analytic estimates, a new {\em gluing-free} approach for the construction of global smooth branches and provides a detailed analysis of the choice of the parameters to be made in order to maximize the chances of performing successfully the computational proofs.

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arXiv2015

Rigorous numerics for nonlinear operators with tridiagonal dominant linear part

We present a method designed for computing solutions of infinite dimensional non linear 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 $Df(\overline x)$ at an approximate solution $\overline x$. We present rigorous computer-assisted calculations showing that $T$ is a contraction near $\overline x$, thus yielding the existence of a solution. Since $Df(\overline x)$ 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.

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arXiv2011

A method to rigorously enclose eigendecompositions of interval matrices

In this paper, a rigorous computational method to enclose eigendecompositions of complex interval matrices is proposed. Each eigenpair $x=(λ,v)$ is found by solving a nonlinear equation of the form $f(x)=0$ via a contraction argument. The set-up of the method relies on the notion of radii polynomials, which provide an efficient mean of determining a domain on which the contraction mapping theorem is applicable.

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arXiv2011

Rigorous numerics in Floquet theory: computing stable and unstable bundles of periodic orbits

In this paper, a new rigorous numerical method to compute fundamental matrix solutions of non-autonomous linear differential equations with periodic coefficients is introduced. Decomposing the fundamental matrix solutions $Φ(t)$ by their Floquet normal forms, that is as product of real periodic and exponential matrices $Φ(t)=Q(t)e^{Rt}$, one solves simultaneously for $R$ and for the Fourier coefficients of $Q$ via a fixed point argument in a suitable Banach space of rapidly decaying coefficients. As an application, the method is used to compute rigorously stable and unstable bundles of periodic orbits of vector fields. Examples are given in the context of the Lorenz equations and the $ζ^3$-model.

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arXiv2009

Recent advances about the uniqueness of the slowly oscillating periodic solutions of Wright's equation

An old conjecture in delay equations states that Wright's equation \[ y'(t)= - αy(t-1) [ 1+y(t)], α\in \mathbb{R} \] has a unique slowly oscillating periodic solution (SOPS) for every parameter value $α>π/2$. We reformulate this conjecture and we use a method called validated continuation to rigorously compute a global continuous branch of SOPS of Wright's equation. Using this method, we show that a part of this branch does not have any fold point nor does it undergo any secondary bifurcation, partially answering the new reformulated conjecture.

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