Delay Equations

Diekmann, Odo; Lunel, Sjoerd M. Verduyn; Gils, Stephan A. van; Walther, Hanns-Otto

doi:10.1007/978-1-4612-4206-2

Cited by 635 publications

(462 citation statements)

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“…(1.1) with r max = 0. It was resolved by the development of socalled sun-star techniques [2], which allow both the system under consideration and its relevant spectral projections to be lifted to the appropriate extended state space C n × L ∞ ([r min , 0], C n ). Unfortunately, these constructions are based on a semigroup approach and therefore break down when r min < 0 < r max , due to the ill-posedness mentioned above.…”

Section: Introductionmentioning

confidence: 99%

Lin's method and homoclinic bifurcations for functional differential equations of mixed type

Hupkes

Lunel²

2009

Indiana Univ. Math. J.

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We extend Lin's method for use in the setting of parameter-dependent nonlinear functional differential equations of mixed type (MFDEs). We show that the presence of M -homoclinic and M -periodic solutions that bifurcate from a prescribed homoclinic connection can be detected by studying a finite dimensional bifurcation equation. As an application, we describe the codimension two orbit-flip bifurcation in the setting of MFDEs.

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Section: Introductionmentioning

confidence: 99%

Lin's method and homoclinic bifurcations for functional differential equations of mixed type

Hupkes

Lunel²

2009

Indiana Univ. Math. J.

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“…We refer the reader to section three of [17] for some motivations of studying the abstract equation or even to the nice text book [4], which addresses this topic for delay differential equations. (7) is satisfied on (t 1 , t 2 ).…”

Section: The Abstract Formulation Of Advance Delay Equationsmentioning

confidence: 99%

The dynamical behaviour near solitary waves in Hamiltonian lattice differential equations

Georgi¹

2009

Indiana Univ. Math. J.

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In this article we consider Hamiltonian lattice differential equations and investigate the existence of travelling waves near solitary wave solutions. We focus on the case where the solitary wave profile induces a homoclinic solution of the associated traveling wave equation, which is a typical scenario in the Fermi-Pasta-Ulam lattice. We will then show the existence of finitely many scalar bifurcation equations, such that zeros of these equations correspond to multi-pulses or periodic traveling waves of the original lattice equation that are located near the primary solitary wave. Compared to previous works, we will have to overcome technical complications which result from the lack of hyperbolicity of the asymptotic steady state.We use our results to prove the existence of periodic travelling waves accompanying a family of stable solitary waves in the Fermi-Pasta-Ulam lattice, where properties of the latter waves have been recently investigated by Pego and Friesecke [8]. As we will show, these waves persist under small reversible perturbations of the FPU lattice, where they induce a family of solitary waves.

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“…The members of this family are examples of a more general and widely studied class called retarded functional differential equations (see [24,25]). Using the delay equation (2) as an abstraction for the retarded-time model that is supposed to be approximated by a system of the form (1), we will discuss an approach for extracting the 'correct' dynamical equations of motion from system (1) that eliminates the runaway solutions.…”

Section: Delay Equations With Small Delaysmentioning

confidence: 99%

Delay equations and radiation damping

et al. 2001

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Abstract. Starting from delay equations that model field retardation effects, we study the origin of runaway modes that appear in the solutions of the classical equations of motion involving the radiation reaction force. When retardation effects are small, we argue that the physically significant solutions belong to the so-called slow manifold of the system and we identify this invariant manifold with the attractor in the state space of the delay equation. We demonstrate via an example that when retardation effects are no longer small, the motion could exhibit bifurcation phenomena that are not contained in the local equations of motion.

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Delay Equations

Cited by 635 publications

References 0 publications

Lin's method and homoclinic bifurcations for functional differential equations of mixed type

Lin's method and homoclinic bifurcations for functional differential equations of mixed type

The dynamical behaviour near solitary waves in Hamiltonian lattice differential equations

Delay equations and radiation damping

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