Continuation of homoclinic orbits in the suspension bridge equation: A computer-assisted proof

Berg, Jan Bouwe van den; Breden, Maxime; Lessard, Jean; Murray, Maxime

doi:10.1016/j.jde.2017.11.011

Cited by 21 publications

(7 citation statements)

References 28 publications

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“…Let us mention that this is far from being the first time that rigorous numerics are used to study connecting orbits, and that there is a rapidly growing literature on the subject, see e.g. [2,15,22,54,61,67,77].…”

Section: Rigorous A-posteriori Validation Methodsmentioning

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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Section: Rigorous A-posteriori Validation Methodsmentioning

confidence: 99%

Rigorous validation of stochastic transition paths

Breden

Kuehn

2019

Journal de Mathématiques Pures et Appliquées

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“…However, when searching for periodic orbits, adding these new constraints comes with the risk of creating an overdetermined system of equations. The content of the next lemma shows, in the spirit of the discussion in [15], how one can add unfolding parameters to balance the system of equations.…”

Section: A Field (E ∂mentioning

confidence: 99%

“…Over the following decades, this argument has been used in various contexts (cf. [13,10,11,12,5,4,7] for ODE, [9,8] for computer-assisted proofs in ODE and [15] for computer-assisted proofs in DDE).…”

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confidence: 99%

On polynomial forms of nonlinear functional differential equations

Hénot¹

2021

JCD

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In this paper we study nonlinear autonomous retarded functional differential equations; that is, functional equations where the time derivative may depend on the past values of the variables. When the nonlinearities in such equations are comprised of elementary functions, we give a constructive proof of the existence of an embedding of the original coordinates yielding a polynomial differential equation. This embedding is a topological conjugacy between the semi-flow of the original differential equation and the semi-flow of the auxiliary polynomial differential equation. Further dynamical features are investigated; notably, for an equilibrium or a periodic orbit and its embedded counterpart, the stable and unstable eigenvalues have the same algebraic and geometric multiplicity.

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“…Such approach is already well known and had been the object of several studies. The interested reader can see for example [39,62,63,99,100,101].…”

Section: Chebyshev Discretization Of the Bvpmentioning

confidence: 99%

Homoclinic dynamics in a spatial restricted four body problem blue skies into Smale horseshoes for vertical Lyapunov families

Murray,

Mireles-James

2020

Preprint

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The set of transverse homoclinic intersections for a saddle-focus equilibrium in the planar equilateral restricted four body problem admit certain simple homoclinic oribts which form the skeleton of the complete homoclinic intersection -or homoclinic web. In the present work the planar restricted four body problem is viewed as an invariant subsystem of the spatial problem, and the influence of this planar homoclinic skeleton on the spatial dynamics is studied from a numerical point of view. Starting from the vertical Lyapunov families emanating from saddle focus equilibria, we compute the stable/unstable manifolds of these spatial periodic orbits and look for intersections between these manifolds near the fundamental planar homoclinics. In this way we are able to continue all of the basic planar homoclinic motions into the spatial problem as homoclinics for appropriate vertical Lyapunov orbits which, by the Smale Tangle theorem, suggest the existence of chaotic motions in the spatial problem. While the saddle-focus equilibrium solutions in the planar problems occur only at a discrete set of energy levels, the cycle-to-cycle homoclinics in the spatial problem are robust with respect to small changes in energy. KeywordsGravitational 4-body problem • blue sky catastrophes • Smale tangle • vertical Lyapunov families • invariant manifolds • boundary value problems

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Continuation of homoclinic orbits in the suspension bridge equation: A computer-assisted proof

Cited by 21 publications

References 28 publications

Rigorous validation of stochastic transition paths

Rigorous validation of stochastic transition paths

On polynomial forms of nonlinear functional differential equations

Homoclinic dynamics in a spatial restricted four body problem blue skies into Smale horseshoes for vertical Lyapunov families

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