17 Result(s)
within
Jean-Philippe Lessard
Mathematics
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Article
Toward Computational Morse–Floer Homology: Forcing Results for Connecting Orbits by Computing Relative Indices of Critical Points
To make progress toward 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 ...
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Article
Open AccessSaddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature
In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial points, referred to as saddle-type blow-up solutions, are studied. Combining...
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Open AccessCorrection to: Rigorous numerics for nonlinear heat equations in the complex plane of time
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Article
Open AccessRigorous numerics for nonlinear heat equations in the complex plane of time
In this paper, we introduce a method for computing rigorous local inclusions of solutions of Cauchy problems for nonlinear heat equations for complex time values. The proof is constructive and provides explici...
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A General Method for Computer-Assisted Proofs of Periodic Solutions in Delay Differential Problems
In this paper we develop a general computer-assisted proof method for periodic solutions to delay differential equations. The class of problems considered includes systems of delay differential equations with ...
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Article
Parameterization of Unstable Manifolds for DDEs: Formal Series Solutions and Validated Error Bounds
This paper studies the local unstable manifold attached to an equilibrium solution of a system of delay differential equations (DDEs). Two main results are developed. The first is a general method for computin...
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Article
A Rigorous Implicit \(C^1\) Chebyshev Integrator for Delay Equations
We present a new approach to validated numerical integration for systems of delay differential equations. We focus on the case of a single constant delay though the method generalizes to systems with multiple ...
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Article
Spatial Relative Equilibria and Periodic Solutions of the Coulomb \((n+1)\) -Body Problem
We study a classical model for the atom that considers the movement of n charged particles of charge \(-1\) ...
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Article
Spontaneous Periodic Orbits in the Navier–Stokes Flow
In this paper, a general method to obtain constructive proofs of existence of periodic orbits in the forced autonomous Navier–Stokes equations on the three-torus is proposed. After introducing a zero finding p...
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Article
Parameterization of Invariant Manifolds for Periodic Orbits (II): A Posteriori Analysis and Computer Assisted Error Bounds
In this paper we develop mathematically rigorous computer assisted techniques for studying high order Fourier–Taylor parameterizations of local stable/unstable manifolds for hyperbolic periodic orbits of analy...
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Article
Computing Discrete Convolutions with Verified Accuracy Via Banach Algebras and the FFT
We introduce a method to compute rigorous component-wise enclosures of discrete convolutions using the fast Fourier transform, the properties of Banach algebras, and interval arithmetic. The purpose of this ne...
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Article
Rapidly and Slowly Oscillating Periodic Solutions of a Delayed Van der Pol Oscillator
In this paper, we introduce a method to prove existence of several rapidly and slowly oscillating periodic solutions of a delayed Van der Pol oscillator. The proof is a combination of pen and paper analytic es...
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Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions
In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite-dimensional nonlinear operators. We compute a simplicial triangulation of the man...
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Article
Analytic enclosure of the fundamental matrix solution
This work describes a method to rigorously compute the real Floquet normal form decomposition of the fundamental matrix solution of a system of linear ODEs having periodic coefficients. The Floquet normal form...
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Article
Computer Assisted Proof of Transverse Saddle-to-Saddle Connecting Orbits for First Order Vector Fields
In this paper we introduce a computational method for proving the existence of generic saddle-to-saddle connections between equilibria of first order vector fields. The first step consists of rigorously comput...
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Article
Global Bifurcation Diagrams 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. 53(4):617–641, 2006] to s...
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Article
Rigorous computation of smooth branches of equilibria for the three dimensional Cahn–Hilliard equation
In this paper, we propose a new general method to compute rigorously global smooth branches of equilibria of higher-dimensional partial differential equations. The theoretical framework is based on a combinati...
