Fourier–Taylor parameterization of unstable manifolds for parabolic partial differential equations: Formalism, implementation and rigorous validation

Reinhardt, Christian; James, J. D. Mireles

doi:10.1016/j.indag.2018.08.003

Cited by 18 publications

(16 citation statements)

References 94 publications

(171 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We have the following theorem, whose proof is found in [56]. Much more general results are found in [74].…”

Section: Representation Of the Local Invariant Manifold By A Conjugatingmentioning

confidence: 81%

“…Validated numerics for stable/unstable manifolds based on functional analytic tools are discussed at length below, and for the moment we only mention the work of Koch and Arioli [51,52] on homoclinic connecting orbits traveling waves, and the Evans function; and also the work of the Breden, Lessard, Reinhardt, and the author on validated numerical methods for stable/unstable manifolds based on the parameterization method [53,54]. Techniques based on the parameterization method lead also to validated numerics for stable/unstable manifolds of periodic orbits, as in the work of Castelli, Lessard and the author [55], and also to validated numerical methods for computing unstable manifolds and heteroclinic connecting orbits for parabolic PDEs [56]. Further discussion of the literature is found in these references.…”

Section: A Brief Survey Of the Surrounding Literaturementioning

confidence: 95%

“…This freedom makes the method very flexible, and it applies to problems as diverse as invariant circles and their stable/unstable manifolds [77,78,79], breakdown/collisions of invariant bundles associated with quasi periodic dynamics [80,81], stable/unstable manifolds of periodic orbits of differential equations and diffeomorphisms [76,82,83,84,85], study slow stable manifolds [74,76] and their invariant vector bundles [86], and invariant tori for differential equations [87,81]. The parameterization methods has also been used to develop KAM strategies not requiring action angle variables [88,89,90,91], as well as to study invariant objects for PDEs [92,93,56] and DDEs [94,95,96]. Moreover the short list above is far from complete.…”

Section: Remark 28 (Validated Numerics For Existence and Localizatiomentioning

confidence: 99%

See 2 more Smart Citations

Validated numerics for equilibria of analytic vector fields; Invariant manifolds and connecting orbits

James¹

2018

Rigorous Numerics in Dynamics

View full text Add to dashboard Cite

The goal of these notes is to illustrate the use of validated numerics as a tool for studying the dynamics near and between equilibrium solutions of ordinary differential equations. We examine Taylor methods for computing local stable/unstable manifolds and also for expanding the flow in a neighborhood of a given initial condition. The Taylor methods discussed here have the advantage that the first N terms in the series are computed by recursion relations. Bounds on the tail of the series follow from a contraction mapping argument. As an application we apply these validated local computations to prove the existence of a heteroclinic connecting orbit in the Lorenz system. Contents

show abstract

“…We have the following theorem, whose proof is found in [56]. Much more general results are found in [74].…”

Section: Representation Of the Local Invariant Manifold By A Conjugatingmentioning

confidence: 81%

Section: A Brief Survey Of the Surrounding Literaturementioning

confidence: 95%

Section: Remark 28 (Validated Numerics For Existence and Localizatiomentioning

confidence: 99%

See 1 more Smart Citation

Validated numerics for equilibria of analytic vector fields; Invariant manifolds and connecting orbits

James¹

2018

Rigorous Numerics in Dynamics

View full text Add to dashboard Cite

show abstract

“…Another intriguing possibility for future work is to apply the parametrization method [5] in the first step of the proof of Theorem 4.1. Recently, the parametrization method has been generalized to allow for the validation of unstable manifolds for parabolic partial differential equations, see for example [39,50]. Using the parametrization method, bounds for the unstable manifold are obtained by computing its expansion in the form of a finite Taylor polynomial.…”

Section: Discussionmentioning

confidence: 99%

Computer-Assisted Proof of Heteroclinic Connections in the One-Dimensional Ohta--Kawasaki Model

Cyranka¹,

Wanner²

2018

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

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 and local and global energy minimizers.The proof of the above statement is conceptually simple, and combines several techniques from some of the authors' and Zgliczyński's works. Central for the verification is the rigorous propagation of a piece of the unstable manifold of the homogeneous state with respect to time. This propagation has to lead to small interval bounds, while at the same time entering the basin of attraction of the stable fixed point. For interesting parameter values the global attractor exhibits a complicated equilibrium structure, and the dynamical equation is rather stiff. This leads to a time-consuming numerical propagation of error bounds, with many integration steps. This problem is addressed using an efficient algorithm for the rigorous integration of partial differential equations forward in time. The method is able to handle large integration times within a reasonable computational time frame, and this makes it possible to establish heteroclinic at various nontrivial parameter values.

show abstract

“…Some works of this kind include the validated numerics for Floquet theory developed in [16], the methods for validated Morse index computations (unstable eigenvalue counts) for infinite dimensional compact maps in [29,37], similar methods for equilibria of parabolic PDEs posed on compact domains in [41,1,44,54,40,55], the validated numerics for stability/instability of traveling waves in [3,2,7,5,6], stability analysis for periodic solutions of delay differential equations [28], the computerassisted proofs of instability for periodic orbits of parabolic partial differential equations found in [22], and the computer-assisted proofs for trapping regions of equilibrium solutions of parabolic PDEs in [18].…”

mentioning

confidence: 99%

A functional analytic approach to validated numerics for eigenvalues of delay equations

Lessard¹,

James²

2020

Journal of Computational Dynamics

View full text Add to dashboard Cite

This work develops validated numerical methods for linear stability analysis at an equilibrium solution of a system of delay differential equations (DDEs). In addition to providing mathematically rigorous bounds on the locations of eigenvalues, our method leads to validated counts. For example we obtain the computer assisted theorems about Morse indices (number of unstable eigenvalues). The case of a single constant delay is considered. The method downplays the role of the scalar transcendental characteristic equation in favor of a functional analytic approach exploiting the strengths of numerical linear algebra/techniques of scientific computing. The idea is to consider an equivalent implicitly defined discrete time dynamical system which is projected onto a countable basis of Chebyshev series coefficients. The projected problem reduces to questions about certain sparse infinite matrices, which are well approximated by N × N matrices for large enough N. We develop the appropriate truncation error bounds for the infinite matrices, provide a general numerical implementation which works for any system with one delay, and discuss computer-assisted theorems in a number of example problems.

show abstract

Fourier–Taylor parameterization of unstable manifolds for parabolic partial differential equations: Formalism, implementation and rigorous validation

Cited by 18 publications

References 94 publications

Validated numerics for equilibria of analytic vector fields; Invariant manifolds and connecting orbits

Validated numerics for equilibria of analytic vector fields; Invariant manifolds and connecting orbits

Computer-Assisted Proof of Heteroclinic Connections in the One-Dimensional Ohta--Kawasaki Model

A functional analytic approach to validated numerics for eigenvalues of delay equations

Contact Info

Product

Resources

About