Nullity bounds for certain Hamiltonian delay equations

Frauenfelder, Urs

doi:10.48550/arxiv.2005.07535

Cited by 2 publications

(2 citation statements)

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“…However, one can think of other useful and interesting delay equations on general manifolds. Some of them stem from a variational formulation (i.e., they are the critical points of an action functional) and may be called Hamiltonian (see [1] and [6]). The idea behind the proof of Theorem 1.1 can be used to cover delay equations on manifolds as well.…”

Section: Introductionmentioning

confidence: 99%

Periodic delay orbits and the polyfold implicit function theorem

Albers

Seifert

2022

Comment. Math. Helv.

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We consider differential delay equations of the form \partial_tx(t) = X_{t}(x(t - \tau)) in \mathbb{R}^n , where (X_t)_{t\in S^1} is a time-dependent family of smooth vector fields on \mathbb{R}^n and \tau is a delay parameter. If there is a (suitably non-degenerate) periodic solution x_0 of this equation for \tau=0 , that is without delay, there are good reasons to expect existence of a family of periodic solutions for all sufficiently small delays, smoothly parametrized by \tau . However, it seems difficult to prove this using the classical implicit function theorem, since the equation above, considered as an operator, is not smooth in the delay parameter. In this paper, we show how to use the M-polyfold implicit function theorem by Hofer–Wysocki–Zehnder (2009, 2021) to overcome this problem in a natural setup.

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Section: Introductionmentioning

confidence: 99%

Periodic delay orbits and the polyfold implicit function theorem

Albers

Seifert

2022

Comment. Math. Helv.

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“…Some of them stem from a variational formulation (i.e. they are the critical points of an action functional) and may be called Hamiltonian (see [AFS20] and [Fra20]). The idea behind the proof of Theorem 1.1 can be used to cover delay equations on manifolds as well.…”

Section: Introductionmentioning

confidence: 99%

Periodic delay orbits and the polyfold implicit function theorem

Albers¹,

Seifert²

2020

Preprint

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We consider differential delay equations of the form ∂tx(t) = Xt(x(t − τ )) in R n , where (Xt) t∈S 1 is a time-dependent family of smooth vector fields on R n and τ is a delay parameter. If there is a (suitably non-degenerate) periodic solution x0 of this equation for τ = 0, that is without delay, there are good reasons to expect existence of a family of periodic solutions for all sufficiently small delays, smoothly parametrized by delay. However, it seems difficult to prove this using the classical implicit function theorem, since the equation above is not smooth in the delay parameter. In this paper, we show how to use the M-polyfold implicit function theorem by Hofer-Wysocki-Zehnder [HWZ09, HWZ17] to overcome this problem in a natural setup.

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Nullity bounds for certain Hamiltonian delay equations

Abstract: In this paper we introduce a class of Hamilton delay equations which arise as critical points of an action functional motivated by orbit interactions. We show that the kernel of the Hessian at each critical point of the action functional satisfies a uniform bound on its dimension.

Cited by 2 publications

References 12 publications

Periodic delay orbits and the polyfold implicit function theorem

Periodic delay orbits and the polyfold implicit function theorem

Periodic delay orbits and the polyfold implicit function theorem

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