Parameterization method for unstable manifolds of delay differential equations

Groothedde, C.M.; James, J. D. Mireles

doi:10.3934/jcd.2017002

Cited by 16 publications

(15 citation statements)

References 60 publications

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“…where the reformulation from (10) to (11) is obtained by exactly factoring out the state vector x(t) from the nonlinear right-hand side a (x(t)) so that all entries of A (x(t)) are well-defined and finite for all reachable states 5 .…”

Section: Verified Simulation Routine For Asymptotically Stable Delay-free Ordinary Differential Equationsmentioning

confidence: 99%

“…Remark 3. For sufficiently smooth system models (10), this approach can be easily extended to include a differential sensitivity analysis with respect to initial conditions and time-invariant parameters (see [22] for details).…”

Section: Remarkmentioning

confidence: 99%

“…In contrast to the existing techniques with result verification for solving delay differential equations [10,35,36] (that employ Taylor methods or radii polynomial approaches), we do not focus on obtaining especially tight enclosures, which is necessary for a computational proof of such properties as periodicity of solutions. Instead, we aim at computing guaranteed outer solution enclosures by an approach that represents state trajectories by simple (exponential) functions in a computationally cheap way.…”

Section: Introductionmentioning

confidence: 99%

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Verified Integration of Differential Equations with Discrete Delay

Rauh

Auer

2022

Acta Cybern

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Many dynamic system models in population dynamics, physics and control involve temporally delayed state information in such a way that the evolution of future state trajectories depends not only on the current state as the initial condition but also on some previous state. In technical systems, such phenomena result, for example, from mass transport of incompressible fluids through finitely long pipelines, the transport of combustible material such as coal in power plants via conveyor belts, or information processing delays. Under the assumption of continuous dynamics, the corresponding delays can be treated either as constant and fixed, as uncertain but bounded and fixed, or even as state-dependent. In this paper, we restrict the discussion to the first two classes and provide suggestions on how interval-based verified approaches to solving ordinary differential equations can be extended to encompass such delay differential equations. Three close-to-life examples illustrate the theory.

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Section: Verified Simulation Routine For Asymptotically Stable Delay-free Ordinary Differential Equationsmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Verified Integration of Differential Equations with Discrete Delay

Rauh

Auer

2022

Acta Cybern

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“…The verification of the error in the approximation is a finite (but long) calculation which can be done using computers taking care of round-off and truncation. Some cases where computer assisted proofs have been used in constant delay equations for periodic orbits and unstable manifolds are [KL12,GMJ17].…”

Section: 1mentioning

confidence: 99%

Persistence and Smooth Dependence on Parameters of Periodic Orbits in Functional Differential Equations Close to an ODE or an Evolutionary PDE

Yang¹,

Gimeno²,

Llave³

2021

Preprint

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We consider functional differential equations(FDEs) which are perturbations of smooth ordinary differential equations(ODEs). The FDE can involve multiple state-dependent delays or distributed delays (forward or backward). We show that, under some mild assumptions, if the ODE has a nondegenerate periodic orbit, then the FDE has a smooth periodic orbit. Moreover, we get smooth dependence of the periodic orbit and its frequency on parameters with high regularity.The result also applies to FDEs which are perturbations of some evolutionary partial differential equations(PDEs).The proof consists in solving functional equations satisfied by the parameterization of the periodic orbit and the frequency using a fixed point approach. We do not need to consider the smoothness of the evolution or even the phase space of the FDEs.

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“…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%

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

James¹

2018

Rigorous Numerics in Dynamics

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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

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Parameterization method for unstable manifolds of delay differential equations

Cited by 16 publications

References 60 publications

Verified Integration of Differential Equations with Discrete Delay

Verified Integration of Differential Equations with Discrete Delay

Persistence and Smooth Dependence on Parameters of Periodic Orbits in Functional Differential Equations Close to an ODE or an Evolutionary PDE

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

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