Computational Proofs in Dynamics

James, J. D. Mireles; Mischaikow, Konstantin

doi:10.1007/978-3-540-70529-1_322

Cited by 10 publications

(9 citation statements)

References 69 publications

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“…given by (52) with the general bounds Y 0 , Z 0 , Z 1 and Z 2 given respectively by (38), (39), (45) and (47), we have all the bounds to define the radii polynomial p(r) as defined in (7). The following results are proved using the radii polynomial approach.…”

Section: The C ∞ But Nowhere Analytic Kernel Functionmentioning

confidence: 97%

“…Proof. The proof is obtained by running the script script proof ex2 th1.m which computes with interval arithmetic the coefficients of the radii polynomial p(r) in (7). For each numerically computed solutionx in the MATLAB data files ex2 smooth non analytic branchj with j ∈ {1, 2, 3}, the code verifies the existence of an interval I = (r min , r max ) such that for each r 0 ∈ I, p(r 0 ) < 0, and so by Theorem 1.5, there exists a uniqueã ∈ B r0 (ā) such that F (ã) = 0 with F given component-wise in (25) Theorem 3.4.…”

Section: The C ∞ But Nowhere Analytic Kernel Functionmentioning

confidence: 99%

“…Indeed, these tools have turned the digital computer into a powerful tool for proving theorems in nonlinear analysis, as is illustrated for example by Lanford's proof of the Feigenbaum conjectures [1] or by Tucker's solution of Smale's 14-th problem [2,3]. A thorough overview of the development of computer assisted analysis is beyond the scope of the present work, and we refer the reader to articles of [4,5,6,7,8] for more discussions of the literature. The interested reader may also want to consult the book of Tucker [9].…”

Section: Introductionmentioning

confidence: 99%

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Computer Assisted Fourier Analysis in Sequence Spaces of Varying Regularity

Lessard¹,

James²

2017

SIAM J. Math. Anal.

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This work treats a functional analytic framework for computer assisted Fourier analysis which can be used to obtain mathematically rigorous error bounds on numerical approximations of solutions of differential equations. An abstract a-posteriori theorem is employed in order to obtain existence and regularity results for C k problems with 0 < k ≤ ∞ or k = ω. The main tools are certain infinite sequence spaces of rapidly decaying coefficients: we employ sequence spaces of algebraic and exponential decay rates in order to characterize the regularity our results. We illustrate the implementation and effectiveness of the method in a variety of regularity classes. We also examine the effectiveness of spaces of algebraic decays for studying solutions of problems near the breakdown of analyticity.

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Section: The C ∞ But Nowhere Analytic Kernel Functionmentioning

confidence: 97%

Section: The C ∞ But Nowhere Analytic Kernel Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computer Assisted Fourier Analysis in Sequence Spaces of Varying Regularity

Lessard¹,

James²

2017

SIAM J. Math. Anal.

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“…The literature on computer-assisted proof as a tool in nonlinear analysis is substantial and growing rapidly. A thorough review would take us far afield, and instead we direct the reader to several expository articles [43,5,44,45] and refer also to the book by Tucker [25] for more scholarly discussion. In what follows we offer only a few clarifying remarks.…”

Section: Computer-assisted Proof In Function Space and In Phase Spacementioning

confidence: 99%

Automatic differentiation for Fourier series and the radii polynomial approach

Lessard

James

Ransford

2016

Physica D: Nonlinear Phenomena

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In this work we develop a computer-assisted technique for proving existence of periodic solutions of nonlinear differential equations with non-polynomial nonlinearities. We exploit ideas from the theory of automatic differentiation in order to formulate an augmented polynomial system. We compute a numerical Fourier expansion of the periodic orbit for the augmented system, and prove the existence of a true solution nearby using an a-posteriori validation scheme (the radii polynomial approach). The problems considered here are given in terms of locally analytic vector fields (i.e. the field is analytic in a neighborhood of the periodic orbit) hence the computer-assisted proofs are formulated in a Banach space of sequences satisfying a geometric decay condition. In order to illustrate the use and utility of these ideas we implement a number of computer-assisted existence proofs for periodic orbits of the Planar Circular Restricted Three-Body Problem (PCRTBP).

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“…For example, it has turned out that the existence of many chaotic attractors could only be established with computerassisted methods [34,63]. For detailed reviews about the history and applications of rigorous numerics, we refer to [53,60,64,69] and the references therein. Understanding the global behavior of a stochastic nonlinear dynamical system is even harder.…”

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confidence: 99%

Rigorous validation of stochastic transition paths

Breden

Kuehn

2019

Journal de Mathématiques Pures et Appliquées

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Global dynamics in nonlinear stochastic systems is often difficult to analyze rigorously. Yet, many excellent numerical methods exist to approximate these systems. In this work, we propose a method to bridge the gap between computation and analysis by introducing rigorous validated computations for stochastic systems. The first step is to use analytic methods to reduce the stochastic problem to one solvable by a deterministic algorithm and to numerically compute a solution. Then one uses fixed-point arguments, including a combination of analytical and validated numerical estimates, to prove that the computed solution has a true solution in a suitable neighbourhood. We demonstrate our approach by computing minimum-energy transition paths via invariant manifolds and heteroclinic connections. We illustrate our method in the context of the classical Müller-Brown test potential.

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Computational Proofs in Dynamics

Cited by 10 publications

References 69 publications

Computer Assisted Fourier Analysis in Sequence Spaces of Varying Regularity

Computer Assisted Fourier Analysis in Sequence Spaces of Varying Regularity

Automatic differentiation for Fourier series and the radii polynomial approach

Rigorous validation of stochastic transition paths

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