Rigorous computation of smooth branches of equilibria for the three dimensional Cahn–Hilliard equation

Gameiro, Marcio; Lessard, Jean‐Philippe

doi:10.1007/s00211-010-0350-3

Cited by 31 publications

(25 citation statements)

References 23 publications

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“…In this case the phase space for the dynamics is infinite dimensional. Again, the method of radii polynomials provides an effective technique for finding fixed points and periodic orbits in this setting [25,18] even in the context of higher dimensional domains [16,17]. The challenging problem is to adapt the parameterization method to this setting in such a way that one can obtain rigorous computational results.…”

Section: Den Berg Mireles-james Lessard and Mischaikowmentioning

confidence: 98%

Rigorous Numerics for Symmetric Connecting Orbits: Even Homoclinics of the Gray–Scott Equation

Berg¹,

James²,

Lessard³

et al. 2011

SIAM J. Math. Anal.

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In this paper we propose a rigorous numerical technique for the computation of symmetric connecting orbits for ordinary differential equations. The idea is to solve a projected boundary value problem (BVP) in a function space via a fixed point argument. The formulation of the projected BVP involves a high order parameterization of the invariant manifolds at the steady states. Using this parameterization, one can obtain explicit exponential asymptotic bounds for the coefficients of the expansion of the manifolds. Combining these bounds with piecewise linear approximations, one can construct a contraction in a function space whose unique fixed point corresponds to the wanted connecting orbit. We have implemented the method to demonstrate its effectiveness, and we have used it to prove the existence of a family of even homoclinic orbits for the Gray-Scott equation.1. Introduction. Equilibria, periodic orbits, connecting orbits, and, more generally, invariant manifolds are the fundamental components through which much of the structure of the dynamics of nonlinear differential equations is explained. Thus it is not surprising that there is a vast literature on numerical techniques for approximating these objects. In particular, the last 30 years have witnessed a strong interest in developing computational methods for connecting orbits [5,10,12,14,15,23]. As mentioned in [13], most algorithms for computing heteroclinic or homoclinic orbits reduce the question to solving a boundary value problem (BVP) on a finite interval where the boundary conditions are given in terms of linear or higher order approximations of invariant manifolds near the steady states. We adopt the same philosophy in this paper. The novelty of our approach is that our computational techniques provide existence results and bounds on approximations that are mathematically rigorous. We hasten to add that a variety of authors have already developed methods that involve a combination of interval arithmetic with analytical and topological tools and provide proofs for the existence of homoclinic and heteroclinic solutions to differential equations [28,22,31,6,32]. However, the combination of techniques we propose appears to be unique, perhaps because our approach is being developed with additional goals in mind. We return to this point later.

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Section: Den Berg Mireles-james Lessard and Mischaikowmentioning

confidence: 98%

Rigorous Numerics for Symmetric Connecting Orbits: Even Homoclinics of the Gray–Scott Equation

Berg¹,

James²,

Lessard³

et al. 2011

SIAM J. Math. Anal.

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“…A natural extension of the methods just mentioned would be to combine them with an infinite dimensional continuation method such as [63,35]. By combining the methods of [64,57,2,50] with the methods of [63,35] it should be possible to study rigorously one parameter branches of connecting orbits.…”

Section: Remark 13 (Other Validated Computations For One Parameter Bmentioning

confidence: 98%

“…By combining the methods of [64,57,2,50] with the methods of [63,35] it should be possible to study rigorously one parameter branches of connecting orbits. However in order to carry out this analysis it will be essential to control the boundary conditions, and even derivatives of the boundary conditions, with respect to parameter.…”

Section: Remark 13 (Other Validated Computations For One Parameter Bmentioning

confidence: 99%

Polynomial approximation of one parameter families of (un)stable manifolds with rigorous computer assisted error bounds

James

2015

Indagationes Mathematicae

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This work describes a method for approximating a branch of stable or unstable manifolds associated with a branch of hyperbolic fixed points or equilibria in a one parameter family of analytic dynamical systems. We approximate the branch of invariant manifolds by polynomials and develop a-posteriori theorems which provide mathematically rigorous bounds on the truncation error. The hypotheses of these theorems are formulated in terms of certain inequalities which are checked via a finite number of calculations on a digital computer. By exploiting the analytic category we are able to obtain mathematically rigorous bounds on the jets of the manifolds, as well as on the derivatives of the manifolds with respect to the parameter. A number of example computations are given.

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“…This leads to methods for validated computations of the linear stable and unstable bundles of periodic orbits in differential equations [24]. Exploiting the isolation bounds as well as continuity of the radii polynomials facilitates the study of problems which depend on parameters via rigorous oneand multi-parameter continuation [25,26,27]. We note that the works just mentioned develop the theory of radii polynomials in the context of a C k function space setup.…”

Section: Introductionmentioning

confidence: 99%

Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach

Hungria

Lessard²,

James

2015

Math. Comp.

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Judicious use of interval arithmetic, combined with careful pen and paper estimates, leads to effective strategies for computer assisted analysis of nonlinear operator equations. The method of radii polynomials is an efficient tool for bounding the smallest and largest neighborhoods on which a Newton-like operator associated with a nonlinear equation is a contraction mapping. The method has been used to study solutions of ordinary, partial, and delay differential equations such as equilibria, periodic orbits, solutions of initial value problems, heteroclinic and homoclinic connecting orbits in the C k category of functions. In the present work we adapt the method of radii polynomials to the analytic category. For ease of exposition we focus on studying periodic solutions in Cartesian products of infinite sequence spaces. We derive the radii polynomials for some specific application problems, and give a number of computer assisted proofs in the analytic framework.

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Rigorous computation of smooth branches of equilibria for the three dimensional Cahn–Hilliard equation

Cited by 31 publications

References 23 publications

Rigorous Numerics for Symmetric Connecting Orbits: Even Homoclinics of the Gray–Scott Equation

Rigorous Numerics for Symmetric Connecting Orbits: Even Homoclinics of the Gray–Scott Equation

Polynomial approximation of one parameter families of (un)stable manifolds with rigorous computer assisted error bounds

Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach

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